TITLE: Problems regarding certain approaches to some certain integrands
QUESTION [1 upvotes]: While solving some problems of integration(indefinite) I came across some integrands which becomes a lot easier to integrate  if numerator and denominator is multiplied by certain expressions. But what are the consequences to it? 3 examples(almost the same issue) are shown below:
Example-1: $$\int{\frac1 {1+\sin x} dx}$$
In this function if we multiply both numerator and denominator with $(1-\sin x)$, then it becomes lot easier to determine the integral. Here the initial function was defined at $x=\frac\pi 2$ but when we are transforming the function to $\frac{1-\sin x} {1-\sin^2 x}$ we are basically multiplying the numerator and denominator by $\frac0 0$ at $x=(4n+1)\frac\pi 2$. Moreover the new function is itself undefined at  $x=(2n+1)\frac\pi 2$. So we have changed the domain of the function by transforming it. What are the consequences to such integrands? Considering this as an indefinite integration, isn't it completely wrong to solve this integral since we dont know the limits even though they represent the same graph but undefined at some more points than it used to?
Again if we transform the function from the expression of $\sin x$ to an expression of $\tan\frac x 2$ isn't it still wrong since at $x=\pi$ the expression becomes undefined?
Example-2:$$\int{\frac{(x^2+1)} {(x^4+1)} dx}$$
I was taught to solve this type of integrals by taking  $x^2$ outside the parentheses as common. But isn't it wrong since we are just dividing the expression inside the parentheses by $x^2$ which could also equal $0$? The new function(transformed) is undefined at $x=0$ whereas for any real value of $x$ the initial function is defined.
Another question: is $f(x)=x^2-1=x(x-1)$ for all real values of $x$? I am asking this question because when we do take any term common or take it outside the parentheses, we usually divide the expression inside the parentheses with that term as I have mentioned above.
Example-3:$$\int{\frac1 {1+3\cos^2 x} dx}$$
Here if we replace with $\cos x$ with $\frac1 {\sec x}$ , will it be a right thing to do since we are again restricting the domain to a lesser number of possible values as the previous 2?
I am facing almost the same problem regarding all the 3 examples shown above. I know that all the transformed functions are graphically the same as their initial counterparts and if these were definite integrals having limits which don't include those points at which the transformed ones are undefined, then they would  yield the same integral. But what if the limits include those points?(Again I am asking this question because I am considering that since the initial functions are defined at those points then there might be a function which has the same slope coinciding with the initial function at those points) Moreover these are indefinite integrals. It is to be noted that I were taught to solve integral in such ways at my school....there might be many other ways to do so but till now I am unaware of those. I just want to know whether this processes are legitimate or not and would yield the same output or no and what are their consequences.

REPLY [2 votes]: What are the consequences to such integrands? Considering this as an indefinite integration, isn't it completely wrong to solve this integral since we dont know the limits even though they represent the same graph but undefined at some more points than it used to?

If two integrands disagree only at finitely many points then the integral is the same. Refer, 2, 3 also a function with finitely many discontinuities in an interval is also integrable refer
P.S: All the trig integral can be solved easily using exponential substitution and the rational function by considering partial fractions on the denominator $x^4 + 1 = (x^2 + i) (x^2 -i)$ .


TITLE: Prove that $\Vert f(x) -f(y)\Vert\geq (1-k)\Vert x-y\Vert,\;\text{and}\;\Vert f'(x)h\Vert\geq (1-k)\Vert h\Vert,\;\forall\,x,y,h\in\Bbb{R^n}$
QUESTION [1 upvotes]: Good day all! I'm preparing for a Graduate exam, so I need to solve this problem.
Let $f:\Bbb{R}^n\to\Bbb{R}^n$ be a function of class $C^{1}$. We suppose that there exists $k\in ]0,1[$ such that $$\Vert g'(x)\Vert\leq k,\forall\;x\in\Bbb{R}^n$$
where $f(x)=g(x)+x,\forall\;x\in\Bbb{R}^n$

Prove that $\Vert f(x) -f(y)\Vert\geq (1-k)\Vert x-y\Vert,\;\forall\,x,y\in\Bbb{R^n}$
Prove that $\Vert f'(x)h\Vert\geq (1-k)\Vert h\Vert,\;\forall\,x,h\in\Bbb{R^n}$
Prove that $f$ is $C^1-$diffeomorphism from $\Bbb{R}^n\to\Bbb{R}^n$.

