text:
\begin{document}

{\Large

\noindent {\bf Unified $(q;\alpha,\beta,\gamma;\nu)$-deformation
of one-parametric}

\noindent{\bf  $q$-deformed oscillator algebras}}

\bigskip

\noindent{\bf I. M. Burban}

\noindent Bogolyubov Institute for Theoretical Physics, Kiev
03680, Ukraine

\noindent E-mail: burban@bitp.kiev.ua

\bigskip

\bigskip
\begin{abstract} We define a generalized
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra and study
the  number of its characteristics. We describe  the structure function of
deformation, analyze the classification of irreducible
representations and  discuss the asymptotic spectrum behaviour of the
Hamiltonian. For a special choice of the deformation parameters we
construct the deformed oscillator with discrete spectrum of its
"quantized coordinate" operator. We establish its connection with
the (generalized) discrete Hermite I polynomials.
\end{abstract}

\noindent PACS numbers: 02.20.Uw, 03.65.G, 03.65.Fd, 05.30.Pr

\bigskip
\bigskip
\bigskip

Short title: Unified deformation
\bigskip\bigskip\bigskip

Corresponding author: Ivan Burban
\medskip

\noindent {\it Postal address}: Bogolyubov Institute for
Theoretical Physics, Metrologichna str. 14b,  Kiev 03680, Ukraine
 \bigskip

\noindent {\it Phone}: 380-44-5213436

\noindent {\it Fax}: 380-44-5265998

\newpage

\noindent
{\bf 1. Introduction}

\medskip

The investigation of an one-parameter deformed canonical
commutation relations in theoretical physics was originated from
the study of the dual resonance models of strong interactions. The
multi-parameter generalizations of these deformations were
actively studied with the appearance quantum groups, quantum
algebras, and quantum spaces. These deformations have found
applications in the study of exotic, different from the standard
Bose-Einstein and Fermi-Dirac, statistics.

On the other hand, many properties of the modified oscillator
algebra have found applications in the study of the integrability
of the two-particle Calogero models. The modified oscillator
algebra has been generalized to $C_{\lambda}$-extended oscillator
algebra \cite{Q} with the hope to exploit these deformations for
construction of new integrable models. With the same aim, a
"hybrid" model \cite{{V},{BHV}} of the modified and $q$-deformed
oscillator algebras has been proposed.

To close this cycle of ideas we consider the generalized
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra as "the
synthesis" of the $(q;\alpha,\beta,\gamma)$-deformed
\cite{{C-C-N-U},{B}} and $\nu$-modified oscillator algebras \cite
{BHV}.

In Section 2 some of the $q$-deformed oscillator algebras are
placed in the order of their extension. We find the structure
function of $(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator
algebra. We show that this algebra is embedded into the deformed
$C_2$-extended oscillator algebra \cite{Q}. In Section 3 we give,
following \cite{Q, KMM}, classification of the representations of
this algebra. In Section 4 for the special choice of the
deformation parameters of the algebra we study the oscillator
\cite {Burb1} with the discrete spectrum of its "quantized
coordinate" $Q.$ This operator is represented by Jacobi matrix. We
find the eigenvalues and eigenfunctions of this operator. The
eigenfunctions are expressed by the (generalized) discrete
$q$-Hermite I polynomial \cite{Koe}. In Section 5 we represent the
Hamiltonian of this model as function of the number operator and
study its asymptotic energy spectrum behaviour.

\noindent

{\bf 2. Oscillator algebra and its generalized deformations}

\medskip

The oscillator algebra of the quantum harmonic oscillator is
defined by canonical commutation relations
\begin{equation}\label{burban: algeb} [a, a] = [a^+, a^+] = 0,\quad [a, a^+] = 1,\quad [N,
a] = - a,\quad[N, a^+]= a^+.
\end{equation} It allows the different types of deformations. Some of them
have been called {\it generalized deformed oscillator algebras}
\cite{{D1},{D2},{C-C-N-U},{MMP}}. Each of them defines an algebra
generated by elements (generators) $\{{\bf 1},a, a^+, N \}$ and
relations
\begin{equation}\label{burban: funct} [N, a]= - a,\quad [N, a^+] = a^+,
\quad a^+a = f(N), \quad aa^+ = f(N+1),
\end{equation} where $f$ is called  {\it a structure
function of the deformation}. Among them - the multiparameter
generalization of one-parameter deformations
\cite{{C-C-N-U},{MMP},{BDY},{BEM},{Q},{Bur},{MLD},{Burb1}}.

Let us recount some of them.

 1. The Arik-Coon $q$-deformed oscillator algebra \cite{AC}
 \[
 a a^+ - q a^+ a = 1,\quad [N,a]= - a,\quad [N, a^+] = a^+,\quad q\in{\mathbb R}_+,
 \]
 \begin{equation} f(n )= \frac{1-q^n}{1-q}.\end{equation}

2. The Biedengarn--Macfarlane $q$-deformed oscillator algebra
\cite{B,M}
\[ aa^+ - q a^+a = q^{- N},\quad aa^+ - q^{-1} a^+a = q^{N},\quad
 [N, a]= - a,\quad [N, a^+] = a^+,\]
 \begin{equation} f(n)=\frac{q^n - q^{-n}}{q - q^{-1}}\quad q\in{\mathbb
 R}_+.\end{equation}

3. The Chung-- Chung--Nam--Um generalized
$(q;\alpha,\beta)$-deformed oscillator algebra \cite{C-C-N-U}
 \[ aa^+ - q a^+a = q^{\alpha N + \beta},\quad
[N, a]= - a,\quad [N, a^+] = a^+,\quad q\in{\mathbb R}_+,\quad
\alpha, \beta \in{\mathbb R},\]\begin{equation} f(n)=
\begin{cases}
q^{\beta}\frac{q^{\alpha n} - q}{q^{\alpha} - q}, &\text {if
$\alpha \ne 1$}\\ nq^{n-1 + \beta}, &\text {if $\alpha = 1.$}\\
\end{cases}
\end{equation}

4. The generalized $(q;\alpha,\beta,\gamma)$-deformed oscillator
algebra \cite{BDY} \[ aa^+ - q^{\gamma} a^+a = q^{\alpha N +
\beta},\quad [N, a]= - a,\quad [N, a^+] = a^+,\quad q\in{\mathbb
R}_+,\alpha, \beta,\gamma \in{\mathbb R},\]
\begin{equation}\label{burban: bor}
f(n)=
\begin{cases}
q^{\beta}\frac{q^{\alpha n}- q^{\gamma n}}{q^{\alpha}- q^{\gamma
n}}, &\text {if $\alpha \ne \gamma $}\\ nq^{n-1 + \beta}, &\text
{if $\alpha = \gamma.$}\\
\end{cases}
\end{equation}

5. The $\nu$-modified oscillator algebra \cite {{V},{BHV}}
\[ [a, a^+]= 1 + 2\nu K,\quad [N, a]= - a,\quad [N, a^+] =
a^+,\]\[\quad a K = - K a,\quad a^+ K = - K a^+,\quad K^2 =
1,\quad \nu \in \mathbb R,\]
\begin{equation}\label{burban: vas}
f(n)=
\begin{cases}
2k + 1 + 2\nu, &\text {if $ n = 2k $}\\ 2k+2, &\text {if $n =
2k+1$}.\\
\end{cases}
\end{equation}
This oscillator, as it has been shown in \cite{BHV}, is connected
with the two-particle Calogero model \cite {Cal}.

6. The deformed $C_{\lambda}$-extended oscillator algebra \cite
{Q} is defined by the relations \[ [a, a^+]_q \equiv aa^+ - q a^+a
= H(N) + K(N)\sum _{k = 0}^{\lambda -1}\nu_k P_k ,\quad [N, a]= -
a,\quad [N, a^+] = a^+,\]\begin{equation}\label{burban: Ques}
\quad a K = - K a,\quad a^+ K = - K a^+,\quad K^2 = 1,\quad \nu_k
\in \mathbb R,\end{equation} where  $\nu_k\in \mathbb R$ and
$H(K),\, K(N)$ are real analytic functions. This algebra admits
the two Casimir operators $C_1= e^{2\pi N} and \quad C_2 = \sum_{k
= 0}^{\lambda -1}e^{-2\pi i(N-k)/\lambda}P_k.$

7. The new $(q;\nu)$-deformed oscillator \cite {BEM} \[ aa^+ - q
a^+ a = (1 + 2\nu K) q^{-N},\quad [N, a]= - a,\]\[ [N, a^+] =
a^+,\quad K a = - a K,\quad K a^+ = - a^+ K,\quad K^2 =
1,\]\begin{equation} f(n)= \Bigl(\frac{q^n - q^{-n}}{q - q^{-1}}+
2\nu \frac {q^n - (-1)^n q^{-n}}{q + q^{-1}}\Bigr)\end{equation}
has been defined by the combination of the idea of Biedenharn--
Macfarlane \cite {{B},{M}} $q$-deformation with Brink, Hanson and
Vasiliev idea \cite {BHV} of the $\nu$-modification of the
oscillator algebra.

8. In order to close this cycle of ideas we consider a
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra --
("hybrid" of the $(q;\alpha,\beta,\gamma)$-deformed (\ref {burban:
bor}) and the $\nu$-modified (\ref{burban: vas}) oscillator
algebras ), or, more exactly, an oscillator defined by generators
$\{I, a, a^+, N, K\}$ and relations
\[ aa^+ - q^{\gamma} a^+a =(1 + 2\nu K) q^{\alpha N +
\beta},\quad [N, a]= - a,\quad [N, a^+] = a^+,
\]
\begin{equation}\label{burban: first}  K a = - a ,\quad K a^+ = - a^+ K,\quad
[N, K] = 0,\quad N^+ = N,\quad K^+ = K,
\end{equation}
where $q \in {\mathbb R}_+,\alpha,\beta \in {\mathbb R}, \nu \in
{\mathbb R}-\{0\}.$ This model unify all deformations 1. - 7. of
the oscillator algebra (\ref{burban: algeb}).

\noindent

{\bf 3. Generalized $(q;\alpha,\beta,\gamma;\nu)$-deformed
oscillator algebra and its simplest properties}

\medskip

{\it (a) $(q;\alpha,\beta,\gamma;\nu)$-deformed structure
function.} Description of an deformed oscillator algebra requires
the determination of the deformation structure function $f(n).$

Equations (\ref{burban: funct}) and (\ref{burban: first}) imply
the recurrence relation
\begin{equation}\label{burban: struct0} f(n+1) - q^{\gamma} f(n) =
\Bigl(1 + 2 \nu(-1)^n\Bigr)q^{\alpha n + \beta}. \end{equation}
Its solution is obtained by method of mathematical induction:

\noindent $0.\quad f(1) = q^{\gamma} f(0) + \Bigr(1 + 2
\nu(-1)^0\Bigr)q^{\alpha 0 + \beta};$
\[1.\quad f(2) = q^{\gamma} f(1) + \Bigr(1 - 2 \nu(-1)^1\Bigr)q^{\alpha 1
+ \beta} = [q^{2\gamma} f(0) + \Bigr(1 + 2
\nu(-1)^0\Bigr)q^{\gamma}q^{\alpha 0 + \beta} + \Bigr(1 - 2
\nu(-1)^1\Bigr)q^{\alpha 1 + \beta};\]
\[2.\quad f(3) = q^{\gamma} f(2) + \Bigr(1 - 2 \nu(-1)^2\Bigr)q^{\alpha
2 + \beta} = q^{3\gamma} f(0) + \Bigr(1 + 2
\nu(-1)^0\Bigr)q^{2\gamma}q^{\alpha 0 + \beta} + \Bigr(1 - 2
\nu(-1)^1\Bigr)q^{\gamma}q^{\alpha 1 + \beta} + \]$$\Bigr(1 - 2
\nu(-1)^2\Bigr)q^{\alpha 2 + \beta}; $$
\[3.\quad f(4) = q^{\gamma} f(3) + \Bigr(1 + 2 \nu(-1)^3\Bigr)q^{\alpha 3
+ \beta} = q^{4\gamma} f(0) + \Bigr(1 + 2
\nu(-1)^0\Bigr)q^{3\gamma}q^{\alpha 0 + \beta} + \Bigr(1 - 2
\nu(-1)^1\Bigr)q^{2\gamma}q^{\alpha 1 + \beta} + \] $\Bigr(1 - 2
\nu(-1)^2\Bigr)q^{\gamma}q^{\alpha 2 + \beta} + \Bigr(1 + 2
\nu(-1)^3\Bigr)q^{\alpha 3 + \beta};$

\noindent $.$

\noindent$.$

\noindent $.$
\[n.\quad f(n) = q^{n\gamma} f(0) + \sum_{k = 0}^{n-1}q^{\gamma(n-k-1)}q^{\alpha k + \beta} +
2 \nu \sum_{k=0}^{n-1}q^{\gamma(n-k-1)}(-1)^{k}q^{\alpha
k+\beta}.\qquad \qquad {}{}{}\] The solution of the equation
(\ref{burban: struct0}) with the initial value $f(0)$ is given by
formula
\begin{equation} \label{burban: struct} f(n) =
\begin{cases} f(0)q^{\gamma n} +
q^{\beta}\Bigl(\frac {q^{\gamma n} - q^{\alpha n}} {q^{\gamma} -
q^{\alpha}} + 2\nu \frac {q^{\gamma n} - (-1)^n q^{\alpha n}}
{q^{\gamma} + q^{\alpha}}\Bigr),&\text{if $\alpha \ne \gamma$}\\
 f(0)q^{\gamma n} + n q^{\gamma(n-1) + \beta} + 2
\nu q^{\gamma(n-1) + \beta} \Bigl(\frac{1 -
(-1)^n}{2}\Bigr),&\text{ if $\alpha = \gamma.$}\end{cases}
\end{equation}

{\it (b) Positivity of $(q;\alpha,\beta,\gamma;\nu)$-deformed
structure function.} We try to define the values of the parameters
$\{\alpha,\beta, \gamma,\nu \},$ where function $ f(n)$ is
positive.

The inequality
\begin{equation}\label{burban: ine}\frac{q^{\gamma n} - q^{\alpha n}}{q^{\gamma} -
q^{\alpha}} + 2\nu \frac{q^{\gamma n} - (-1)^n q^{\alpha n}}
{q^{\gamma} + q^{\alpha}} > 0,\end{equation} can be rewritten as
\begin{equation}
(q^{\gamma n}- q^{\alpha n})\Bigl(\frac{1}{q^{\gamma} -
q^{\alpha}} + 2 \nu\frac{1}{q^{\gamma} + q^{\alpha}}\Bigr) > 0
\end{equation}
for even $n$ and
\begin{equation}\frac{q^{\gamma n}- q^{\alpha n}}{q^{\gamma n}+ q^{\alpha
n}} \frac{q^{\gamma }+ q^{\alpha}}{q^{\gamma }- q^{\alpha }} + 2
\nu
> 0
\end{equation}
for odd $n.$

It follows \begin{equation}\label{burban: fre}1 + 2 \nu > 0
\end{equation} if $q
> 1, \gamma - \alpha > 0 \,( q < 1,\, \gamma - \alpha < 0)$

\begin{equation}\label{burban: frest} -1 < 2 \nu < -\frac{q^{\gamma} +
q^{\alpha}}{q^{\gamma} - q^{\alpha}}
\end{equation}
if $q < 1, \gamma - \alpha
> 0 \,( q > 1,\, \gamma - \alpha < 0).$

The conditions (\ref{burban: fre}) and (\ref{burban: frest}) are
sufficient to ensure that the Fock representation of the relation
(\ref{burban: first}) has the Hermitian properties.