I started this:
$$\Vert g'(x)\Vert\leq k,\forall\;x\in\Bbb{R}^n,\;\text{for some}\;k$$
$$\Vert f'(x)-1\Vert\leq k,\forall\;x\in\Bbb{R}^n,\;\text{for some}\;k$$
$$\Vert \lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}-1\Vert\leq k,\forall\;x\in\Bbb{R}^n,\;\text{for some}\;k$$
But I couldn't proceed. Please, I need an instant help. Proofs and references are welcome!

REPLY [1 votes]: (Just an hint for 1, to show how to proceed.)
Your assumption on $g$ says that $g$ is $k$-Lipschitz, i.e.
$$
\|g(x) - g(y)\| \leq k \|x-y\|
\qquad\forall x,y\in\mathbb{R}^n.
$$
Hence,
$$
\|f(x) - f(y)\| = \|g(x) + x - g(y) - y\| \geq \|x-y\| - \|g(x)-g(y)\| \geq (1-k)\|x-y\|.
$$


TITLE: write an explicit formula for the sum $\sum_{i=1}^n {(3i+1)}$
QUESTION [3 upvotes]: I've been shown that  :
$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$
Now I need to write an explicit formula for the sum:
$$\sum_{i=1}^n {(3i+1)}$$
I've come up with an answer that is:
$$\sum_{i=1}^n  {(3i+1)} = \frac{9n^2 + 6n + 3}{2}$$
However I am not sure if this is right or not. Thanks in advance for any help.

REPLY [2 votes]: Of course, egreg has outlined the optimal way of obtaining your closed-form expression, but if you are learning about sequences and series, then I imagine the "intended" solution was via the $n$th partial sum for arithmetic sequences, namely $S_n=\frac{n(a+a_n)}{2}$, where $a_n$ is the $n$th term ($a_n=a+(n-1)d$, where $d$ is the common difference) and $a$ the first time. Using the information you have given, it is clear that $a=4$ and $a_n=4+(n-1)3=1+3n$. Hence,
$$
S_n=\frac{n[4+(1+3n)]}{2}=\frac{n(5+3n)}{2}=\frac{3n^2+5n}{2},
$$
confirming the "slicker" answers provided by others already.


text:
\begin{document}

\title{Efficient Soft-Input Soft-Output Detection of Dual-Layer MIMO Systems}
\author{Ahmad Gomaa and Louay M.A. Jalloul,~\IEEEmembership{Senior~Member,~IEEE}
\thanks{The authors are with the Mobile and Wireless Group, Broadcom Corporation, Sunnyvale, CA 94086 U.S.A.}}

\maketitle

\begin{abstract}
A dual-layer multiple-input multiple-output (MIMO) system with multi-level modulation is considered. A computationally efficient soft-input soft-output receiver based on the \textit{exact} max-log maximum a posteriori (max-log-MAP) principle is presented in the context of iterative detection and decoding. We show that the computational complexity of our exact max-log-MAP solution grows linearly with the constellation size and is also less than that of the best known methods of Turbo-LORD that only provide approximate solutions. Using decoder feedback to change the decision thresholds of the constellation symbols, we show that the exhaustive search operation boils down to a simple slicing operation.
\end{abstract}

\setlength{\textfloatsep}{10pt}
\setlength{\floatsep}{8pt}

\section{Introduction}\label{sec_Introduction}
Iterative detection and decoding (IDD) techniques have been widely used \cite{Tuchler02,Haykin02,SIOF2010,Mikami09} to improve the performance of multiple-input multiple-output (MIMO) systems. The detector utilizes the feedback from the decoder to enhance the accuracy of its output statistics. In \cite{Tuchler02,Haykin02}, the detector was designed as a linear minimum mean square error equalizer, accepting soft input from the channel decoder. The soft input was used to cancel the interference from other streams and to adapt the equalization (weight) vector by modifying the variance of the canceled streams. In \cite{SIOF2010}, the detector was designed as a decision feedback equalizer with successive cancellation at the symbol level before passing the log-likelihood ratios (LLRs) of the code bits to the decoder. In \cite{Mikami09}, IDD was used to mitigate the effect of inter-cell interference in orthogonal frequency division multiplexing (OFDM) systems. In \cite{TLORD_TWC,T-LORD,K-best}, a maximum a posteriori (MAP) approximating algorithm was proposed as an improvement over the layered orthogonal lattice detector (LORD) approach \cite{LORD-Alerton,LORD}. In \cite{Junti-IDD}, list detectors were proposed in addition to iterative channel estimation in OFDM systems. Other MAP approximation algorithms were proposed in \cite{MAP-approx-Choi,MAP-approx-Hassibi,Tellambura12,Hochwald03} where modified sphere detection techniques were used.

Dual-layer transmission schemes are widely used in current cellular systems where user equipments cannot easily support more than two antennas. The solution presented in this paper is an \textit{exact} solution of the max-log MAP detector for dual-layer systems and uses fewer metric computations than the approximate solution provided in \cite{TLORD_TWC,T-LORD}. To generate the LLRs for one layer, we use the a-priori LLRs generated by the turbo decoder for the other layer to modify its decision thresholds and then use the slicer as a simple search device.

The rest of the paper is organized as follows. The system model is described in Section \ref{sec_SysModel}, and the exact max-log MAP solution is derived in Section \ref{sec_exact_MaxLogMAP}. In Section \ref{sec_no_decesion_regions}, we prove that the a-priori probabilities can lead to constellation symbols with empty decision regions. In Section \ref{sec_algo}, we provide the complete algorithm and describe it in pseudo code. We analyze the algorithm computational complexity and compare its complexity with other algorithms in Section \ref{sec_complexity}, and the paper is concluded in Section \ref{sec_Conclusion}.

\textit{Notations}: Unless otherwise stated, lower case and upper case bold letters denote vectors and matrices, respectively, and $\bI_m$ denotes the identity matrix of size $m$. Furthermore, $\left|\;\right|$ and $\|\;\|$ denote the absolute value and the $l_2$-norm, respectively, while $(\,)^H$ denotes the complex conjugate transpose operation.