{\it (c) Useful formulas.} The two formulae for the study of this
algebra will be useful. One of them
\begin{equation}\label{burban: useful} a(a^+)^n - q^{\gamma n }\, (a^+)^n a =
[n;\alpha,\gamma;\nu K](a^+)^{n-1}q^{\alpha
N+\beta},\end{equation} where $n\ge 1,$
\begin{equation}  [n;\alpha,\gamma;\nu K] =
\begin{cases}
\Bigl(\frac {q^{\gamma n} - q^{\alpha n}} {q^{\gamma} -
q^{\alpha}}+ 2\nu K\frac {q^{\gamma n} - (-1)^n q^{\alpha n}}
{q^{\gamma} + q^{\alpha}}\Bigr),&\text{if $\alpha \ne \gamma$;}\\
 n q^{\alpha(n-1)} + 2\nu K q^{\alpha(n-1)} \Bigl(\frac{1 - (-1)^n}{2}\Bigr),&\text{
if $\alpha = \gamma$}\end{cases}
\end{equation}
is deduced by method of mathematical induction.

 Indeed, for $n = 1$ the relation (\ref{burban: useful}) is true by
definition. The assumption that it is true for some $n$ implies
\[ \{ a(a^+)^{n+1} = a(a^+)^n a^+ =
\Bigl( q^{\gamma n}(a^+)^n a+[n;\alpha,\gamma;\nu K
](a^+)^{n-1}q^{\alpha N + \beta}\Bigr) \, a^+ = \]
\[ q^{\gamma n}(a^+)^n a a^+ + (a^+)^{n}[n;\alpha,\gamma;-\nu K]q^{\alpha
(N+1)+ \beta} = \] \[q^{\gamma n}(a^+)^n \Bigl(q^{\gamma} a^+ a +
(1 + 2\nu K )q^{\alpha N + \beta}\Bigr) + (a^+)^{n}
[n;\alpha,\gamma;-\nu K] q^{\alpha (N+1) + \beta} = \] \[
q^{\gamma (n+1)}(a^+)^{(n+1)}a + q^{\gamma n}(a^+)^n \Bigl(1 + 2
\nu K \Bigr) q^{\alpha N + \beta} + q^{\alpha}(a^+)^n
[n;\alpha,\gamma;-\nu K] q^{\alpha N + \beta}.\] The direct
calculations leads to (\ref{burban: useful}).

The second formula gives the generated function for
$[n;\alpha,\gamma;\nu K]:
$
\begin{equation}
\sum_{n = 0}^{\infty}[n;\alpha,\gamma;\nu K]z^n =
\begin{cases}
\frac{z}{1-q^{\gamma}z}\Bigl(\frac{1}{1-q^{\alpha}z}+2\nu K
\frac{1}{1+q^{\alpha}z}\Bigr) , &\text {if $\alpha \ne \gamma$}\\
\frac{z}{(1-q^{\gamma}z)^2}+ 2\nu K\frac{z}{1-q^{2\gamma}z^2} ,
&\text {if $\alpha = \gamma.$}\\
\end{cases}
\end{equation}

{\it (d) Deformed $C_2$-extended and
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebras.}
 The defining relations of the deformed $C_2$-extended oscillator
are given by $$ [N, a^+]= a^+,\quad [N,P_{k}] = 0,\quad a^+ P_{k}
= P_{k +1}a^+,\quad P_1 + P_2 = I,\quad P_{k} P_{l} =
\delta_{k,l}P_{l},$$
\begin{equation}\label{burban: deform}
aa^+ - q^{\gamma} a^+ a = H(N)+ \nu\Bigl( E(N+1) + q^{\gamma}
E(N)\Bigr)( P_0 - P_1),
\end{equation}
where $q, \nu \in {\mathbb R},k,l =1, 2,$ and $E(N), H(N)$ are
real analytic functions. As we saw above the deformed extended
oscillator algebra $C_{\lambda}$ admits the two Casimir operators
$C_1, C_2.$ In case of the $C_2$-extended oscillator algebra they
have the form
\begin{equation} C_1 = e^{2\pi i N},\quad C_2 =
\Sigma_{k = 0}^1e^{-2\pi i(N-k)/2}= e^{i\pi N}K.
\end{equation}
Let us define the operator \begin{equation}\label{burban: casim}
{\tilde C}_3 = q^{-\gamma N}\Bigl(D(N) + \nu E(N) K -
a^+a\Bigr),\end{equation} where $D(N),\, E(N)$ some analytic
functions of $N$. The operator ${\tilde C}_3$ will be the Casimir
operator of the oscillator algebra (\ref{burban: deform}) if the
only one condition $[{\tilde C}_3, a] = 0$ holds.  It amounts to
determination of the solution of the equations
\begin{equation}\label{burban: addi}
D(N+1)- q^{\gamma}D(N) = H(N),\quad E(N+1)\beta_{k+1}-
q^{\gamma}E(N)\beta_{k} = K(N)\nu_{k},
\end{equation} where $\nu_0 = - \nu_1 =\nu, \beta_0 = 0,\beta_2 =
0, \beta_1 = \nu,\quad k = 0, 1.$ Putting the solution $ E(N) = 2
q^{\alpha N + \beta}/({q^{\gamma} + q^{\alpha}})$ of the equation
of (\ref{burban: addi}) and $ H(N) = q^{\alpha N + \beta}$ in
(\ref{burban: deform}), we obtain the commutation relations of the
$ (q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra
(\ref{burban: first}). Moreover, the solution $$ D(N) =
\begin{cases} q^{\beta}\Bigl(\frac{q^{\gamma N } - q^{\alpha N
}}{q^{\gamma} - q^{\alpha}} + 2\nu\frac{q^{\gamma N}}{q^{\gamma} +
q^{\alpha}}\Bigr)&\text{if $\gamma \ne \alpha
$}\\q^{\beta}(q^{\gamma(N-1)}N + \nu q^{-\gamma})&\text{if $\gamma
=\alpha $}\end{cases} $$ of the first equation (\ref{burban:
addi}) gives the explicit form of the Casimir operator
\begin{equation}\label{burban: casimir}
{\tilde C}_3 = \begin {cases}
q^{-\gamma N }\Bigl((\frac{q^{\gamma N} - q^{\alpha N
}}{q^{\gamma} - q^{\alpha}} + 2 \nu \frac{q^{\gamma N} - (-1)^N
q^{\alpha N}}{q^{\gamma} + q^{\alpha}})q^{\beta} - a^+a
\Bigr)&\text {if $\alpha\ne\gamma$}\\ q^{-\gamma N}\Bigl(N + \nu(1
+ (-1)^N)q^{\gamma N + \beta}- a^+a\Bigr)&\text{if $\alpha =
\gamma $}.\end{cases}\end{equation}

\noindent

{\bf 3. Classification of representations of unified
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra}

\medskip

As has been shown in the previous Section the
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra allows
nontrivial center which means that it has irreducible
non-equivalent representations \cite {{Rid},{Ch}}. We give a
classification of these representations by method similar to the
one in the articles \cite{Q}, \cite {KMM}.

Due to the relations (\ref{burban: first}) there exists a vector
$|0\rangle$ such that $$a^+a|0\rangle = \lambda_0 |0\rangle, \quad
aa^+ |0\rangle = \mu_0|0\rangle,\quad N |0\rangle = \varkappa_0
|0\rangle,\quad K|0\rangle = \omega e^{-i\pi
\varkappa_0}|0\rangle, $$ where $\omega$ is the value of the
Casimir operator $C_2$ on the given irreducible representation. By
using the formula (\ref{burban: useful}) we find that vectors
\begin{equation}
|n\rangle = \begin{cases} (a^+)^n|0\rangle, &\text{if $n\ge 0$}\\
(a)^{-n}|0\rangle, &\text{if $n < 0$}\\
\end{cases}
\end{equation}
are eigenvectors of the operators $a^+a$ and $aa^+:$
\[ a^+ a|n\rangle = \lambda_n |n\rangle, \quad aa^+ |n\rangle = \mu_n |n\rangle.\]

Let us define new system of the orthonormal vectors
$\{|n\rangle\}_{n = - \infty}^{n = \infty},$ by
\begin{equation}
|n\rangle = \begin{cases}\Bigl(\prod_{k = 1}^{n} \lambda_k
\Bigr)^{-1/2} (a^+)^n|0\rangle, &\text{if $n\ge 0$}\\
\Bigl(\prod_{k = 1}^{-n}\lambda_{n+k}\Bigr)^{-1/2}
(a)^{-n}|0\rangle, &\text{if $n < 0$}.\\
\end{cases}
\end{equation}
Then the relations (\ref{burban: first}) are represented by
operators
\[
a^+|n\rangle = \sqrt{\lambda_{n+1}}|n+1\rangle, \quad a |n\rangle
= \sqrt{\lambda_n}|n-1\rangle, \]
\begin{equation}\label{burban: rep} N|n\rangle = (\varkappa_0 +
n)|n\rangle, \quad K |n\rangle =
\frac{(-1)^n}{2\nu}B|n\rangle,\end{equation} where $B = 2 \nu
\omega e^{-i\pi \varkappa_0}\in {\mathbb R}.$ Due to
non-negativity of the operators $a^+a,$ $aa^+$ we have
$\lambda_n\ge 0$ and $\mu_n\ge 0.$ By using (\ref{burban: first}),
we find (\ref{burban: rep}) and $\lambda_n =\mu_{n-1}$ we obtain
the recurrence relation
\begin{equation}\label{burban: resur}
\lambda_{n+1} - q^{\gamma}\lambda_n = \Bigl(1 + (-1)^n B
\Bigr)q^{\alpha(n +\varkappa_0)+ \beta}.
\end{equation}
Take into account the relation (\ref{burban: struct}) the solution
of equation (\ref{burban: resur}) can be represented by
\begin{equation} \label{burban: solu} \lambda_n =
\begin{cases} \lambda_0 q^{\gamma n} +
q^{\alpha\varkappa_0 + \beta}\Bigl(\frac {q^{\gamma n} - q^{\alpha
n}} {q^{\gamma} - q^{\alpha}} + B \frac {q^{\gamma n} - (-1)^n
q^{\alpha n}} {q^{\gamma} + q^{\alpha}}\Bigr),&\text{if $\alpha
\ne \gamma$;}\\ \lambda_0 q^{\gamma n} + n q^{\gamma(n +
\varkappa_0 -1) + \beta } + B q^{\gamma(n + \varkappa_0 -1)+
\beta} \Bigl(\frac{1-(-1)^n}{2}\Bigr),&\text{ if $\alpha =
\gamma.$}\end{cases}\end{equation}

Equivalently, using the expression of the Casimir operator
(\ref{burban: casimir}) and the representation (\ref{burban: rep})
the solution $\lambda_n$ of equation (\ref{burban: resur}) can be
presented by means of the eigenvalue $c_3$ of the Casimir operator
$C_3$ in irreducible representation of the relation (\ref{burban:
first})
\begin{equation}\label{burban: coins}
\lambda_n =
\begin{cases}  - q^{\gamma n}c_3 +
q^{\beta}\Bigl(\frac {q^{\gamma (n + \varkappa_0)} - q^{\alpha( n
+ \varkappa_0)}} {q^{\gamma} - q^{\alpha}} + 2\nu \frac
{q^{\gamma(n +\varkappa_0) } - (-1)^{n + \varkappa_0} q^{\alpha
(n+\varkappa_0)}} {q^{\gamma} + q^{\alpha}}\Bigr),&\text{if
$\alpha \ne \gamma$;}\\ \lambda_0 q^{\gamma n}c_3 + n q^{\gamma(n
+ \varkappa_0 -1) + \beta } + B q^{\gamma(n + \varkappa_0 -1)+
\beta} \Bigl(\frac{1-(-1)^n}{2}\Bigr),&\text{ if $\alpha =
\gamma.$}\end{cases}\end{equation} It is easy see that $\lambda_n$
in (\ref{burban: solu}) coincides with the value of (\ref{burban:
coins}), where $B = (-1)^{\varkappa_{0}}2\nu.$

According to \cite {Q} a representations of the generalized
oscillator algebra is reduced to the four classes of unireps:

1. Representations {\it bounded from below}. They are defined by
(\ref{burban: rep}) and $n_1\in {\mathbb Z}_{\le 0}$ such that
\begin{equation}\label{burban: clas1} \lambda_n =
\begin{cases}
\lambda_{n_1} = 0 &\text {if $n_1\in \{\ldots, -2, -1, 0
\}$}\\\lambda_n > 0 &\text {if $n \in \{n_1+1, n_1+2,
\ldots\}$}.\\
\end{cases}
\end{equation}

(i) Let $\lambda_n$ be defined by formula (\ref{burban: solu}) for
$\gamma - \alpha = 0,$ $q > 0$. The non-negativity of $\lambda_n$
implies
\begin{equation}\label{burban: sabst} \lambda_n = \lambda_0 + n
q^{\gamma( \varkappa_0 -1) + \beta} + B q^{\gamma( \varkappa_0 -1)
+ \beta} \Bigl(\frac{1 - (-1)^n}{2}\Bigr)\ge 0.
\end{equation}
It follows (\ref{burban: clas1}) and due to (\ref{burban: rep}) $a
|n_1\rangle = 0.$

After possible renumbering of vectors $|n\rangle$ we obtain
$a|0\rangle = 0$ and due to (\ref{burban: rep}) one gets
$\lambda_0 = 0.$ Therefore the representations are given by
formulae (\ref{burban: rep}) with
\begin{equation}\lambda_n = n q^{\gamma(n + \varkappa_0 -1) +
\beta} + B q^{\gamma(n + \varkappa_0 -1) + \beta}\Bigl(\frac{1 -
(-1)^n}{2}\Bigr),\end{equation} where $ n \ge 0$. The arbitrary
values of the parameter $\varkappa_0,$ and $B\ge 0$ correspond to
nonequivalent representations of (\ref{burban: first}).

(ii) Let $\lambda_n$ be defined by (\ref{burban: solu}) for
$\gamma - \alpha >0,$ $q > 1$ $(\gamma-\alpha < 0, 0 < q < 1).$

The assumption of the positivity at last one of the numbers
$\frac{1}{q^{\gamma} - q^{\alpha}} \pm \frac{ B}{q^{\gamma} +
q^{\alpha}}$ and the nonnegativity of  $\lambda_n$
\begin{equation}\label{burban: cs1}
\Bigl(\lambda_0 q^{-(\alpha \varkappa + \beta)}+
\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{B}{q^{\gamma} +
q^{\alpha}}\Bigr)\ge q^{-(\gamma
-\alpha)n}\Bigl(\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{(-1)^n
B}{q^{\gamma} + q^{\alpha}} \Bigr)
\end{equation} for $n\ge 0$
implies (\ref{burban: clas1}) and and due to (\ref{burban: rep})
$a |n_1\rangle = 0.$

After possible renumbering of vectors $|n\rangle$ we obtain
$a|0\rangle = 0$ and due to (\ref{burban: rep}) one gets
$\lambda_0 = 0.$ Therefore the representations are given by
(\ref{burban: rep}) with
\begin{equation}\label{burban: so1}
\lambda_n = q^{\alpha\varkappa_0  + \beta + n\gamma}\Bigl(\frac{1
- q^{(\alpha -\gamma)n}}{q^{\gamma} - q^{\alpha}}+ B\frac{1
-(-1)^nq^{(\alpha - \gamma)n}}{q^{\gamma} +
q^{\alpha}}\Bigr),\quad n \ge 0.
\end{equation}
The nonnegativity condition for $\lambda_n$ implies $B \ge -1.$

The arbitrary values of the parameter $\varkappa_0,$ and
$\lambda_0 = 0$ and $B > -1$ defined nonequivalent
infinite-dimensional representations of (\ref{burban: first}).

iii) Let $\lambda_n$ be given by (\ref{burban: solu}) for $\gamma
- \alpha > 0,$ $0 < q < 1 (\gamma-\alpha < 0, q > 1).$

It can be represented by $$\lambda_n = q^{\alpha\varkappa_0 +
\beta + \gamma n}\times $$
\begin{equation}\label{burban: fin}
\Bigl\{\Bigl(\lambda_0 q^{-(\alpha \varkappa + \beta)}+
\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{B}{q^{\gamma} +
q^{\alpha}}\Bigr) - q^{-(\gamma
-\alpha)n}\Bigl(\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{(-1)^n
B}{q^{\gamma} + q^{\alpha}} \Bigr)\Bigr\}.\end{equation}

The nonnegativity of $\lambda_n\ge 0,$ negativity of
$\Bigl(\lambda_0 q^{-(\alpha \varkappa + \beta)} +
\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{B}{q^{\gamma} +
q^{\alpha}}\Bigr)< 0,$ and the non-positivity of
$\Bigl(\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{(-1)^n
B}{q^{\gamma} + q^{\alpha}} \Bigr)$ implies (\ref{burban: clas1})
and $a |n_1\rangle = 0.$ The same arguments as in the items i),
ii) give $a|0\rangle = 0$ and due to (\ref{burban: rep})
$\lambda_0 = 0.$ Therefore the representation is given by (\ref
{burban: rep}) with $\lambda_n$ as (\ref{burban: so1}). The
nonnegativity condition for $\lambda_n$ gives a restriction for
possible values of $B:$
\begin{equation}- 1 \le B < -\frac{q^{\gamma} + q^{\alpha}}{q^{\gamma} +
q^{\alpha}}
\end{equation}
 The arbitrary values of the
parameter $\varkappa_0,$ and $ -1\le B < - \frac{q^{\gamma}+
q^{\alpha}}{q^{\gamma} + q^{\alpha}}$ and $\lambda_0 = 0 $
distinguished irreducible representation of the relations
(\ref{burban: first}).