\section{System Model}\label{sec_SysModel}
We consider dual-layer transmission schemes, where two layers (streams) are transmitted over $N_t\geq 2$ antennas using the precoding matrix $\bW$ of size $N_t\times 2$. The receiver detects the transmitted streams using $N_r \geq 2$ receive antennas. The input-output relation is given by
\begin{equation}\label{eqn_I/P_O/P}
\by = \bar{\bH} \bW \bs +\bn \triangleq \bH\bs+\bn= \bh_1 s_1 + \bh_2 s_2 +\bn
\end{equation}
where $\by$, $\bs$, $\bn$ and $\bar{\bH}$ denote the $N_r\times 1$ received signal, $2\times 1$ transmitted symbols, $N_r \times 1$ background noise plus inter-cell interference, and $N_r \times N_t$ channel matrix, respectively. Furthermore, $\bh_i$ is the $i$-th column vector of the equivalent channel matrix $\bH=\bar{\bH}\bW$, and $s_i$ is the $i$-th transmitted symbol chosen from the $M$-QAM constellation $\chi$. The $i\text{-th}$ $M$-QAM symbol, $s_i$, represents $q=\text{log}_2(M)$ code bits $\bc_i=\begin{bmatrix}c_{i1}&c_{i2}&\ldots& c_{iq}\end{bmatrix}$. The above model suits single-carrier systems over flat fading channels and OFDM systems over frequency-selective channels where the relation in \eqref{eqn_I/P_O/P} applies to every subcarrier. In IDD, the detector computes the LLRs of the code bits and passes them to the channel decoder, which computes the extrinsic LLRs and feeds them back to the detector. The detector uses the a priori LLRs computed by the decoder to generate more accurate LLRs for the channel decoder and so forth. Assuming known channel and zero-mean circularly symmetric complex Gaussian noise $\bn$ of covariance matrix $\bC_{\bn\bn}=\bQ^{-1}$, we write the log MAP a posteriori detector LLR of the bit $c_{1k}$ as follows:
\begin{align}
&L(c_{1k})=\text{log}\nonumber\\
&\left(\frac
{\sum\limits_{\bar{s}_1\in \chi_{k,1}} \hspace{-0.1cm}P_{\bar{s}_1}\sum\limits_{\bar{s}_2\in \chi} \text{exp}\left(-\|\by-\bh_1 s_1 -\bh_2 s_2 \|_{\bQ}^2\right)\hspace{-0.05cm}P_{\bar{s}_2}}
{\sum\limits_{\bar{s}_1\in \chi_{k,0}} \hspace{-0.1cm}P_{\bar{s}_1}\sum\limits_{\bar{s}_2\in \chi} \text{exp}\left(-\|\by-\bh_1 s_1 -\bh_2 s_2 \|_{\bQ}^2\right)\hspace{-0.05cm}P_{\bar{s}_2}}\right) \label{eqn_logmap_LLR}
\end{align}
where $\|\bx \|_\bQ^2\equiv\bx^H\bQ\bx$, $\chi_{k,1}$ and $\chi_{k,1}$ denote the constellation sets where the $k$-th bit are '1' and '0', respectively, and $P_{\bar{s}_1}$ and $P_{\bar{s}_2}$ denote the a priori probabilities that $s_1=\bar{s}_1$ and $s_2=\bar{s}_2$, respectively. The max-log MAP approximation of the LLR of $c_{1k}$ is given by:
\begin{align}
&L(c_{1k})\nonumber\\
&\hspace{-0.1cm}=\hspace{-0.1cm}\max\limits_{\bar{s}_1\in \chi_{k,1}} \hspace{-0.15cm}\left(\hspace{-0.03cm}\text{log} P_{\bar{s}_1}\hspace{-0.1cm}+\max\limits_{\bar{s}_2\in \chi} \left(\text{log} P_{\bar{s}_2}\hspace{-0.1cm}-\hspace{-0.1cm}\|\by\hspace{-0.05cm}-\hspace{-0.05cm}\bh_1 \bar{s}_1 \hspace{-0.05cm}-\hspace{-0.05cm}\bh_2 \bar{s}_2 \|_{\bQ}^2\right)\hspace{-0.1cm}\right)\nonumber\\
&\hspace{-0.1cm}-\hspace{-0.1cm}\max\limits_{\bar{s}_1\in \chi_{k,0}} \hspace{-0.15cm}\left(\hspace{-0.03cm}\text{log} P_{\bar{s}_1}\hspace{-0.1cm}+\max\limits_{\bar{s}_2\in \chi} \left(\text{log} P_{\bar{s}_2}\hspace{-0.1cm}-\hspace{-0.1cm}\|\by\hspace{-0.05cm}-\hspace{-0.05cm}\bh_1 \bar{s}_1 \hspace{-0.05cm}-\hspace{-0.05cm}\bh_2 \bar{s}_2 \|_{\bQ}^2\right)\hspace{-0.1cm}\right)\label{eqn_LLR_s1_MaxLog}
\end{align}
Similarly, the max-log MAP LLR of $c_{2k}$ is:
\begin{align}
&L(c_{2k})\nonumber\\
&=\hspace{-0.1cm}\max\limits_{\bar{s}_2\in \chi_{k,1}} \hspace{-0.1cm}\left(\hspace{-0.03cm}\text{log} P_{\bar{s}_2}\hspace{-0.1cm}+\max\limits_{\bar{s}_1\in \chi} \left(\text{log} P_{\bar{s}_1}\hspace{-0.1cm}-\hspace{-0.1cm}\|\by\hspace{-0.05cm}-\hspace{-0.05cm}\bh_1 \bar{s}_1 \hspace{-0.05cm}-\hspace{-0.05cm}\bh_2 \bar{s}_2 \|_{\bQ}^2\right)\hspace{-0.1cm}\right)\nonumber\\
&-\hspace{-0.1cm}\max\limits_{\bar{s}_2\in \chi_{k,0}} \hspace{-0.1cm}\left(\hspace{-0.03cm}\text{log} P_{\bar{s}_2}\hspace{-0.1cm}+\max\limits_{\bar{s}_1\in \chi} \left(\text{log} P_{\bar{s}_1}\hspace{-0.1cm}-\hspace{-0.1cm}\|\by\hspace{-0.05cm}-\hspace{-0.05cm}\bh_1 \bar{s}_1 \hspace{-0.05cm}-\hspace{-0.05cm}\bh_2 \bar{s}_2 \|_{\bQ}^2\right)\hspace{-0.1cm}\right)\label{eqn_LLR_s2_MaxLog}
\end{align}
The brute force solution of \eqref{eqn_LLR_s1_MaxLog} (and similarly \eqref{eqn_LLR_s2_MaxLog}) requires the computation of $M^2$ metrics where, for each instance of $\bar{s}_1$, the metric $\left(\text{log} P_{\bar{s}_2}\hspace{-0.1cm}-\hspace{-0.1cm}\|\by-\bh_1 \bar{s}_1 -\bh_2 \bar{s}_2 \|_{\bQ}^2\right)$ is computed for all instances of $\bar{s}_2$. However, we show in Section \ref{sec_exact_MaxLogMAP} how we obtain the \textit{exact} max-log MAP solution for the LLRs using fewer than $4M$ (rather than $M^2$) metrics computations.