 2. Representations {\it bounded from above.}
They are defined by formulae (\ref{burban: rep}) and $n_2\in
{\mathbb N}$ such that
\begin{equation} \label{burban: clas2} \lambda_n =
\begin{cases}
\lambda_{n_2} = 0 &\text {if $n_2\in \{1, 2, 3, \ldots \}$}
\\\lambda_n
> 0 &\text {if $n \in \{n_2-1, n_2-2, \ldots\} $}.\\
\end{cases}
\end{equation}
Let $\lambda_n$ be given by formula (\ref{burban: solu}) $0 < q <
1, (\alpha - \gamma) > 0 \quad (q > 1, (\alpha - \gamma) < 0 $)
and both values $\frac{1}{q^{\gamma} - q^{\alpha}} \pm
\frac{B}{q^{\gamma} + q^{\alpha}}$ are non-positive (at last one
of them must be strictly negative). From the non-negativity
condition $\lambda_n\ge 0$ it follows the existence of $n_2$ such
that (\ref{burban: clas2}) holds. From formulae (\ref{burban:
rep}) we obtain $a^+ |n_2\rangle = 0,$ and after possible
renumbering we may assume that $a^+ |0\rangle = 0,$ and due to
(\ref{burban: rep}) $\lambda_1 = 0.$

The formula (\ref{burban: sabst}) implies $\lambda_0 = -
q^{\alpha\varkappa_0 + \beta - \gamma}(1+B).$  The condition
$\lambda_0\ge 0$ is equivalent to $B \le -1.$ It follows
\begin{equation}\label{burban: solu2}
\lambda_n =  q^{\alpha \varkappa_0 + \beta + \gamma
n}\Bigl(\Bigl(\frac{1- q^{(\alpha - \gamma)n}}{q^{\gamma} -
q^{\alpha}} - q^{-\gamma}\Bigr) + B \Bigl(\frac{1- q^{(\alpha
-\gamma)n}(-1)^n}{q^{\gamma} + q^{\alpha}} - q^{-\gamma} \Bigr)\ge
0,
\end{equation}
$n \le 0.$ Therefore the representation is given by formulae (\ref
{burban: rep}) with $\lambda_n$ as (\ref{burban: solu2}). If $B <
-1$ the nonnegativity condition  for $\lambda_n$ gives a
restriction for possible values of $B:$
\begin{equation} B \le -\frac{q^{\gamma}+ q^{\alpha}}{q^{\gamma}+
q^{\alpha}}
\end{equation}
 The arbitrary values of parameter $\varkappa_0$ and $\lambda_0 = -
q^{\alpha\varkappa_0 + \beta - \gamma}(1+B),$ $B <
\frac{q^{\gamma}+ q^{\alpha}}{q^{\gamma}+ q^{\alpha}} $
distinguish irreducible representation of (\ref{burban: first}).

3. The {\it finite dimensional representation.} They defined by
formulae (\ref{burban: rep}) and $n_1\in\{\ldots,-2,-1,0\},$
$n_2\in \{ 1,2,3,\ldots\}$ such that
\begin{equation}\label{burban: findim1} \lambda_n =
\begin{cases}
\lambda_{n_1} = \lambda_{n_2} = 0 &\text {if $n_2\in \{1, 2, 3,
\ldots \}, n_1\in \{ \ldots, -2, -1, 0\}$}
\\ \lambda_n > 0 &\text {if $n \in\{n_1 +1, n_1+2, \ldots, n_2-1\}$}.\\
\end{cases}
\end{equation}

i) One-dimensional representations. These representations are
given by formulae (\ref{burban: rep}) with $\lambda_n$ calculated
from (\ref{burban: so1}), (\ref{burban: fin}) and (\ref{burban:
solu2}) at $B = -1$
\begin{equation}
\lambda_n = q^{-\alpha \varkappa_0 + \beta }\Bigl(\frac{q^{\gamma
n }-q^{\alpha n}}{q^{\gamma}-q^{\alpha}} - \frac{q^{\gamma n} -
(-1)^n q^{\alpha n}}{q^{\gamma} + q^{\alpha}}\Bigr).
\end{equation}
In this case $n_1 = 0$ $(n_1 = -1)$ and $n_2 = 1$ $(n_2 = 0)$ and
the representations (\ref{burban: rep}) are defined by
\begin{equation}\label{burban: onedim}
a = a^+ = 0, \quad N = \varkappa_0,\quad K = - \frac{1}{2\nu}.
\end{equation}
These representations are one-dimensional. Their are parametrized
by $\varkappa_0.$

ii) Two-dimensional representations. Let $\lambda_n$ be defined
(\ref{burban: solu}) for $\gamma - \alpha > 0,$ $0 < q < 1$
$(\gamma - \alpha < 0, q > 1).$ The values of $\lambda_n, n \ge 0$
in (\ref{burban: fin}) are nonnegative if we assume
$\Bigl(\lambda_0 q^{-(\alpha \varkappa + \beta)} +
\frac{1}{q^{\gamma} - q^{\alpha}} + \frac{B}{q^{\gamma} +
q^{\alpha}}\Bigr) = 0$ and both values $\Bigl(\frac{1}{q^{\gamma}
- q^{\alpha}} + \frac{(-1)^n B}{q^{\gamma} + q^{\alpha}} \Bigr)$
are nonpositive. At $ B = \pm\frac{q^{\gamma} +
q^{\alpha}}{q^{\gamma} - q^{\alpha}}$ we obtain
\begin{equation}\label{burban: eveod}\lambda_n
= \frac{q^{\alpha(\varkappa_0 + n)+ \beta}}{q^{\alpha} -
q^{\gamma}}\Bigl(1 \pm (-1)^n\Bigr). \end{equation}

 The value $\lambda_n =
\frac{q^{\alpha(\varkappa_0 + n)+ \beta}}{q^{\alpha} -
q^{\gamma}}\Bigl(1 + (-1)^n)\Bigr)$  implies (\ref{burban:
findim1}) for $n_1 = - 1,$ $n_2 = 1$ and $\lambda_0 =
\frac{2q^{\alpha\varkappa_0 + \beta}}{q^{\alpha} - q^{\gamma}}.$

The vector space of this representation spanned by the
two-dimensional vectors $$ \left(
\begin{array}{c}
\psi_{-1} \\ \psi_0
\end{array}\right)
$$ and due to (\ref{burban: rep}) the representation are defined
by $$ a = \left(
\begin{array}{cc}
0&\sqrt{\frac{2 q^{\alpha\varkappa_0 + \beta}}{q^{\alpha} -
q^{\gamma}}}\\ 0& 0
\end{array}\right),\qquad
a^+ = \left(
\begin{array}{cc}
0& 0\\ \sqrt{\frac{2 q^{\alpha\varkappa_0 + \beta}}{q^{\alpha} -
q^{\gamma}}}& 0
\end{array}\right), $$
\begin{equation}
N = \left(
\begin{array}{cc}\chi_0 - 1
& 0\\0& \chi_0
\end{array}\right),\qquad K = \frac{1}{2\nu}\frac{q^{\gamma}+
q^{\alpha}}{q^{\alpha} - q^{\gamma}}\left(
\begin{array}{cc}
1&0\\0& 1
\end{array}\right).
\end{equation}
These representations are distinguished by the arbitrary values
$\varkappa_0,$ and $B = \pm\frac{q^{\gamma} +
q^{\alpha}}{q^{\gamma}- q^{\alpha}},$ $\lambda_0 =
\frac{2q^{\alpha(\varkappa_0 ) + \beta}}{q^{\alpha} -
q^{\gamma}}.$

The value $\lambda_n = \frac{q^{\alpha(\varkappa_0 + n) +
\beta}}{q^{\alpha} - q^{\gamma}}\Bigl(1 - (-1)^n\Bigr)$ implies
(\ref{burban: findim1}) for $n_1 = 0,$ $n_2 = 2.$ Moreover
$\lambda_1 = \frac {2 q^{\alpha(\varkappa_0 + 1) +
\beta}}{q^{\alpha} - q^{\gamma}}.$

The vector space of this representation spanned by the
two-dimensional vectors $$ \left(
\begin{array}{c}
\psi_0 \\ \psi_{-1}
\end{array}\right)
$$ and due to (\ref{burban: rep}) the representations are given by
$$ a = \left(
\begin{array}{c
c}
0&0\\\sqrt{\frac{q^{\alpha(\varkappa_0+1) + \beta}}{q^ {\alpha} -
q^{\gamma}}}& 0
\end{array}\right),\qquad
a^+ = \left(
\begin{array}{cc}
 0& \sqrt{\frac{q^{\alpha(\varkappa_0+1) + \beta}}{q^
{\alpha} - q^{\gamma}}}\\0& 0
\end{array}\right),
$$
\begin{equation}
N = \left(
\begin{array}{cc}
\chi_0& 0\\0& \chi_0 + 1
\end{array}\right),\qquad K = \frac{1}{2\nu}\frac{q^{\gamma}+
q^{\alpha}}{q^{\alpha} - q^{\gamma}}\left(
\begin{array}{cc}
-1&0\\0& 1
\end{array}\right).\qquad
\end{equation}
These representations are defined by arbitrary values of
$\varkappa_0$, and  $\lambda_0 = 0,$ $B = - \frac {q^{\gamma} +
q^{\alpha}}{q^{\gamma}- q^{\alpha}}$.

4. {\it Unbounded representations.} They defined by (\ref{burban:
rep}) with $\lambda_n \ge 0$ for all $ n\in{\mathbb Z}.$ As
follows from (\ref{burban: fin}) they are realized if $0 < q < 1,
(\alpha - \gamma)
> 0 \quad ( q > 1, (\alpha - \gamma) < 0 )$ and
\begin{equation}\label{burban: inf1} \lambda_0 q^{-\alpha \varkappa_0 - \beta} +
\frac{1}{q^{\gamma} - q^{\alpha}} + \frac {B}{q^{\alpha} +
q^{\gamma}}\ge 0,\quad |B| < - \frac{q^{\gamma} +
q^{\alpha}}{q^{\gamma} - q^{\alpha}},
\end{equation} or
\begin{equation} \label{burban: inf2} \lambda_0 q^{-\alpha \varkappa_0 - \beta} +
\frac{1}{q^{\gamma} - q^{\alpha}} + \frac {B}{q^{\alpha} +
q^{\gamma}}> 0, \quad |B|= -\frac{q^{\gamma} +
q^{\alpha}}{q^{\gamma} - q^{\alpha}}. \end{equation} Under the
transformations $$ B\to B'= (-1)^n B,\quad \varkappa_0\to
\varkappa' = \varkappa + n, $$
\begin{equation}\lambda_0\to \lambda'_0 =
\lambda_0 q^{\gamma n} + q^{\alpha\varkappa_0 + \beta}\Bigl(\frac
{q^{\gamma n} - q^{\alpha n}} {q^{\gamma} - q^{\alpha}} + B \frac
{q^{\gamma n} - (-1)^n q^{\alpha n}}{q^{\gamma} +
q^{\alpha}}\Bigr)
\end{equation}
the conditions (\ref{burban: inf1}) and (\ref{burban: inf2}) are
preserved
\begin{equation}
\lambda'_n = \lambda_0 q^{2\gamma n} + q^{\alpha\varkappa_0 +
\beta}\Bigl(\frac {q^{2\gamma n} - q^{2\alpha n}} {q^{\gamma} -
q^{\alpha}} + B \frac {q^{2\gamma n} - q^{2\alpha n}} {q^{\gamma}
+ q^{\alpha}}\Bigr),
\end{equation}
\begin{equation}
\lambda'_0 q^{-\alpha \varkappa'_0 - \beta} + \frac{1}{q^{\gamma}
- q^{\alpha}} + \frac {B'}{q^{\alpha} + q^{\gamma}} =
q^{(\gamma-\alpha)n}\Bigl(\lambda_0 q^{-\alpha \varkappa_0 -
\beta} + \frac{1}{q^{\gamma} - q^{\alpha}} + \frac {B}{q^{\alpha}
+ q^{\gamma}}\Bigr)
\end{equation}
and {\it unbounded representations} pass to themselves.

\noindent

{\bf 4. Spectrum of "quantized coordinate" $Q$}

\medskip

According to previous section  if we put $\omega = 1, \varkappa_0
= 0,$ in (\ref{burban: so1}) we obtain the Fock representation of
the relations (\ref{burban: first})
\[N|n\rangle = n|n\rangle,\quad K|n\rangle = (-1)^n|n\rangle, $$ $$
a^+|2n\rangle = q^{\beta/2} \Bigl(\frac{q^{\gamma (2n+1)} -
q^{\alpha (2n+1)}}{q^{\gamma} - q^{\alpha}} + 2\nu\frac{q^{\gamma(
2n+1)}+ q^{\alpha (2n+1)}}{q^{\gamma} + q^{\alpha
}}\Bigr)^{1/2}|2n+1\rangle,\]
\begin{equation} \label{burban: rep2} a^+|2n-1\rangle = q^{\beta/2} \Bigl(\frac{q^{2\gamma n} -
q^{2\alpha n}}{q^{\gamma} - q^{\alpha}} + 2\nu \frac{q^{2\gamma n}
- q^{2\alpha n}}{q^{\gamma} +
q^{\alpha}}\Bigr)^{1/2}|2n\rangle.\end{equation}

The properties of the quantum harmonic oscillator are closely
related to Hermite polynomials. Different $q$-deformed oscillators
generate distinct $q$-deformed Hermite polynomials and their
generalization. Unlike to quantum harmonic oscillator, generalized
q-deformed oscillators frequently define the position and momentum
operators with the discrete spectrum and their eigenfunctions are
represented by means of a q-Hermite polynomials. For some
q-deformed oscillators this problem was studied in \cite{BK},
\cite{CK}, \cite{K}. In \cite{Burb1} the generalized q-deformed
oscillators connected with the discrete (generalize) q-Hermite I
and q-Hermite II polynomials have been build. In this Section we
show that corresponding oscillator algebras of this models are
embedded in the $(q;\alpha,\beta,\gamma:\nu)$-deformed oscillator
algebra.