\section{Exact Max-Log MAP solution}\label{sec_exact_MaxLogMAP}
The main strategy of our solution is to convert the $l_2$-norms in \eqref{eqn_LLR_s1_MaxLog} and \eqref{eqn_LLR_s2_MaxLog} into simple absolute values fitted for the slicing operations. Then, we exploit the a-priori LLRs to control the the thresholds of the slicers. We begin by whitening the noise to get $\tilde{\by}$=$\sqrt{\bQ}\by$ and $\tilde{\bh}_i$=$\sqrt{\bQ}\bh_i$. We then rewrite the bottleneck maximization problem $\max\limits_{\bar{s}_2\in \chi}\left(\text{log} P_{\bar{s}_2}\hspace{-0.1cm}-\hspace{-0.1cm}\|\by-\bh_1 \bar{s}_1 -\bh_2 \bar{s}_2 \|_{\bQ}^2\right)$ as follows:
\begin{align}
&\max\limits_{\bar{s}_2\in \chi}\left(\text{log} P_{\bar{s}_2}-\|\tilde{\by}-\tilde{\bh}_1 \bar{s}_1 -\tilde{\bh}_2 \bar{s}_2 \|^2\right)\nonumber\\
=&\max\limits_{\bar{s}_2\in \chi}\left(\text{log} P_{\bar{s}_2}-\|\tilde{\bh}_2\|^2\left\|\frac{\left(\tilde{\by}-\tilde{\bh}_1 \bar{s}_1\right)}{\|\tilde{\bh}_2\|}
 -\bB \begin{bmatrix}  \bar{s}_2\\\bO \end{bmatrix} \right\|^2\right)
\end{align}
where $\bO$ is all-zero vector of length $N_r-1$, $\bB=\begin{bmatrix} \frac{\tilde{\bh}_2}{\|\tilde{\bh}_2\|} & \bE \end{bmatrix}$ is an $N_r\times N_r$ unitary matrix as $\bE$ is an $N_r \times (N_r-1)$ matrix chosen such that $\bE^H\tilde{\bh}_2=0$ and $\bE^H\bE=\bI$. The reason we write the matrix $\bB$ in this form is to exploit its unitary structure and take it as a common factor out of the norm without affecting its value. This will lead to converting the $l_2$-norm into a single absolute value as follows. Since $\bB^H\bB=\bI_2$, we rewrite the maximization as follows:
\begin{align}
&\max\limits_{\bar{s}_2\in \chi}\left(\text{log} P_{\bar{s}_2}-\|\tilde{\bh}_2\|^2\left\|\frac{\bB^H\left(\tilde{\by}-\tilde{\bh}_1 \bar{s}_1\right)}{\|\tilde{\bh}_2\|}
 - \begin{bmatrix}  \bar{s}_2\\\bO \end{bmatrix} \right\|^2\right) \nonumber\\
 =&\max\limits_{\bar{s}_2\in \chi}\left(\frac{\text{log} P_{\bar{s}_2}}{\|\tilde{\bh}_2\|^2}-\left| Z_{(\bar{s}_1)}
 - \bar{s}_2 \right|^2\right)\label{eqn_max_over_s2}
\end{align}
where $Z_{(\bar{s}_1)}=\tilde{\bh}_2^H\left(\tilde{\by}-\tilde{\bh}_1 \bar{s}_1\right)/\|\tilde{\bh}_2\|^2$. If the a-priori probability term ($\text{log} P_{\bar{s}_2}/\|\tilde{\bh}_2\|^2$) were not there \cite{French_paper_ML} (i.e., ML instead of MAP), then the solution of the maximization in \eqref{eqn_max_over_s2} would be a simple slicer, and only $2M$ metrics (enumeration over $\bar{s}_1$ and $\bar{s}_2$ in \eqref{eqn_LLR_s1_MaxLog} and \eqref{eqn_LLR_s2_MaxLog}) were to be computed to obtain the LLRs of the code bits corresponding to $s_1$ and $s_2$. With the a-priori probability term, we obtain the \textit{exact} solution of \eqref{eqn_max_over_s2} with a reasonable increase in the number of metrics computations which is, interestingly, less than that of the approximate solution in \cite{TLORD_TWC,T-LORD}. In modern communications standards \cite{LTE-211}, the real and imaginary parts of $\bar{s}_2$ correspond to two orthogonal $L$-PAM constellations, $\psi$, where $L=\sqrt{M}$\footnote{Assuming square constellation, without loss of generality.}. In Fig. \ref{fig_constellation}, we show the $L$-PAM one-dimensional constellation corresponding to the real or imaginary of any complex QAM constellation. Hence, we rewrite \eqref{eqn_max_over_s2} as follows:
\begin{figure}
\centering
\epsfig{file=One_Dim_Constellation.eps,height=0.5in,width=3.45in}
\caption{$L$-PAM constellation for the real (or imaginary) part of the $M$-QAM constellation.} \label{fig_constellation}
\end{figure}
\begin{align}
&\max\limits_{\bar{s}_2\in \chi}\left(\frac{\text{log} P_{\bar{s}_2}}{\|\tilde{\bh}_2\|^2}-\left| Z_{(\bar{s}_1)}
 - \bar{s}_2 \right|^2\right)\nonumber\\
 &=\max\limits_{\bar{s}_{2,r}\in \psi}\left(\frac{\text{log} P_{\bar{s}_{2},r}}{\|\tilde{\bh}_2\|^2}-\left| Z_{\bar{s}_1,r}
 - \bar{s}_{2,r} \right|^2\right) \nonumber\\
 &+\max\limits_{\bar{s}_{2,I}\in \psi}\left(\frac{\text{log} P_{\bar{s}_{2},I}}{\|\tilde{\bh}_2\|^2}-\left| Z_{\bar{s}_1,I}
 - \bar{s}_{2,I} \right|^2\right)\label{eqn_max_s2_split}
\end{align}
where $Z_{\bar{s}_1,r}$ and $Z_{\bar{s}_1,I}$ denote the real and imaginary parts of $Z_{(\bar{s}_1)}$, respectively. Furthermore, $P_{\bar{s}_{2},r}$ and $P_{\bar{s}_{2},I}$ denote the a-priori probabilities that the real and imaginary parts of $s_2$ equal $\bar{s}_{2,r}$ and $\bar{s}_{2,I}$, computed using the a-priori LLRs of the bits corresponding to the real and imaginary parts, respectively.