 Let us consider the operator ("quantized coordinate") $Q
= a^+ + a,$ or
\begin{equation}\label{burban: coord} Q|n\rangle = r_n|n+1\rangle + r_{n-1}|n-1\rangle,
\quad r_n = f^{1/2}(n+1).
\end{equation}

The self-adjointness and spectral properties of this operator is
defined by polynomials $P_n(x)$ of first kind for the Jacobi
matrix $Q.$

Defining the generalized eigenfunction $Q |x\rangle = x|x\rangle
$, where $|x\rangle = \sum_{n = 0}^{\infty} P_n(x)|n\rangle,$ we
obtain the recurrence relation
\begin{equation}\label{burban: jac}
f^{1/2}(n) P_{n-1}(x) + f^{1/2}(n+1) P_{n+1}(x) = x P_n(x),
\end{equation}
where $ P_{-1}(x) = 0, P_{0} = 1.$

In this Section we consider of the oscillator (\ref{burban:
first}) for special choice of parameters
\begin{equation}
\alpha = 2 a,\quad \gamma = 2 a + c - 1 \quad \beta = 2 a + b,\nu
= 0.
\end{equation}
We obtain the relation \cite {Burb1}:
\begin{equation}
\label{burban: grelation1} a a^+ - q^{2a + c-1} a^+ a = q^{2 a (N
+ 1) + b},
\end{equation}
or equivalent
\begin{equation}
\label{burban: grelation2} a a^+ - q^{2a} a^+ a = q^{2 a (N + 1) +
b}q^{(c-1)N} \end{equation} which together with other relations of
(\ref{burban: first}) form the corresponding oscillator algebra.

The self-adjoints of the operator (\ref{burban: coord}) can be
established by the Theorem 1.3 Chapter VII in \cite {Ber}. For
this purpose it is sufficient to prove the divergence of the
series
\begin{equation}\label{burban: karl}
\sum_{n = 0}^{\infty} \frac{1}{r_n} = \infty.
\end{equation}
The operator $Q$ (\ref{burban: coord}) is symmetric one. If
\begin{equation}
r_n\ge 0,\quad r_{n-1}r_{n+1}\le r_n^2,\quad \sum_{n=0}^{\infty}
\frac{1}{r_n} < \infty.
\end{equation}
by the Theorem 1.5 Chapter VII in \cite {Ber} its closure ${\bar
Q}$ is not self-adjoint operator.

In our case the property $ r_n\ge 0,\quad r_{n-1}r_{n+1}\le r_n^2$
is satisfied automatically, therefore, the self-adjointness of the
operator $Q$ are reduced to the proving of the convergence of the
series (\ref{burban: karl}). We obtain the cases
\begin{equation}
\label{burban:cas1} q < 1,\quad  \begin {cases} a < 0,\quad {c-1}
> 0,& \text{divergent,}\\ a > 0,\quad c-1 > 0, &\text{
convergent,}\\ a + \frac{c-1}{2} < 0,\quad c-1 <
0,&\text{divergent,}\\ a + \frac{c-1}{2} > 0,\quad c-1< 0, &
\text{convergent.}
\end{cases}
\end{equation}
The equation (\ref{burban: jac}) for this case take the form $$ x
P_n{(x;q)} = \Bigl(\frac{1}{1-q')}\Bigr)^{1/2} q^{a n +
b/2}(1-q'^{n})^{1/2}P_{n-1}(x;q)$$
\begin{equation}
\label{burban:equl1} + \Bigl(\frac{1}{1-q'}\Bigr)^{1/2}q^{{a(n +
1)} + b/2}(1-q'^{(n + 1)})^{1/2}P_{n + 1}(x;q),
\end{equation}
where $q'=q^{c-1}.$ If we  change and rescale of the variables $y
= (1-q')^{1/2} x, $ $ P_n(x;q)= \psi_n ((1 - q')^{1/2}x;q)$ then
it follows  $$ x \psi_n(x;q)$$
\begin{equation}
\label{burban:equer} = q^{a (n + 1) + b/2}(1 -
q'^{n})^{1/2}\psi_{n+1}(x;q) + q^{a n + b}(1 -
q'^{n})^{1/2}\psi_{n-1}(x;q).
\end{equation}
Representing $\psi_n(x;q)$ as $$\psi_n(x;q) = \frac {q^{a
n^2/2}}{q^{(a + b) n/2}(q';q')_n^{1/2}}h_n(x;q)$$ we get from
(\ref{burban:equer}) the recurrence relation for the (generalized)
discrete q-Hermite I polynomials $h_n(x;q):$
\begin{equation}
\label{burban: discrhermit} x h_n(x;q)= h_{n+1}(x;q)+ q^{2a n + b}
(1-q'^{n})h_{n-1}(x;q).
\end{equation}
The solution of this equation is given by anzatz
\begin{equation}
\label{burban:finitaI} h_n(x;q)= \sum_{k = 0}^{[n/2]}
\frac{(q';q')_n}{\Bigl((a_n,c_n); (1,q'^d)\Bigr)_k}(-1)^k q^{(2a n
+ b)k} q'^{k(k-n)} x^{n-2k}
\end{equation}
where $a_n, c_n, d$ are unknown quantities.  The notation
\begin{equation}
\Bigl((a, c );(p,q)\Bigr)_k = \begin {cases} 1,& \text{if $ k = 0
;$}\\ (a - c)(a p - c q)\ldots(a p^{k-1}- c
q^{k-1}),&\text{otherwise }\\
\end{cases}
\end{equation} we use from \cite{JR}.
The substitution (\ref{burban:finitaI}) into (\ref{burban:
discrhermit}) leads to the equation
\begin{equation}
1-q'^{n+1 }- q^{2a n + b}(a_n-c_n q'^{d(k-1)}) q'^{-2(k-1)} = 1 -
q'^{n-2k+1}
\end{equation}
The solutions of this equation
\begin{equation}
a_n = q^{-2a n-b}q'^{n-1}, c_n = q^{-2a n-b}q'^{n+1}, d+2
\end{equation}
give the solutions of (\ref{burban: discrhermit})
\begin{equation} h_n(x;q)= \sum_{k = 0}^{[n/2]}
\frac{(q';q')_n}{(q'^2 ; q'^2)_k (q';q')_{n-2k}} (-1)^k q^{(2a n +
b)k} q'^{k(k-n)} x^{n-2k}.
\end{equation}
They can be represented in terms of the basic hypergeometric
function as
\begin{equation}
h_n(x;q)= x^n{}_2\phi_0\Bigl( \begin{array}{cc|c}
q'^{-n},&q'^{-n+1}&\\ {}&-&\end{array}q'^2;\frac{q^{2a n + b}
q'^n}{x^2} \Bigr).
\end{equation}
 As has been shown in \cite {Burb1} these solutions at $a = \frac{1}{2}, b = - 1, c =
2$ is reduced to discrete q-Hermite I polynomials and the
relations (\ref{burban: grelation1}) (or (\ref{burban:
grelation2})) to equation of the corresponding $q$-deformed
oscillator.

 The solutions $P_n(x;q)$ of the equations
(\ref{burban:equl1}) with the initial conditions $P_{-1}(x;q) =
0,\quad P_0(x;q) = 1$ can be written as polynomials of degree $n$
in $x:$
\begin{equation} \label{burban:resI} P_n(x;q) =
\frac{q^{- a n^2/2}}{q^{\frac{a + b}{2} n}(q';q')_n^{1/2}}h_n
(\sqrt{1-q'}x;q).
\end{equation}
\medskip
Now we restrict ourselves by the condition $a = (c-1)/2$ in
(\ref{burban:resI}). Then \begin{equation} P_n(x;q)= \frac{q^{-a
n(n-1)/2}}{(q';q')_n ^{1/2}}h_n^0 (q^{-(2a + b)/2}
\sqrt{1-q'}x;q').
\end{equation}
These polynomials are orthogonal with respect to the discrete
measure $$ d\omega(x) = \frac {q^{-(2a + b)}\sqrt{1-q'}
}{2}(q';q'^2)_{\infty}\delta (x - \frac {q'^0}{ q^{-(2 a + b
)}\sqrt{1-q'}})d x $$ $$+ \sum_{k > 0} \frac{q^{-(2 a +
b)/2}\sqrt{1-q'}|x|}{2}\frac{(q^{-(2 a + b)}(q'^2(1-q')x^2,q';
q^2)_{\infty}}{(q';q')_{\infty}} \,\delta (x - \frac {q'^k}{q^{-(2
a + b)/2}\sqrt{1-q'}})d x $$
\begin{equation}
+ \sum_{k > 0} \frac{q^{-(2a +
b)/2}\sqrt{1-q'}|x|}{2}\frac{(q^{-(2 a + b)}(q'^2(1-q')x^2,q';
q^2)_{\infty}}{(q';q')_{\infty}} \,\delta ( x + \frac {q'^k}{q^{-(
2 a + b)/2}\sqrt{1-q'}})d x.
\end{equation}
The orthogonality relation has the form $$
\frac{\delta_{mn}}{(q';q')_n} =
\frac{1}{2}\frac{(q';q')_{\infty}}{(q'^2;q'^2)_{\infty}}
P_m(1;q)P_n(1;q)  $$
\begin{equation}
+ \sum_{k > 0}\{P_m (q'^k;q)P_n(q'^k;q) + P_m (- q'^k;q)P_n(-
q'^k;q)\}\frac{q'^k}{2}\frac{(q'^{2k+2},q';q'^2)_{\infty}(q';q'^2)_{\infty}}
{(q'^2;q'^2)_{\infty}}.\end {equation} It follows that spectrum of
the position operator $Q$ is
\begin{equation} Sp\,Q =
\Bigl\{ \frac{\pm q^{(2a + b)/2}}{\sqrt{1-q'}},\quad\frac{\pm
q^{(2a + b)/2}q'}{\sqrt{1-q'}},\ldots, \frac{\pm q^{(2 a +
b)/2}q'^k}{\sqrt{1-q'}},\ldots; k\ge 0 \Bigr\}.
\end{equation}

\noindent

{\bf 4. Hamiltonian and energy spectrum of
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator}

\medskip

{\it (g) Energy spectrum of free
$(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator.} In general, a
Hamiltonian $H(a^+,a,N)$ of an oscillator is a function of the
operators $a^+, a, N.$ As an example, we consider the free
standard Hamiltonian
\begin{equation}\label{burban: ham} H =
\frac{\hbar\omega_0}{2}\{a^+, a\} = \frac{\hbar\omega_0}{2}(aa^+ +
a^+a)\end{equation} of the $(q;\alpha,\beta,\gamma;\nu)$-deformed
oscillator.

Due to the structure function of deformation the free Hamiltonian
$H$ of the $(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator in
the Fock representation (\ref{burban: rep2}) can be written only
as function of number operator $N$
\[ H = \frac{\hbar\omega_0}{2}q^{\beta}\Bigl\{
\Bigl(\frac{q^{\gamma N} - q^{\alpha N}}{q^{\gamma} - q^{\alpha}}
+ 2 \nu \frac{q^{\gamma N} - (-1)^N q^{\alpha N}}{q^{\gamma} +
q^{\alpha}}\Bigr) +
\]
\begin{equation} \Bigl(\frac{q^{\gamma (N+1)} -
q^{\alpha (N+1)}}{q^{\gamma} -
q^{\alpha}} + 2 \nu \frac{q^{\gamma (N+1)}-(-1)^{(N+1)} q^{\alpha
(N+1)}}{q^{\gamma} + q^{\alpha}}\Bigr)\Bigr\}.
\end{equation}

For a study of the spectrum and the nonlinear frequency of this
oscillator, it is convenient to use new parametrization
\begin{equation}\label{burban: newpar}
q = e^{\tau},\quad \alpha = \rho-\mu,\quad \gamma = \rho
+\mu.\end{equation}
We have
\[ E_n = \frac{\hbar\omega_0}{2}e^{\tau(\beta + \rho n)}\times \]
\[ \Bigl\{\frac{\sinh(\tau\mu(n+1))}{\sinh(\tau\mu)} +
e^{-\tau\rho}\frac{\sinh(\tau\mu n)}{\sinh(\tau \mu)} +
2\nu\frac{1 -
(-1)^n}{2}\Bigl(\frac{\sinh(\tau\mu(n+1))}{\cosh(\tau \mu)} +
e^{-\tau \rho} \frac{\cosh(\tau \mu n)}{\cosh(\tau \mu)}\Bigr) +
\]
\begin{equation}\label{burban: hamo}
2\nu\frac{1 + (-1)^n}{2}\Bigl(\frac{\cosh(\tau \mu (n +
1))}{\cosh( \tau \mu)} + e^{-\tau\rho}\frac{\sinh(\tau\mu
n)}{\cosh(\tau \mu)}\Bigr) \Bigr\},
\end{equation}
or
\[ E_n = \frac{\hbar\omega_0}{2}e^{\tau(\beta - \rho)}
\Bigl\{ \frac{1 + e^{\tau(\rho + \mu)}}{2}\frac{e^{\tau(\rho +
\mu)n}}{\sinh(\tau\mu)} - \frac{1 + e^{\tau(\rho -
\mu)}}{2}\frac{e^{\tau(\rho - \mu)n}}{\sinh(\tau\mu)}
\]
\[ + 2 \nu
\frac{1 - (-1)^n}{2}\Bigl( \frac{1 + e^{\tau(\rho +
\mu)}}{2}\frac{e^{\tau(\rho + \mu)n}}{\cosh(\tau \mu)} + \frac{1 -
e^{\tau(\rho - \mu)}}{2}\frac{e^{\tau(\rho - \mu)n}}{\cosh(\tau
\mu)}\Bigr)+\]
\begin{equation}\label{burban: hami}2\nu \frac{1 + (-1)^n}{2}\Bigl(\frac{1 + e^{\tau(\rho +
\mu)}}{2}\frac{e^{\tau(\rho + \mu)n}}{\cosh(\tau\mu)} - \frac{1 -
e^{\tau(\rho - \mu)}}{2}\frac{e^{\tau(\rho -
\mu)n}}{\cosh(\tau\mu)}\Bigr)\Bigr\}.
\end{equation}
It follows that the number of the energy levels is infinite, and
the eigenvalues $E_n$ for $\tau \ne 0, n\to\infty$ depend on the
parameters $\alpha, \gamma$ via the exponential factor
$e^{\tau(\rho - |\mu|)}.$

It is convenient to consider the eigenvalue $E_n$ of $H$ for $n$
even and odd separately:
\[
E_n = \frac{\hbar\omega_0}{2}e^{\tau(\beta - \rho)}\Bigl\{\frac{1
+ e^{\tau(\rho + \mu)}}{2}\frac{e^{\tau(\rho +
\mu)n}}{\sinh(\tau\mu)} - \frac{1+e^{\tau(\rho -
\mu)}}{2}\frac{e^{\tau(\rho - \mu)n}}{\sinh(\tau\mu)} + \]
\begin{equation}
\label{burban: odd} 2\nu\Bigl(\frac{1 + e^{\tau(\rho +
\mu)}}{2}\frac{e^{\tau(\rho + \mu)n}}{\cosh(\tau\mu)}+\frac{1 -
e^{\tau(\rho - \mu)}}{2}\frac{e^{\tau(\rho -
\mu)n}}{\cosh(\tau\mu)}\Bigr)\Bigr\}
\end{equation}
for $n$ odd, and
\[ E_n = \frac{\hbar \omega_0}{2}e^{\tau(\beta - \rho)}\Bigl\{\frac{1
+ e^{\tau(\rho + \mu)}}{2}\frac{e^{\tau(\rho +
\mu)n}}{\sinh(\tau\mu)} - \frac{1 + e^{\tau(\rho -
\mu)}}{2}\frac{e^{\tau(\rho - \mu)n}}{\sinh(\tau\mu)}+ \]
\begin{equation}\label{burban: even}
2\nu\Bigl(\frac{1 + e^{\tau(\rho + \mu)}}{2}\frac{e^{\tau(\rho +
\mu)n}}{\cosh(\tau\mu)} - \frac{1 - e^{\tau(\rho -
\mu)}}{2}\frac{e^{\tau(\rho - \mu)n}}{\cosh(\tau\mu)}\Bigr)\Bigr\}
\end{equation}
for $n$ even. It follows the spectrum of this oscillator is not
equidistant and the spacing is equal

\begin{equation}
E_{2n+1} - E_{2n} = \frac{\hbar
\omega_0}{2}\frac{\sinh2\mu\tau}{\sinh(\tau\mu)} e^{\tau(\beta
+\rho)} e^{2\gamma\tau n}.
\end{equation}

{\it (h) Asymptotic behaviour of energy spectrum of Hamiltonian
$H$.}

 According to the analysis given above, we have
to give the estimation of the spectrum of Hamiltonian (\ref
{burban: hami}) for the parameter $-1 < 2\nu ,$ if $\tau(\gamma -
\alpha) > 0$ and $-1 < 2 \nu < \coth(\tau(\gamma - \alpha)/2)$ if
$\tau(\gamma - \alpha) < 0.$
 Three cases must be distinguished:
$\tau(\rho + |\mu|)>0,\tau(\rho + |\mu|)< 0,\tau(\rho + |\mu|) =
0.$

In special case $\mu = 0,$ $(\alpha = \gamma = \rho) $ we have
\begin{equation}
E_n =\frac{\hbar\omega_0}{2}e^{\tau(\beta + \rho
n)}\Bigl[(n+\nu)(1 + e^{-\tau\rho})+(-1)^n\nu(1 - e^{-\tau\rho}) +
1 \Bigr].
\end{equation}

If $(\tau(\rho + |\mu|)
> 0$ and $ -1 < 2\nu$, then from (\ref{burban: hami}) it implies that
the energy grows to infinity with increasing $n.$

If $\tau(\rho + |\mu|) = 0,$ then in the case ($\rho = \mu $) we
have
\begin{equation}
E_n = \frac{\hbar \omega_0}{2}e^{\tau \beta}e^{\beta -
\rho}\Bigl\{ \frac{e^{2\tau\rho n} -1}{e^{\tau\mu}-e^{-\tau\mu}} +
\frac{e^{2\tau\rho (n+1)} -1}{e^{\tau\mu} - e^{-\tau\mu}} + 2 \nu
\frac{e^{2\rho\tau n} - e^{2\rho\tau (n+1)}}{e^{\tau\mu}-
e^{-\tau\mu}}\Bigr\}.
\end{equation}
The same holds for $\rho +\mu = 0.$

It follows that $E_n$ monotonically increase to upper bounds that
depends on parameters

$$E_{\max} = \frac{\hbar\omega_0}{2}\frac{e^{\tau (\beta -
\rho)}}{\sinh\tau|\mu|}.$$

If $\tau(\rho + |\mu|) < 0$ and $ -1 < 2\nu$ then the energies
(\ref{burban: odd}) and (\ref{burban: even}) at first increase and
then go to zero as $n\to\infty.$ In the particular case $\mu = 0$
in (\ref{burban: odd}) and (\ref{burban: even}) we have

\[E_n = \frac{\hbar \omega_0}{2}e^{\tau(\beta + \rho n)}\Bigl\{n(1+
e^{- \tau\rho}) + 2 \nu + 1\Bigr\},\] for $n$ odd and
\[ E_n = \frac{\hbar \omega_0}{2}e^{\tau(\beta + \rho n)}\Bigl\{n (1 + e^{-\tau\rho}) +
2 \nu e^{- \tau\rho} + 1 \Bigr\}\] for $n$ even.

\subsection*{Acknowledgements}

This research was partially supported by Grant 14.01/016 of the
State Foundation of Fundamental Research of Ukraine.


TITLE: parentheses in free and bound variable in lambda calculus
QUESTION [1 upvotes]: I'm now starting the syntax of the lambda calculus, and have a very simple question about the parentheses for free and bound variable.
Are " x1:= λx.x x " and " x2:= (λx.x) x " represent the same? In other word, is the second "x" in x1 free? I know the second "x" in x2 is free for sure, I just wonder what the purpose put parentheses there.
BTW, x3:= λx.x y and x4:= (λy.x) y, I think "y" in x3 is free and "x" in x4 is free, anyone could check for me ? Thanks

REPLY [1 votes]: The lambda calculus syntax rules/conventions are:
1) application has higher precedence than abstraction. This is:
λx.AB = λx.(AB) ≠ (λx.A)B

2) application is left associative. This means:
ABC = (AB)C ≠ A(BC)

In your specific examples, the semantic difference would be:

"x1:= λx.x x" Here x1 is a function that takes a value x and returns x x.
So if we apply the function x1 to a free variable z we get: x1 z -reduces to-> z z
"x2:= (λx.x) x" Here we have the identity function (λx.x takes an x and returns it as is) applied to the free variable x. So this will reduce to x.

Your guesses about free variables inside x3 and x4 are correct.


TITLE: Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.
QUESTION [3 upvotes]: Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01.
Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over k}$. Then: ${\ln(1+x)\over x}=\sum_{k=1}^{\infty}{(-1)^{k-1}x^{k-1}\over k}$ and therefore: $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx=\sum_{m=0}^{\infty}\int_{0}^{1\over 2}{(-1)^{m}x^{m}\over m+1}dx=\sum_{m=0}^{\infty}{(-1)^{m}x^{m+1}\over (m+1)^2}|_{0}^{1\over 2}=\sum_{m=0}^{\infty}{(-1)^{m}\over 2^{m+1}(m+1)^2}$ by the uniqueness of the Taylor series, the last expression is the Taylor series of $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$. Now I have to find out the extent to which I need to sum up the series addends, so that I maintain the required accuracy. How can I do so?

REPLY [1 votes]: After what you wrote and just as user84413 commented, for a given tolerance equal to $\epsilon$ you need to find the first $n$ such that $\frac{1}{2^{n}n^2}<\epsilon$ or $2^n n^2 >\frac{1}\epsilon$. In the case where $\epsilon$ is not too small, inspection could suffice to find the appropriate value of $n$ where to stop.
But, just for your curiosity, let us consider potentially very small values of the tolerance.
Consider the function $$f(n)=2^n n^2 -\frac{1}\epsilon$$ It is very stiff while its transform (using logarithms) $$g(n)=n\log(2)+2\log(n)+\log(\epsilon)$$ is "nicer". Moreover, this equation as an explicit solution $$n=\frac{2 }{\log (2)}W\left(\frac{1}{2} \sqrt{\frac{1}{\epsilon }} \log (2)\right)$$ where $W(a)$ is Lambert function.
For large values of argument $a$, Lambert function can be approximated using
$$W(a)=L_1-L_2+\frac {L_2}{L_1}+\frac {L_2(L_2-2)}{2L_1^2}+\cdots$$ where $L_1=\log(a)$ and $L_2=\log(L_1)$.
Using this approximation, we find $n=3.10$ for $\epsilon=0.01$, $n=5.19$ for $\epsilon=0.001$, $n=7.50$ for $\epsilon=0.0001$.
Going further, writing $ \epsilon=10^{-k}$, you could show that $n\approx -2+\frac 52 k$.


TITLE: Are maximal compact subgroups of connected groups connected?
QUESTION [14 upvotes]: Assume $G$ is a connected locally compact group and $M$ is a maximal compact subgroup of $G$. Is $M$ connected too?

REPLY [19 votes]: Disclaimer: Locally compact groups are absolutely not my field of expertise. I hope an expert can check my statements below, and perhaps add some details and references.
The Malcev–Iwasawa theorem implies that any connected, locally compact group $G$ satisfies:

$G$ has a maximal compact subgroup;
there exists $n\in\mathbb{N}$ such that for any maximal compact subgroup $K$ of $G$, the underlying space of $G$ is homeomorphic to $K\times\mathbb{R}^n$.

In particular, every maximal compact subgroup of a connected, locally compact group is itself connected.
References: The following references state the necessary results without proof.

Theorem 32.5 of Markus Stroppel's book "Locally compact groups".
The article "Compact subgroups of Lie groups and locally compact groups" (DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1166357-9), published in the Proceedings of the American Mathematical Society, volume 120, number 2, in February 1994 (pages 623-634). See the statements of theorems A, B, and C in the introduction to this article. According to the discussion there, the theorems hold for connected, locally compact groups: they follow from the analogous results for Lie groups as soon as one knows that a connected, locally compact group is a projective limit of Lie groups.


text:
\begin{document}

   \title{Unevenly-sampled signals: a general formalism of the Lomb-Scargle periodogram}
   \author{R. Vio\inst{1}
    \and
           P. Andreani\inst{2}
    \and
           A. Biggs\inst{3}
          }
   \institute{Chip Computers Consulting s.r.l., Viale Don L.~Sturzo 82,
              S.Liberale di Marcon, 30020 Venice, Italy\\
              \email{robertovio@tin.it},
         \and
		  ESO, Karl Schwarzschild strasse 2, 85748 Garching, Germany\\
                  INAF-Osservatorio Astronomico di Trieste, via Tiepolo 11, 34143 Trieste, Italy\\
		  \email{pandrean@eso.org}
         \and
		  ESO, Karl Schwarzschild strasse 2, 85748 Garching, Germany\\
                  \email{abiggs@eso.org}
             }

\date{Received .............; accepted ................}

\abstract {The periodogram is a popular tool that tests whether a
  signal consists only of noise or if it also includes other
  components. The main issue of this method is to define a
  critical detection threshold that allows identification of a
  component other than noise, when a peak in the periodogram exceeds
  it. In the case of signals sampled on a regular time grid,
  determination of such a threshold is relatively simple.  When the
  sampling is uneven, however, things are more complicated. The most
  popular solution in this case is to use the {{\it Lomb-Scargle}}
  periodogram, but this method can be used only when the noise is the
  realization of a zero-mean, white (i.e.\ flat-spectrum) random
  process. In this paper, we present a general formalism based on
  matrix algebra, which permits analysis of the statistical
  properties of a periodogram independently of the characteristics of
  noise (e.g.\ colored and/or non-stationary), as well as the
  characteristics of sampling.}

\keywords{Methods: data analysis -- Methods: statistical}
\titlerunning{Periodogram analysis of unevenly-sampled signals}
\authorrunning{R. Vio, P. andreani and A. Biggs}
\maketitle

\section{Introduction}

Spectral analysis is a popular tool for testing whether a given
experimental time series $\{ x(t_0), x(t_1), \ldots, x(t_{M-1}) \}$
contains only noise, i.e.\ $x(t_j) = n(t_j)$, or whether some other
component $s(t)$ is present, i.e.\ $x(t) = s(t) + n(t)$. The classic
approach is to fit the time series with the model function
\begin{equation} \label{eq:model}
x(t_j) = \sum_{k=0}^{N-1} a_k \cos{(2 \pi f_k t_j)} + b_k \sin{(2 \pi f_k t_j}) + n(t_j),
\end{equation}
$j=0, 1, \ldots, M-1$. If, for example, a periodic component $s(t)$ is
present with a frequency $f_l$ close to one in the set $\{ f_k \}$,
then the periodogram $\{ \ph_k \}$,
\begin{equation} \label{eq:period1}
\ph_k = a_k^2 + b_k^2, \qquad k=0, 1, \ldots, N-1,
\end{equation}
will show a prominent peak close to $k=l$. If $s(t)$ is semi-periodic
or even non-periodic, the situation is more complicated since more
peaks are expected. The main problem with the use of this technique
is the definition of a detection threshold that fixes the contribution
of noise in such a way that, when a peak exceeds it, the presence of a
component $s(t)$ can be claimed. In the case of signals sampled on a
regular time grid, the determination of such a threshold is a
relatively simple procedure, but this is not the case when the
condition of regularity does not apply. In this respect, several
solutions have been proposed \citep[see][and references
  therein]{lom76,fer81, sca82, gil87, ree07, zec09} that, however,
work only under rather restrictive conditions (e.g.\ white and/or
stationary noise) and are difficult to extend to more general
situations.

In this paper, a general formalism is presented that allows
analysis of the statistical properties of periodograms independently
of the specific characteristics of the noise and the sampling of the
signal. In Sec.~\ref{sec:formalization} the formalism is presented
for the case of even sampling and its extension to arbitrary sampling
in Sec.~\ref{sec:generalization}. The usefulness of the proposed
formalism is illustrated in Sec.~\ref{sec:irregular}, where the case
of white noise with a mean different from zero and that of colored
noise is calculated. In Sec.~\ref{sec:fit} the relationship between
the periodogram and the least-squares method is considered. Finally,
on the basis of simulated signals and a real time series, we discuss in
Sec.~\ref{sec:discussion} whether the use of algorithms
specifically developed for computing the periodogram of
unevenly-sampled time series is really advantageous.

\section{Periodogram analysis in the case of even sampling}
\label{sec:formalization}

If a continuous signal $x(t)$ is sampled on a set of $N$ equispaced
time instants $t_0, t_1, \ldots, t_{N-1}$, a time series $x_j$, $j=0,
1, \ldots, N-1$ is obtained. Its discrete Fourier transform (DFT) is
given by
\begin{equation} \label{eq:DFT}
\xh_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} x_j {\rm e}^{ - i 2 \pi k j / N}, \qquad k=0, 1, \ldots, N-1,
\end{equation}
with $i = \sqrt{-1}$. The sequence $\{ x_j \}$ can be recovered from
$\{ \xh_k \}$ by means of
\begin{equation} \label{eq:IDFT}
x_j = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \xh_k {\rm e}^{ 2 \pi k j/ N}, \qquad j=0, 1, \ldots, N-1.
\end{equation}
The set $\{ k/N \}_{k=0}^{N-1}$ provides the so-called {\it Fourier
  frequencies}. Implicit in the use of DFT is the assumption that $\{
x_j \}$ is a periodic sequence with period $N \Delta t$ where $\Delta
t = t_{j+1} - t_j$.