Next, we use the a-priori probabilities (LLRs) to modify the decision regions of the $L$-PAM real and imaginary symbols; correspondingly apply the slicer to the real and imaginary parts of $Z_{(\bar{s}_1)}=\tilde{\bh}_2^H\left(\tilde{\by}-\tilde{\bh}_1 \bar{s}_1\right)/\|\tilde{\bh}_2\|^2$, respectively, to find the solution of \eqref{eqn_max_s2_split}; and then compute the metrics in \eqref{eqn_LLR_s1_MaxLog} and \eqref{eqn_LLR_s2_MaxLog}, which can be significantly simplified using \eqref{eqn_max_s2_split}. To develop the method of modifying the decision boundaries, we derive the decision region of the symbol $x_1$ in Fig. \ref{fig_constellation} by writing the conditions on $Z_{\bar{s}_1,r}$ such that
\begin{align}
\frac{\text{log} P_{x_1}}{\|\tilde{\bh}_2\|^2}-\left| Z_{\bar{s}_1,r}
 \hspace{-0.1cm}-\hspace{-0.1cm} x_1 \right|^2 > \frac{\text{log} P_{x_j}}{\|\tilde{\bh}_2\|^2}-\left| Z_{\bar{s}_1,r}
 \hspace{-0.1cm}-\hspace{-0.1cm} x_j \right|^2, \forall j\neq 1 \label{eqn_cond_x1}
\end{align}
Simplifying \eqref{eqn_cond_x1}, we get the decision region of $x_1$ as follows:
\begin{align}
Z_{\bar{s}_1,r} > \max\limits_{j>1} \left(\frac{x_1+x_j}{2}-\frac{\text{log} \left(P_{x_1}/P_{x_j}\right)}{2\left(x_1-x_j\right)\|\tilde{\bh}_2\|^2}\right)
\end{align}
Similarly, the decision region of $x_k$ is given by
\begin{align}\label{eqn_DecReg_xk}
\max\limits_{j>k} D_{kj} <  Z_{\bar{s}_1,r} < \min\limits_{j<k} D_{kj}, \qquad  1 < k < L
\end{align}
and the decision region of the last symbol $x_L$ is given by
\begin{gather}
Z_{\bar{s}_1,r} < \min\limits_{j<L} D_{Lj}\\
\text{where}\quad D_{kj}=D_{jk}=\frac{x_k+x_j}{2}-\frac{\text{log} \left(P_{x_k}/P_{x_j}\right)}{2\left(x_k-x_j\right)\|\tilde{\bh}_2\|^2}\label{eqn_Dkj}
\end{gather}
is called the \textit{probabilistic boundary} between the constellation symbols $x_j$ and $x_k$. Equation \eqref{eqn_Dkj} shows that the boundary between two neighboring symbols moves towards the symbol with the lower a-priori probability, tending to shrink its decision region while extending that of the symbol with the higher a-priori probability. Equation \eqref{eqn_Dkj} also shows that without a-priori LLRs (i.e., $P_{x_k}=\frac{1}{L},\;\forall k$), the boundaries between symbols return to their original values (the average of constellation symbols amplitudes).