In matrix notation, Eqs.~(\ref{eq:DFT}) and (\ref{eq:IDFT}) can be
written in the form
\begin{equation} \label{eq:FDFT}
\xhb = \Fb \xb
\end{equation}
and
\begin{equation} \label{eq:IFDFT}
\xb = \Fb^* \xhb.
\end{equation}
Here, $\xb$ and $\xhb$ are column arrays that contain, respectively,
the sequences $\{ x_j \}$ and $\{ \xh_k \}$, and $\Fb$ is the so-called ``{\it Fourier matrix}'',  which is an $N \times
N$ square, symmetric matrix whose $(k,j)$-entry \footnote{In the following,
the element in the $n$th row and $m$th column
of an $N \times M$ matrix $\Ab$ will be indicated with $A_{mn}$ or alternatively with
$(\Ab)_{mn}$, $n=0, 1, \ldots N-1$, $m=0, 1,\ldots, M-1$.} is given by
\begin{equation} \label{eq:F}
F_{k j} = \frac{1}{\sqrt{N}} {\rm e}^{ - i 2 \pi k j /N}.
\end{equation}
The superscript ``${}^*$'' denotes the {\it complex conjugate transpose}.
Matrix $\Fb$ is unitary, i.e.,
\begin{equation} \label{eq:inverse}
\Fb \Fb^* = \Fb^* \Fb = \Ib,
\end{equation}
with $\Ib$ the identity matrix. Another useful property is that
\begin{equation} \label{eq:FF}
\Fb \Fb = \Fb^T \Fb = \Fb \Fb^T = \Fb^T \Fb^T = \Hb
\end{equation}
with
\begin{equation} \label{eq:Permut}
\Hb =
\left(
\begin{array}{ccccccc}
1 & 0 & 0 & \cdots & 0 & 0 & 0 \\
0 & 0 & 0 & \cdots & 0 & 1 & 0 \\
0 & 0 & 0 & \cdots & 1 & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0  & 1 & \cdots & 0 & 0 & 0  \\
0 & 1 & 0 & \cdots & 0 & 0 & 0 \\
\end{array}
\right).
\end{equation}
By means of Eq.~(\ref{eq:FF}) it can be shown that
\begin{equation} \label{eq:orth}
\Fb_{\Rmatc} \Fb_{\Imatc} = \Fb^T_{\Rmatc} \Fb_{\Imatc} = \Fb_{\Rmatc} \Fb^T_{\Imatc} = \Fb^T_{\Rmatc} \Fb^T_{\Imatc} = \zerob,
\end{equation}
where $\Fb_{\Rmatc} \equiv \Rmatc[\Fb]$ and $\Fb_{\Imatc} \equiv \Imatc[\Fb]$ and where $\Rmatc[.]$ and $\Imatc[.]$ are the real and the
imaginary parts of a complex quantity. Indeed,
\begin{equation} \label{eq:prop}
\Fb \Fb = \Fb_{\Rmatc} \Fb_{\Rmatc} + \Fb_{\Imatc} \Fb_{\Imatc} + i 2  \Fb_{\Rmatc} \Fb_{\Imatc} = \Hb,
\end{equation}
and $\Hb$ is a real matrix, hence Eq.~(\ref{eq:orth}) has to hold.
From Eq.~(\ref{eq:DFT}) it is also possible to see that $\xh_0$ is a
real quantity. In the case that $N$ is an even number, the same holds
for $\xh_{N_{\dag}+1}$ where $N_{\dag} = \lceil N/2 \rceil$ and
$\lceil z \rceil$ represents the smallest integer greater than
$z$. Finally, dealing with $\xhb$, only half of this array can be
considered, since $\xh_{N-k} = \xh^*_k$, $k=1, 2, \ldots,
N_{\dag}-\delta_{2 N_{\dag}}^N$, where $ \delta_i^j = 1$ if $i=j$ and
zero otherwise. With this notation, the periodogram of $\xb$ is
defined as
\begin{equation} \label{eq:period2}
\ph_k = 2 (\Rmatc[\xhu_k]^2 + \Imatc[\xhu_k]^2) = 2 | \xhu_k |^2, \qquad k=0,1,\ldots, N_{\dag}-1,
\end{equation}
where $\xhu_k$ is the $k$th entry of the array
$\xhbu^T = [\xh_0, \xh_1, \ldots, \xh_{N_{\dag}-1}]$ \footnote{From now on, if $\rb$ is an $N \times 1$ column array, then
$\rbu$ is a column array that contains the first $N_{\dag} = \lceil N/2 \rceil$ entries of $\rb$, i.e.\ $\rbu = [r_0, r_1, \ldots, r_{N_{\dag}-1}]^T$.
Similarly, if $\Ab$ is an $N \times M$ matrix,
then $\Abu$ is a matrix that contains the first $N_{\dag}$ rows of $\Ab$.}
and $| . |$ denotes the {\it Euclidean} norm.
By means of
\begin{equation}
\zhb = \left(
\begin{array}{c}
\xhbu_{\Rmatc} \\
\xhbu_{\Imatc}
\end{array}
\right),
\end{equation}
a column array obtained by the column concatenation of
$\xhbu_{\Rmatc} \equiv \Rmatc[\xhbu]$ and $\xhbu_{\Imatc} \equiv \Imatc[\xhbu]$, Eq.~(\ref{eq:period2}) can be rewritten in the form
\begin{equation} \label{eq:period3}
\ph_k = 2 (\zh_k^2 + \zh_{N_{\dag}+k}^2),  \qquad k=0,1,\ldots, N_{\dag}-1.
\end{equation}
In Sect.~\ref{sec:fit} it is shown that this periodogram is equivalent to that obtainable by means the least-squares fit of
model~(\ref{eq:model}) with $M=N$, $t_j = j$ and $f_k = k/N$.

An important point to stress is that, if $\xb$ is the realization of a
(not necessarily Gaussian) random process, then each $\xhu_k$ is given
by the sum of $N$ random variables. This is because of the linearity
of the {\it Fourier operator} $\Fb$.  Thanks to the {\it central limit
  theorem}, therefore, the entries of $\xhb$ can be expected to be Gaussian
random quantities. As a consequence, the entries of $\zhb$ can also be expected to be Gaussian random quantities
with covariance matrix $\Cb_{\zhb}  = {\rm E}[\zhb \zhb^T]$ given by
\begin{equation} \label{eq:Ce}
\Cb_{\zhb} = \left(
\begin{array}{cc}
\Fbu_{\Rmatc} \Cb_{\xb} \Fbu^T_{\Rmatc} & \Fbu_{\Rmatc} \Cb_{\xb} \Fbu^T_{\Imatc} \\
\Fbu_{\Imatc} \Cb_{\xb} \Fbu^T_{\Rmatc} & \Fbu_{\Imatc} \Cb_{\xb} \Fbu^T_{\Imatc}
\end{array}
\right).
\end{equation}
Here, ${\rm E}[.]$ denotes the {\it expectation operator},  $\Cb_{\xb} = {\rm E}[\xb \xb^T]$ is the covariance matrix of
$\xb$, and $\Fbu$
the matrix obtained by the first $N_{\dag}$ rows of the {\it Fourier matrix} $\Fb$.  From
Eqs.~(\ref{eq:Ce}) and (\ref{eq:orth}), it is easy to deduce that, if
$\xb$ is the realization of a standard white-noise process,
i.e., $\Cb_{\xb} = \Ib$, then $\Cb_{\zhb}$ is a diagonal matrix with $(\Cb_{\zhb})_{11}=1$,
$(\Cb_{\zhb})_{N_{\dag} N_{\dag}}=0$
and $(\Cb_{\zhb})_{kk}=0.5$. In other words, the entries of $\zhb$ are
mutually uncorrelated. In turn, this means that $\Rmatc[\xhu_k]$ is
uncorrelated with $\Imatc[\xhu_k]$. If $\xb$ is a colored (not
necessarily stationary) noise process, i.e.\ $\Cb_{\xb}$ is not a diagonal matrix,
then this holds also for $\Cb_{\zhb}$. However, from
$\xb$ it is possible to obtain an array $\yb$ containing mutually
uncorrelated entries by means of the transformation
\begin{equation} \label{eq:transf}
\yb = \Cb_{\xb}^{-1/2} \xb.
\end{equation}
The matrix $\Cb_{\xb}^{-1/2}$ can be computed via
\begin{equation} \label{eq:eig}
\Cb_{\xb}^{-1/2} = \Ub^T \Sigmab^{-1/2} \Ub
\end{equation}
with $\Ub$ the orthogonal matrix whose columns contain
the eigenvectors of $\Cb_{\yb}$ and $\Sigmab$ a diagonal matrix containing the corresponding eigenvalues $\{ \lambda_l \}_{l=0}^{N - 1}$.
This decomposition is particularly simple and computationally efficient if $\Cb_{\xb}$ is a {\it circulant} matrix since it
can be diagonalized according to
\begin{equation} \label{eq:diag}
\Cb_{\xb} = \Fb^* {\rm diag}[\Fb \cb] \Fb,
\end{equation}
where ``${\rm diag}[\qb]$'' denotes a diagonal matrix whose diagonal
entries are given by the array $\qb$ and $\cb$ is the first column of
$\Cb_{\xb}$. Because of this, the decomposition~(\ref{eq:eig}) can be
directly computed with $\Ub = \Fb$ and $\Sigmab = {\rm diag}[\cb]$;
hence, $\yb = \Fb^* \Sigmab^{-1/2} \Fb$. This means that the whitening
operation can be performed in the harmonic domain through the
following procedure: a) computation of $\xhb$, i.e.\ the DFT of $\xb$;
b) computation of $\{ \yh_k \} = \{\xh_k / \lambda^{1/2}_k \}$; and c)
the inverse DFT of $\yhb$. A potential difficulty in using
transformation~(\ref{eq:transf}) is that $\Cb_{\xb}$ is required to be
of full rank. Such a condition can be expected to be satisfied in most
practical applications. Some problems can arise if the time step of
the sampling is much shorter than the decorrelation time of
$\nb$ \footnote{The decorrelation time of a random signal $n(t)$ is
  the time interval $\Delta t$ such that two values $n(t_1)$ and
  $n(t_2)$, with $t_2 - t_1 = \Delta t$, can be considered as
  uncorrelated.}. Indeed, some columns (and therefore some rows) of
$\Cb$ could almost be identical i.e., this matrix could become
numerically {\it ill-conditioned}. In this case, the most {\it natural} solution
consists of averaging the data that are close in time.

When the periodogram is used to test whether $\xb = \nb$ vs. $\xb =
\ssb + \nb$, a threshold $L_{\ph_k}$ has to be defined such that, with
a prefixed probability, a peak in $\ph_k$ that exceeds $L_{\ph_k}$
can be expected to not arise because of the noise. This requires
knowledge of the statistical properties of $\phb$ under the hypothesis
that $\xb = \nb$. The simplest situation is when $\nb$ is the
realization of a standard white-noise process. In fact, since the
entries of $\zhb$ are uncorrelated random Gaussian quantities, from
Eq.~(\ref{eq:period3}) it can be derived that the entries of $\phb$
are (asymptotically) independent quantities distributed according to a $\chi^2_2$
distribution\footnote{$\chi^2_2$ denotes the chi-square distribution
  with two degrees of freedom.}. As a consequence, independent of
the frequency $k$, a threshold $L_{{\rm Fa}}$ can be determined that
corresponds to the level that a peak due to the noise would exceed
with a prefixed probability $\alpha$ when a number $N_f$ of ({\it statistically
  independent}) frequencies are inspected. More specifically, $L_{{\rm
    Fa}}$ is the highest value for which
$1-[1-\exp{(-L_{\ph_k})}]^{N_f} \le \alpha$ \citep{sca82}, in formula
\begin{equation} \label{eq:false}
L_{{\rm Fa}} = \underset{ L_{\ph_k} }{\sup} \left\{ 1-[1-\exp{(-L_{\ph_k})}]^{N_f} \le \alpha \right\}.
\end{equation}
If the entire periodogram is inspected, then $N_f = N_\dag$. Commonly, $L_{{\rm
    Fa}}$ is called the {\it level of false alarm}.

Threshold~(\ref{eq:false}) is not applicable when the noise is
colored. However, the requirement to fix a different level for each
frequency can be avoided if the original signal $\xb$ is transformed
into
\begin{equation} \label{eq:whitey}
\yb = \Cb^{-1/2}_{\nb} \xb.
\end{equation}
Indeed, under the hypothesis that $\xb=\nb$, the entries of $\yb$ are
uncorrelated and unit-variance random quantities. As seen above,
this operation should not be difficult. Some problems emerge if the
decomposition~(\ref{eq:eig}) is carried out by means of the efficient DFT approach described above
(a necessary approach in the case of very long sequences of data). This is because, in
forming $\Cb_{\nb}$, it is necessary to take the
periodicity of the sequence $\xb$ forced by the DFT into account. In other words,
it is necessary to impose a spurious correlation among the first and
the last entries in $\nb$. For instance, for a stationary noise process,
$\Cb_{\nb}$ is a {\it Toeplitz} matrix, but it has to be approximated
with a {\it circulant} one. When dealing with sampled signals, this is
an unavoidable problem that no technique can completely solve. A
classical solution to relieving this situation is the {\it windowing
  method}, i.e., the substitution of $x_j$ in Eq.~(\ref{eq:DFT}) with
$\eta_j x_j$, where $\{ \eta_j \}$ is some prefixed discrete function
(window) that makes the signal gently reduce to zero at the extreme of
the sampling interval \citep[e.g. see][]{opp89}. This method can be
expected to work satisfactorily only when the decorrelation time of
noise is shorter than the length of the signal.

\section{Periodogram analysis in the case of uneven sampling} \label{sec:generalization}

If a signal $x(t)$ is sampled on an uneven set of time instants, some
problems emerge: it is no longer possible to define a set of "{\it
  natural}" frequencies such as those obtained by the {\it Fourier}
transform. In turn, this implies some ambiguities in the definition of
the Nyquist frequency that, loosely speaking, corresponds to the
highest frequencies that contain information on the signal of interest
\citep[e.g. see][]{vio00, koe06}. As a consequence,
Eq.~(\ref{eq:DFT}) has to be modified. In the following, with no loss
of generality, it is assumed that $\xb(t)$ is sampled at $M$
arbitrary time instants $t_0, t_1, \ldots, t_{M-1}$ with $t_0 = 0$,
$t_{M-1} = M-1$ and the remaining $t_j$ arbitrarily distributed within
this interval. Moreover, a set of $N$ frequencies $k =0, 1, \ldots,
N-1$ is considered with $N \gtreqless M$. Such a set corresponds
to the frequencies that are typically inspected when looking for a
periodicity. However, others can be chosen. With these conditions, a
transformation corresponding to the one given by Eq.~(\ref{eq:FDFT}) is
\begin{equation} \label{eq:irr4}
\xhb = \Fmatcb \xb,
\end{equation}
where
\begin{equation} \label{eq:irr3}
\Fmatc_{k j} = \frac{1}{\sqrt{M}} {\rm e}^{ - i 2 \pi k \tilde{t}_j / N},
\end{equation}
$\tilde{t}_j = t_j/ \Delta_m t$, $\Delta_m t = \gamma \min{ [
    \{t_{j+1} - t_j\}]}$ and $\gamma$ is a real positive number. Because of the normalization
by $\Delta_m t$, time $\tilde{t}_j$ is expressed in units of the
shortest sampling time interval. Apart from the substitution of the {\it Fourier} matrix $\Fb$
with $\Fmatcb$, the uneven sampling of signals does not modify the formalism introduced in the previous section.
In particular, the covariance matrix $\Cb_{\zhb}$ defined in Eq.~(\ref{eq:Ce}) becomes
\begin{equation} \label{eq:Cei}
\Cb_{\zhb} = \left(
\begin{array}{cc}
\Fmatcbu_{\Rmatc} \Cb_{\xb} \Fmatcbu^T_{\Rmatc} & \Fmatcbu_{\Rmatc} \Cb_{\xb} \Fmatcbu^T_{\Imatc} \\
\Fmatcbu_{\Imatc} \Cb_{\xb} \Fmatcbu^T_{\Rmatc} & \Fmatcbu_{\Imatc} \Cb_{\xb} \Fmatcbu^T_{\Imatc}
\end{array}
\right).
\end{equation}

The $N \times M$ {\it Fourier matrix} $\Fmatcb$ does not have the
properties~(\ref{eq:inverse})-(\ref{eq:prop}). As consequence, and
also in the case that $\xb$ is the realization of a standard
white-noise process i.e.\ $\Cb_{\xb} = \Ib$, matrix $\Cb_{\zhb}$,
\begin{equation} \label{eq:CI1}
\Cb_{\zhb} = \left(
\begin{array}{cc}
\Fmatcbu_{\Rmatc} \Fmatcbu^T_{\Rmatc} & \Fmatcbu_{\Rmatc} \Fmatcbu^T_{\Imatc} \\
\Fmatcbu_{\Imatc} \Fmatcbu^T_{\Rmatc} & \Fmatcbu_{\Imatc} \Fmatcbu^T_{\Imatc}
\end{array}
\right),
\end{equation}
is not diagonal. In general, $\Cb_{\zhb}$ is not even diagonalizable.
For example, this happens when the periodogram of a time series containing $M$ data
is computed on $N$ frequencies with $N > M$ (a typical situation in practical applications).
This implies that
the entries of $\zhb$ cannot be made mutually uncorrelated. Obviously,
the same holds for the entries of $\phb$ as given by
Eq.~(\ref{eq:period3}). As a consequence, although a number $N$ of frequencies are considered in $\phb$, at most only $M/2$
of them are statistically independent\footnote{This is because,
if $N > M$, the rank of the $N \times N$ matrix $\Cb_{\zhb}$ is smaller than or equal to $M$.
This implies that the array ${\zhb}$ has at most $M$ degrees of freedom.
Since each entry of $\phb$ is given by the sum
of two entries of ${\zhb}$, then a periodogram has at most $M/2$ degrees of freedom.}.
Particularly troublesome is that, for a given
frequency $k$, $\Rmatc[\xhu_k]$, and $\Imatc[\xhu_k]$ (i.e.\ $\zh_k$
and $\zh_{N_{\dag}+k}$) are also correlated. This makes it difficult
to fix the statistical characteristics of $\ph_k$. In this respect,
two choices are possible. The first consists in the determination, for
each frequency, of the PDF of $\ph_k$. Actually, this is a rather involved
approach, because $\zh_k$ and $\zh_{N_{\dag} + k}$ have
variance $(C_{\zhb})_{kk}$ and $(C_{\zhb})_{N_{\dag} + k, N_{\dag} +
  k}$, respectively, and covariance $(C_{\zhb})_{k, N_{\dag} +
  k}$. Therefore, once $\zh_k$ and $\zh_{N_{\dag} + k}$ are normalized to unit variance,
each $\ph_k$ is given by the sum of two correlated $\chi_1^2$ random
quantities. Although available in analytical form \citep{sim06}, the resulting PDF is rather complex
and hence difficult to handle \citep[for an alternative approach, see][]{ree07}.
Moreover, there is the additional problem that
$L_{\ph_k}$ changes with $k$. A simpler alternative is the use of two
uncorrelated and unit-variance random quantities, $\uph_k$ and
$\uph_{N_{\dag} + k}$, obtained through the transformation
\begin{equation} \label{eq:norm}
\uphb = \Sigmab^{-1/2}_\star \Ub_\star \zhb_\star.
\end{equation}
Here, $\uphb^T = [\uph_k, \uph_{N_{\dag} + k}]$, $\zhb_\star^T =
[\zh_k, \zh_{N_{\dag} + k}]$, $\Sigmab_{\star}^{-1/2}$ is a diagonal matrix
whose entries are given by the reciprocal of the square root of
the non-zero eigenvalues of the covariance matrix
\begin{equation} \label{eq:step3}
\Cb_\star = \left(
\begin{array}{cc}
(C_{\zhb})_{kk} & (C_{\zhb})_{k, N_{\dag} + k} \\
(C_{\zhb})_{k, N_{\dag} + k}  & (C_{\zhb})_{N_{\dag} + k, N_{\dag} + k}
\end{array}
\right),
\end{equation}
zero otherwise,
and $\Ub_\star$ is an orthogonal matrix that contains the
corresponding eigenvectors\footnote{The diagonal elements $\lambda_1$
  and $\lambda_2$ of $\Sigma_*$ can be trivially computed through the
  solution of the quadratic equation $\lambda^2 - {\rm tr}[\Cb_\star]
  + {\rm det}[\Cb_\star] = 0$, with ${\rm tr[.]}$ and ${\rm det}[.]$
  denoting the {\it trace} and {\it determinant} operators. The arrays
  $\ub_1$ and $\ub_2$, which constitute the columns of $\Ub_*$, can be
  obtained by solving the equations $(\Cb_\star - \lambda_l \Ib) \ub_l
  = \zerob$, $l=1,2$.}. Indeed, if the periodogram is defined as
\begin{equation} \label{eq:step1}
\ph_k = \uph_k^2 + \uph^2_{N_{\dag} + k}, \qquad k=0,1, \ldots, N_{\dag} -1,
\end{equation}
then each $\ph_k$ is given by the sum of two independent,
unit-variance, Gaussian random quantities. As a consequence, the
corresponding PDF is, independently of $k$, a $\chi^2_2$ whose {\it
  cumulative distribution function} (CDF) is the exponential
function. This permits determining the statistical significance of $\ph_k$
for a {\it specified} frequency $k$. Things become more
complex if $N_f$ frequencies are
inspected when looking for a peak. Indeed, also after the operation~(\ref{eq:norm}), it
happens that ${\rm E}[\ph_k \ph_l] \neq 0$ for $k \neq l$, i.e., the
frequencies of the periodogram remain mutually correlated. This is an
unavoidable problem. Because of it, $N_f$ does not correspond to the
number of {\it independent frequencies}, so the {\it level of
  false alarm}~(\ref{eq:false}) cannot be computed. However,
since $L_{\ph_k}$ is the same for all the frequencies, an
upper limit can be fixed for $L_{{\rm Fa}}$ by setting $N_f = \lceil
M/2 \rceil$. The periodogram obtained by means of Eq.~(\ref{eq:step1})
corresponds to the original {\it Lomb-Scargle} periodogram.

Since the transformation~(\ref{eq:transf}) does not depend on the
characteristics of the signal sampling, the strategy of following in the
case that $\xb$ is the realization of (not necessarily stationary)
colored noise is simply the one in
Sec.~\ref{sec:formalization} i.e.\ transformation of $\xb$ to an
array $\yb$ with uncorrelated entries. After that, the {\it
  Lomb-Scargle} periodogram can be computed. It is worth noticing that
this simple result has been possible thanks to a formulation of the
problem in the time domain and the use of the matrix notation. The
same results could have been obtained by following the popular
approach of working in the harmonic domain but at the price of a much
more difficult derivation.

\section{Two examples} \label{sec:irregular}

To illustrate the usefulness and the simplicity of the proposed
formalism in handling different situations from the classical ones, we
show two examples in this section.

The first consists of a periodogram of a mean-subtracted time
series. The evaluation of the reliability of a peak in the periodogram
of a signal $\xb$ requires that (under the {\it null} hypothesis $\xb = \nb$) $\nb$ be the realization of a
zero-mean noise process. In most experimental situations, this
condition is not fulfilled and one works with a centered
(i.e.\ mean-subtracted) version $\chib$ of $\xb$. This, however,
introduces some (often neglected) problems. The case where $\xb$ is
the realization of a discrete white noise process with variance
$\sigma^2_{\xb}$ has been considered several times in the literature.
An example is the paper by \citet{zec09} where a rather elaborate
solution is presented. With the approach proposed here, a simpler
solution can be obtained if one takes into consideration that the
subtraction of the mean from $\xb$ forces a spurious correlation among
the entries of $\chib$ in such a way that the covariance matrix $\Cb_{\chib}=E[ \chib \chib^T ]$ is given by
\begin{equation} \label{eq:chib}
\Cb_{\chib} = \sigma^2_{\xb} \left( \Ib - \frac{\large{\oneb}}{M} \right),
\end{equation}
where $M$ is number of entries of $\xb$ and $\large{\oneb}$ an $M \times M$ matrix with every entry equal to unity.
Since this matrix is singular, it cannot be diagonalized and therefore $\chib$ cannot be whitened.
In any case, if in Eq.~(\ref{eq:Cei}) matrix $C_{\xb}$ is substituted for $\Cb_{\chib}$ and one sets
\begin{equation}
\zhb = \left(
\begin{array}{c}
\chihbu_{\Rmatc} \\
\chihbu_{\Imatc}
\end{array}
\right),
\end{equation}
then it is a trivial matter to decorrelate
$\zh_k$ and $\zh_{N_{\dag} + k}$ by means of
Eqs.~(\ref{eq:norm})-(\ref{eq:step3}) and to compute the
periodogram through Eq.~(\ref{eq:step1}). This result can be easily
extend to the case where, because of measurement errors, each entry of
$\xb$ has its own variance $\sigma^2_{x_j}$ and a weighted mean is
subtracted from the data sequence i.e.\ $\chi_j = x_j - \sum _l
\eta_j x_l / \sum_l \eta_l$, with $\eta_l = 1/
\sigma^2_{x_l}$. Indeed, it is sufficient to substitute $\Cb_{\chib}$
as given by Eq.~(\ref{eq:chib}) with
\begin{equation}
\Cb_{\chib} = {\rm diag}[\sigmab^2] - \frac{\large{\oneb}}{\sum_l \eta_l},
\end{equation}
where $\sigmab^2=[\sigma^2_{x_0}, \sigma^2_{x_1}, \ldots, \sigma^2_{x_{N-1}}]^T$.
The rest of the procedure remains unmodified.

The second example consists of zero-mean colored noise. The
improvement in the quality of the results obtainable with the approach
presented in the previous section is visible in
Fig.~\ref{fig:color}. The top left panel shows a discrete signal $x_j
= 0.5 \sin{(2 \pi f j)} + n_j$, $f=0.127$, simulated on a regular grid
of $120$ time instants but with missing data in the ranges $[31~70]$
and $[76~115]$. Here, $\nb$ is the realization of a discrete,
zero-mean, colored noise process whose autocovariance function is
given in the top right panel. From the bottom left panel, it is
evident that {\it Lomb-Scargle} periodogram of the original sequence $\xb$
provides rather ambiguous results concerning the presence of a
sinusoidal component. On the other hand, such component is well visible
in the bottom right panel that shows the {\it Lomb-Scargle} periodogram of the sequence
$\yb = \Cb_{\nb}^{-1/2} \xb$.

\section{Periodogram and least-squares fit of sinusoids} \label{sec:fit}

The formalism proposed here is also useful in the context of more
theoretical questions (but with important practical implications).
For example, a point often overlooked in the astronomical literature is
the relationship between the periodogram and the least-squares fit of
sine functions. Often these two methods are believed to be
equivalent. Actually, this is true only when the sampling is regular
and the frequencies of the sinusoids are given by the {\it Fourier}
ones. Indeed, if $t_j = \tilde{t}_j$ and $f_k = k / N$, $k=0, 1,
\ldots, N-1$, then Eq.~(\ref{eq:model}) can be written in the
form
\begin{equation} \label{eq:ls}
\xb -  \Fmatfb^T \ab = \nb,
\end{equation}
with
\begin{equation}
\Fmatfb = \frac{2}{\sqrt{N}} \left(
\begin{array}{c}
\Fmatcb_{\Rmatc} \\
\Fmatcb_{\Imatc}
\end{array}
\right),
\end{equation}
and $\ab = [a_0, a_1, \ldots, a_{N-1}, b_0, b_1, \ldots, b_{N-1}]^T$. The least-squares solution $\bar{\ab}$ of system~(\ref{eq:ls}) is given by
\begin{equation} \label{eq:lsfit}
\bar{\ab} = (\Fmatfb \Fmatfb^T)^+ \Fmatfb \xb,
\end{equation}
where ``${}^+$'' denotes {\it Moore-Penrose pseudo-inverse} \citep{bjo96}. In the case of even sampling,
i.e.\ when $\Fmatcb_{\Rmatc} = \Fb_{\Rmatc}$ and $\Fmatcb_{\Imatc} = \Fb_{\Imatc}$, it happens that
\begin{equation}
\bar{\ab} = \Fmatfb \xb.
\end{equation}
In other words, coefficients $\{ a_k \}$ and $\{ b_k \}$, as given by
the least-squares approach, can be obtained through the DFT of $\xb$,
because, as shown by means of
Eqs.~(\ref{eq:orth}), $(\Fmatfb \Fmatfb^T)^+ \Fmatfb = \Fmatfb$. In
the case of uneven sampling, this identity is not fulfilled. Any kind
of periodogram computed through Eq.~(\ref{eq:irr4}) and the
least-squares fit of sine functions has to be expected to give
different results. Moreover, as only under the
two above-mentioned conditions do the sine functions constitute an
orthonormal basis, the least-squares fit of a single sine function per
time does not in general provide the same result as the simultaneous
fit of all the sinusoids as in Eq.~(\ref{eq:lsfit})
\citep[e.g. see][page 450]{ham73}. In particular, if an
unevenly-sampled signal is given by the contribution of two or more
sinusoids, the one-at-a-time fit of a single sine function provides
biased results. This also holds for the {\it Lomb-Scargle}
periodogram, which is equivalent to the least-squares fit of a
single sinusoid with a specified frequency, with the constraint that
the corresponding coefficients ``$a$'' and ``$b$'' are uncorrelated
\citep{sca82, zec09}.

\section{Discussion} \label{sec:discussion}

As demonstrated in Sec.~\ref{sec:generalization}, when noise has
arbitrary statistical characteristics, the computation of the
periodogram of an unevenly-sampled signal requires two steps:
\begin{itemize}
\item Whitening and standardization of the noise component (in this way a signal $\yb$ is obtained);
\item Computation of the {\it Lomb-Scargle} periodogram of $\yb$.
\end{itemize}
The first step, unavoidable even in the case of regular sampling, is a
computationally expensive operation. Therefore, for time series
containing more than a few thousand data points, dedicated
algorithms exploiting the specific structure of $\Cb_{\nb}$
(e.g.\ often this matrix is of banded type) have to be developed for
implementing Eq.~(\ref{eq:whitey}). However, this problem is
beyond the aim of the present paper. The second step is much less
time consuming. Indeed, in the case of time series containing some
thousands of points and when the periodogram has to be computed on a
similar number of frequencies, the direct implementation of
Eqs.~(\ref{eq:irr4})-(\ref{eq:step1}) results in a few seconds of
computation time only. In other words, in many practical situations,
no dedicated algorithm is really necessary. However, fast algorithms have been proposed
for very long time series \citep{pre92}.

The last issue that has to be considered is to which extent the use of
the {\it Lomb-Scargle} periodogram is really advantageous. Indeed, the
action of the algorithms dealing with uneven sampling is essentially
directed, for each frequency $k$, to force $\ph_k$ to be the sum of
two independent Gaussian random quantities. However, although not
clearly emphasized, it has already been pointed out
that this operation is not critical \citep[e.g. see][]{sca82}. It can
be expected that a periodogram computed simply through
\begin{equation} \label{eq:pow}
\ph_k = 2 | \yhu_k |^2,
\end{equation}
with $\yhbu = \Fmatcbu \yb$, is often very close to the one given by
Eq.~(\ref{eq:step1}). A rigorous demonstration of this fact is
difficult because of its strict dependence on the specific
sampling. However, with the help of some numerical experiments and of
the formalism introduced here, some insights are possible. In
particular, we consider the covariance matrix $\Cb_{\zhb}$ of a
white noise signal when sampled on different uneven time grids.

\subsection{Numerical simulations}

In our simulations, we take the realization of a zero-mean,
unit-variance, Gaussian white-noise process $\nb$ sampled on $M_s=120$ time
regularly-spaced instants, but with $80$ missing data (i.e.\ $M=40$). The available
signal $\xb$ can be written in the form
\begin{align}
\xb &= \Wb \nb, \label{eq:window1}\\
\Wb & = {\rm diag}[\wb], \label{eq:window2}
\end{align}
where $\Wb = {\rm diag}[\wb]$ and $\wb$ is an array whose entries are equal to one in
correspondence to a value of $\nb$ that is available and zero
otherwise. In this case, Eq.~(\ref{eq:DFT}) can be written as
\begin{align}
\xhb & = \Fb \Wb \nb \label{eq:irr1} \\
& = \Fmatcb \nb,
\end{align}
with $\Fmatcb = \Fb \Wb$. Three different cases have been considered
where the missing data have time indices in the ranges a) $[31~110]$,
b) $[31~70]$ and $[76~115]$, c) $[6~25]$, $[36~75]$ and $[96~115]$,
whereas they are randomly distributed in a fourth case . The related
covariance matrices $\Cb_{\zhb}$, computed through Eq.~(\ref{eq:CI1})
with $N=M_s$, are shown in Fig.~\ref{fig:cov}, whereas
Fig.~\ref{fig:covd} displays the corresponding main diagonal of the
blocks $\Fmatcbu_{\Rmatc} \Fmatcbu^T_{\Rmatc}$ and $\Fmatcbu_{\Rmatc}
\Fmatcbu^T_{\Imatc}$. Especially from Fig.~\ref{fig:covd} it is
evident that, for an arbitrary frequency $k$, the covariance between
$\Rmatc[\xhu_k]$ and $\Imatc[\xhu_k]$ is quite close to zero. This
means that each entry of $\phb$, as given by Eq.~(\ref{eq:pow}), can
be assumed to be distributed according to a $\chi^2_2$.  From
Fig.~\ref{fig:cov} it is also evident that, in the case of gaps
present in the sampling pattern, there are nearby frequencies $k$ and
$l$ for which not only $\Rmatc[\xhu_k]$ and $\Rmatc[\xhu_l]$ are
mutually correlated, but also $\Rmatc[\xhu_k]$ with $\Imatc[\xhu_l]$ and
$\Rmatc[\xhu_l]$ with $\Imatc[\xhu_k]$. These correlations,
especially those between the real and the imaginary components,
disappear in the case of random sampling.

\subsection{Interpretation of the results of the simulations}

To understand these results, it is necessary to take into
account that Eq.~(\ref{eq:irr1}) can be rewritten in the form
\begin{align}
\xhb & = \Fb \Wb \Fb^* \Fb \nb, \\
& = \Whb \nhb.
\end{align}
As $\Wb$ is a (singular) diagonal matrix, $\Whb$ is a (singular)
circulant matrix. This implies that $\xhb$ is given by the {\it
  circular convolution} of $\nhb$ (the DFT of the original signal,
inclusive of the missing data) with the {\it spectral window} $\whb$
(the DFT of the sampling pattern). Since for two arbitrary
frequencies, say $k$ and $l$, $\Rmatc[\nhu_k]$, $\Imatc[\nhu_k]$,
$\Rmatc[\nhu_l]$, and $\Imatc[\nhu_l]$ are mutually independent, any
correlation existing between $\Rmatc[\xhu_k]$, $\Imatc[\xhu_k]$,
$\Rmatc[\xhu_l]$, and $\Imatc[\xhu_l]$ is induced by the
correlation between the entries of $\Rmatc[\whb]$ and
$\Imatc[\whb]$. Now, as shown in Fig.~\ref{fig:windows}, in the
presence of gaps the real and the imaginary parts of $\whb$ are both
only significantly different from zero in a narrow interval of
frequencies surrounding the origin, i.e., corresponding to the lowest
frequencies ($\wb$ can be interpreted as low-pass filter). This
produces the correlations observed among $\Rmatc[\xhu_k]$,
$\Rmatc[\xhu_l]$, $\Rmatc[\xhu_l]$ and $\Imatc[\xhu_k]$ for nearby
frequencies $k$ and $l$. On the other hand, in the case of random
sampling $\whb$ mimics the behaviour of white noise, making the
correlations less important. Moreover, Fig.~\ref{fig:cross}
shows that, in the case of sampling with gaps, the cross-correlation
between $\Rmatc[\whb]$ with $\Imatc[\whb]$ is significant but not
centered at zero lag. This explains why the quantities
$\Rmatc[\xhu_k]$ and $\Imatc[\xhu_l]$, $k \neq l$ can be strongly
correlated in spite of the small correlation between
$\Rmatc[\xhu_k]$ and $\Imatc[\xhu_k]$.
Similar arguments also hold for the case of
sampling with periodic gaps (rather common in astronomical experiments).
Indeed, in practical applications the period of these gaps is somewhat  smaller than the mean sampling time step
of an uninterrupted sequence of data. As a consequence, both
$\Rmatc[\whb]$ and $\Imatc[\whb]$ present a sharp and narrow peak in correspondence to
a frequency close to the origin. Actually, the {\it spectral window} of a sampling with periodic
gaps is also characterized by the presence of {\it aliases}. However, these aliases too are
sharp and narrow, and their importance decreases for increasing frequencies.
The combination of these facts leads again to
the quantities $\Rmatc[\xhu_k]$, $\Rmatc[\xhu_l]$, $\Rmatc[\xhu_l]$, and $\Imatc[\xhu_k]$
almost being uncorrelated if
the frequencies $k$ and $l$ are not sufficiently close enough.
Moreover, since with periodic gaps the cross-correlation
between $\Rmatc[\whb]$ with $\Imatc[\whb]$ can also be significant, but not
centered on zero lag, then for each frequency $k$
the correlation between $\Rmatc[\xhu_k]$ and $\Imatc[\xhu_k]$ is negligible.
These considerations are confirmed by Figs.~\ref{fig:covpr} and \ref{fig:covp}.
Figure~\ref{fig:covpr} displays the covariance matrix $\Cb_{\zhb}$, computed through Eq.~(\ref{eq:CI1}),
of a discrete zero-mean, unit-variance, and white-noise process sampled on $600$
time instants randomly distributed (i.e.\ not rebinned) along six
cycles (i.e.\ $100$ points per cycle) of the sampling pattern shown in
the bottom panel of Fig.~\ref{fig:covp}; $1200$ frequencies have been
considered. The top panel of Fig.~\ref{fig:covp} displays the
main diagonal of the blocks $\Fmatcbu_{\Rmatc} \Fmatcbu^T_{\Rmatc}$
and $\Fmatcbu_{\Rmatc} \Fmatcbu^T_{\Imatc}$.

The numerical simulations indicate that the consequences of
unevenly-sampled data seem to concern the number of independent
frequencies in $\phb$ rather than the correlation between $\Rmatc[\xhu_k]$
and its imaginary counterpart $\Imatc[\xhu_k]$. At present, no
general method has been developed to deal with this
problem. However, it has been pointed out in the literature that the
number of independent frequencies is not a critical parameter to test
the significance level of a peak in $\phb$.
In particular, empirical arguments indicate that this number can be
safely set to $M/2$ \citep[e.g. see][]{pre92}.  The conclusion is that forcing each entry
of $\phb$ to be the sum of two independent Gaussian quantities only has
minor effects. This is shown by Fig.~\ref{fig:example}
where the top panel shows that the
{\it Lomb-Scargle} periodogram of the time series with periodic gaps
considered above
is quite similar to that provided simply by
$\ph_k = 2 |\xhu_k|^2$ with $\xhbu = \Fmatcbu \xb$. This is
evident in the bottom panel of the same figure where the similarity of
the two periodiograms is demonstrated by their absolute difference.

A final point to underline, which has important practical
implications, is that for long time series the small differences
visible in Fig.~\ref{fig:example} should decrease. Indeed, as seen
above, $\ph_k$ will be significantly correlated with $\ph_l$ only if
the two frequencies $k$ and $l$ are close enough. The only exception
is represented by the frequencies at the extremes of the frequency
domain where the assumption of periodic signal intrinsic to DFT forces
a spurious correlation. For longer and longer time series, this
spurious correlation will affect a smaller and smaller fraction of
frequencies and, as consequence, a larger and larger fraction of them
will be mutually independent. This is shown in Fig.~\ref{fig:long}
where, in the context of the previous experiment, the mean absolute
difference between the two periodograms is plotted as a function of $N_s$,
the number of cyclic sampling
patterns ($N_s = 6$ in Fig.~\ref{fig:example}). This argument also
explains why in many practical situations the number of independent frequencies
can be safely fixed to $M/2$. In conclusion, only in
the case of signals that contain a small number of data, the {\it Lomb-Scargle}
periodogram can be expected to exhibit noticeable differences from the
periodogram given by
Eq.~(\ref{eq:pow}). Often, using it does not change anything. Comparable
results can be expected with less sophisticated approaches.

\subsection{Application to an astronomical time series}

As an example of an unsophisticated method able to produce results
similar to those obtainable with the {\it Lomb-Scargle} periodogram,
we consider the rebinning of the original time series on an
arbitrarily dense regular time grid (in this way a signal with a
regular sampling is obtained but some data are missing) followed by
applying any of the {\it fast Fourier transform} (FFT)
algorithms available nowadays. Figure~\ref{fig:data1} shows an
experimental (mean-subtracted) time series versus its rebinned
version. This time series, which is characterized by rather irregular
sampling, was obtained with the VLA array \citep{big01} and consists
of polarisation position angle measurements at an observing frequency
of 15~GHz for one of the images of the double gravitational lens system
B0218+357. The original sequence contains only $M=45$ data and it is
rebinned on a regular grid of $M_r = 92$ time instants. In spite of
this, as visible in Fig.~\ref{fig:data2}, the corresponding
periodograms, computed on $N=M_r$ equispaced frequencies by means of
the {\it Lomb-Scargle} and the FFT approach, are remarkably similar.
Here, the highest frequency approximately corresponds to the Nyquist
frequency that is related to the shortest sampling time step. The
main conclusion of this example is to point out that, although
in the previous section we stated that use of the {\it Lomb-Scargle} periodogram
can be expected to be effective only for time series that contain a small number of data, this is not a sufficient
condition to guarantee that the method is truly useful.

\section{Conclusions}

In this paper we worked out a general formalism, based on the matrix
algebra, that is tailored to analysis of the statistical
properties of the {\it Lomb-Scargle} periodogram independently of the characteristics of
the noise and the sampling. With this formalism it has become possible
to develop a test for the presence of components of interest in a signal in more general
situations than those considered
in the current literature (e.g. when noise is colored and/or
non-stationary). Moreover, we were able to clarify the relationship between the {\it Lomb-Scargle} periodogram
and other techniques (e.g.\ the least-squares fit of sinusoids)
and to fix the conditions under which the use of such method can be expected to be effective.

config: arxiv


TITLE: How many powers of 2 are easy to double?
QUESTION [12 upvotes]: Possible Duplicate:
Is 2048 the highest power of 2 with all even digits (base ten)?

Numbers written in base $10$ are easiest to double when their digits lie in the range $0, \ldots, 4$, so that no carries are performed.  For instance, ...

$1$ is easy to double, yielding $2$;
$2$ is also easy to double, yielding $4$;
$4$ is also also easy to double, yielding $8$;
but $8$ is not easy to double, since $8+8 = 16$ involves a carry.

The only numbers $2^n$ which are easy to double that I'm aware of are the three listed above, along with $2^5 = 32$ and $2^{10} = 1024$.  My suspicion is that $2^{10}$ is the last such power of $2$, and that's my question:

My question is: Is $2^{10} = 1024$ the last example of a power of $2$ which is easy to double?  If it isn't the last example, how many more are there?  Is there a(n infinite) family of examples which is easy to produce?

There's an theorem of Kummer which can be used to solve a "base $p$ analogue" of this question for $p$ a prime:

Theorem (Kummer): In $p$-adic arithmetic, the number of times a carry is performed when adding the numbers $n$ and $m$ (in base $p$) is equal to the number of times $p$ divides the binomial coefficient $\binom{n+m}{m}$.

I haven't found any analogue that holds in base $10$.  An essential obstacle is that $\binom{5}{3} = 10$ shows that $2 + 3$ has carries when written both in base $2$ and base $5$, but not in base $10$.  Because of this sort of phenomenon, I suspect that this isn't the right way to approach my problem, but it helps form a complete question and I thought someone might find it cute.

REPLY [1 votes]: If you do it in binary, all of them are very easy to double: Just append another $0$. For example, $\{1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, \ldots\}$.


TITLE: Iterated Integral with variable substitution
QUESTION [1 upvotes]: I need to calculate the double integral of the function $f(x,y) = (x+y)^9(x-y)^9$:
$\int_0^{1/2} \int_x^{1-x}  (x+y)^9(x-y)^9 dydx$
I have a solution but I definitely arrived at it after a sloppy attempt. I got -0.0025. To begin with, I know you need to separate the variables. I tried integration by parts, but got lost in it. How can I separate the variables to continue the integration?

REPLY [1 votes]: It's convenient to solve this integral with substitution. Since we want to separate the variables in the Integral
\begin{align*}
\int_0^{1/2} \int_x^{1-x}  (x+y)^9(x-y)^9 dy\,dx\tag{1}
\end{align*}
it's reasonable to use  the substitution
\begin{align*}
\left.
\begin{matrix}
u=x+y\\
v=x-y\\
\end{matrix}
\right\}
\quad\Longleftrightarrow\quad
\left\{
\begin{matrix}
x=\frac{1}{2}(u+v)\\
y=\frac{1}{2}(u-v)
\end{matrix}
\right.\tag{2}
\end{align*}

According to the Change of variable theorem we want to calculate
  \begin{align*}
\int_{x_0}^{x_1}\int_{y_0}^{y_{1}}f(x,y)\,dy\,dx
=\int_{u_0}^{u_1}\int_{v_0}^{v_{1}}f(x(u,v),y(u,v))
\left|\operatorname{det}
\begin{pmatrix}
x_u&x_v\\
y_u&y_v
\end{pmatrix}
\right|
\,dv\,du\tag{3}
\end{align*}
Jacobian
At first we calculate the absolute value of the determinant of the Jacobian matrix
  \begin{align*}
\left|\operatorname{det}
\begin{pmatrix}
x_u&x_v\\
y_u&y_v
\end{pmatrix}
\right|=
\left|\operatorname{det}
\begin{pmatrix}
\frac{1}{2}&\frac{1}{2}\\
\frac{1}{2}&-\frac{1}{2}
\end{pmatrix}
\right|
=\frac{1}{2}
\end{align*}
Area transformation
We observe from (1) the region of integration is
  \begin{align*}
&0\leq x\leq \frac{1}{2}\\
&x\leq y\leq 1-x
\end{align*}
This is the area of a triangle with three lines as boundary lines,  the $y$-axis  $x=0$, the  major diagonal $x=y$ and a parallel to the minor diagonal through $(1,0)$, $y=1-x$.
Since the transformation (2) is linear, lines are transformed to lines. So, the transformed $(u,v)$-area is again enclosed by three lines.
\begin{array}{lclcl}
x=0\qquad&\rightarrow&\qquad \frac{1}{2}(u+v)=0
&\qquad\rightarrow&u=-v\\
x=y\qquad&\rightarrow&\qquad \frac{1}{2}(u+v)=\frac{1}{2}(u-v)
&\qquad\rightarrow&v=0\\
y=1-x\qquad&\rightarrow&\qquad \frac{1}{2}(u-v)=1-\frac{1}{2}(u+v)
&\qquad\rightarrow&u=1\\
\end{array}
We see the $(u,v)$-triangle area is given by the three lines $u=-v, v=0$ and $u=1$ which can be written as
  \begin{align*}
0\leq u \leq 1\\
-u\leq v\leq 0
\end{align*}

$$ $$

Integration
Now we have all ingredients to perform the integration according to (3). We obtain
\begin{align*}
\int_0^\frac{1}{2}\int_x^{1-x}  (x+y)^9(x-y)^9 dy\,dx
&=\int_0^1\int_{-u}^0(uv)^9\cdot\frac{1}{2}\,dv\,du\\
&=\frac{1}{2}\int_0^1u^9\left(\int_{-u}^0v^9\,dv\right)\,du\\
&=\frac{1}{2}\int_0^1u^9\left(\left.\frac{1}{10}v^{10}\right|_{-u}^0\right)\,du\\
&=-\frac{1}{20}\int_0^1u^{19}\,du\\
&=-\frac{1}{20}\left.\left(\frac{1}{20}u^{20}\right)\right|_{0}^1\\
&=-\frac{1}{400}
\end{align*}