\section{Symbols With Empty Decision Regions}\label{sec_no_decesion_regions}
We prove that the a-priori probability distribution can lead to constellation symbols with empty decision regions that will not be chosen by the slicer regardless of $Z_{\bar{s}_1,r}$ (or $Z_{\bar{s}_1,I}$).

\begin{theorem}
When computing the lower bound of the decision region for the constellation symbol $x_k$, given by $\max_{j>k} D_{kj}$, the following can occur:
\begin{equation}
j^*> k+1, \quad \text{where}\quad j^*\equiv arg \max\limits_{j>k} D_{kj}
\end{equation}
meaning that the lower bound of the symbol $x_k$ is not determined by its boundary with the adjacent symbol $x_{k+1}$, but determined instead by its boundary with a farther symbol $x_{j^*} < x_{k+1}$. In this case, all symbols lying between $x_k$ and $x_{j^*}$ (i.e., the constellation symbols $x_m$, where $k<m<j^*$) do not have decision regions and will not be chosen regardless of the decision statistic value.
\end{theorem}

\begin{proof}
From \eqref{eqn_DecReg_xk}, the decision boundaries for $x_m$, where $k<m<j^*,$ are given by
\begin{align}
\max &\left( D_{m(m+1)},..,D_{mj^*},..,D_{mL} \right) < Z_{\bar{s}_1,r}\nonumber\\
&< \min \left( D_{m(m-1)},..,D_{mk},..,D_{m1} \right) \label{eqn_ineq_proof}
\end{align}
However, there is no value for $Z_{\bar{s}_1,r}$ that satisfies \eqref{eqn_ineq_proof} if
\begin{equation}\label{eqn_cond_proof}
D_{mk} < D_{mj^*}
\end{equation}
In the sequel, we prove that the condition in \eqref{eqn_cond_proof} is satisfied if  $j^*=arg \max\limits_{j>k} D_{kj}$, i.e., $D_{kj^*} > D_{km}$ and, hence,
\begin{gather}
x_{j^*} \hspace{-0.05cm}-\hspace{-0.05cm} x_m \hspace{-0.05cm}>\hspace{-0.05cm}
 \frac{\text{log} \left(P_{x_k}/P_{x_{j^*}}\right)}{2\left(x_k-x_{j^*}\right)\|\tilde{\bh}_2\|^2}
 - \frac{\text{log} \left(P_{x_k}/P_{x_m}\right)}{2\left(x_k-x_m\right)\|\tilde{\bh}_2\|^2} \\
 (m \hspace{-0.1cm}-\hspace{-0.1cm} j^*) \,d_{cs} > \frac{\text{log} \left(\left(P_{x_k}/P_{x_{j^*}}\right)^{m\hspace{-0.05cm}-k}\left(P_{x_m}/P_{x_k}\right)^{j^*\hspace{-0.1cm}-k}\right)}
 {(j^*-k)(m-k)d_{cs}\|\tilde{\bh}_2\|^2}\label{eqn_inequality_basic}
\end{gather}
where $x_k - x_{j^*} = (j^* - k)\,d_{cs}$ and $d_{cs}$ is the separation between adjacent real (or imaginary) constellation symbols as shown in Fig. \ref{fig_constellation}. Since $k<m<j^*$, we define
\begin{align}\label{eqn_definitions}
m = k + f,\qquad j^* = m + g = k + f + g
\end{align}
where $f,g\in\{0,\mathbb{Z}^+\}$. We rewrite \eqref{eqn_inequality_basic} as follows:
\begin{equation}\label{eqn_inequality_2}
fg(f + g)d_{cs}^2 \|\tilde{\bh}_2\|^2 \hspace{-0.05cm}<\hspace{-0.05cm} \text{log} \left(\left(P_{x_{j^*}}/P_{x_m}\right)^f\left(P_{x_k}/P_{x_m}\right)^g\right)
\end{equation}
Next, we rewrite the condition in \eqref{eqn_cond_proof} as follows:
\begin{align}
D_{mk} \hspace{-0.05cm}-\hspace{-0.05cm} D_{mj^*} \hspace{-0.05cm}=\hspace{-0.05cm} \frac{(f\hspace{-0.05cm}+\hspace{-0.05cm}g)\,d_{cs}}{2}-\hspace{-0.05cm}
\frac{\text{log}\hspace{-0.05cm} \left(\hspace{-0.05cm}\left(\frac{P_{x_k}}{P_{x_m}}\right)^g\left(\frac{P_{x_{j^*}}}{P_{x_m}}\right)^f\right)}
 {2fg\,d_{cs}\|\tilde{\bh}_2\|^2}
\end{align}
Using the inequality in \eqref{eqn_inequality_2}, we bound $D_{mk} \hspace{-0.05cm}-\hspace{-0.05cm} D_{mj^*}$ as follows:
\begin{align}
D_{mk} - D_{mj^*} &< \frac{(f+g)\,d_{cs}}{2} - \frac{fg(f+g)d_{cs}^2\|\tilde{\bh}_2\|^2}{2fg\,d_{cs}\|\tilde{\bh}_2\|^2}\nonumber\\
D_{mk} - D_{mj^*} &<0,\qquad D_{mk} < D_{mj^*}
\end{align}
which concludes the proof.
\end{proof}
The practical importance of this theorem is that it can reduce the algorithm complexity and further speed it up. For example, if the lower boundary of $x_1$ is determined by $x_4$ then we do not need to compute the decision boundaries of $x_2$ and $x_3$ because they will have empty decision regions.

\section{Algorithm and Computational Complexity}\label{sec_algo}
In the sequel, we summarize the algorithm and show the receiver model in Fig. \ref{fig_system_model}.
\\
{\bf Preprocessing}: Compute $\bH=\bar{\bH} \bW$ and whiten the noise by computing $\tilde{\by}=\sqrt{\bQ}\by$, $\tilde{\bh}_1=\sqrt{\bQ}\bh_1$, and $\tilde{\bh}_2=\sqrt{\bQ}\bh_2$.

\\
{\bf Procedure}:
\begin{enumerate}
\item\label{item_1} Get the decision regions for $Z_{\bar{s}_1,r}, Z_{\bar{s}_1,I}, Z_{\bar{s}_2,r},$ and $Z_{\bar{s}_2,I}$ using the corresponding a priori LLRs as follows:
\\
Initialize $k=1$.
\\
{\bf While} $k<=L$
\\
A) Compute the lower and upper thresholds of the $k$-th constellation symbol as $\max_{j>k} D_{kj}$ and $\min_{j<k} D_{kj}$, respectively, where
\begin{equation}\label{eqn_Dkj_LLRs}
D_{kj}=\frac{x_k+x_j}{2}-\frac{\sum_{n=1}^{q/2} \left((b_{n,k}-b_{n,j})L_a(c_{i\,n})\right)}{2\left(x_k-x_j\right)\|\tilde{\bh}_i\|^2}
\end{equation}
where $i\in\{1,2\}$, $L_a(c_{i\,n})$ denotes the a priori LLR of the code bit $c_{in}$, and $\left\{b_{n,k},b_{n,j}\right\}_{n=1}^{q/2}\in \left\{0,1\right\}$ are the bit vectors corresponding to the constellation symbols $x_k$ and $x_j$, respectively. The transition from the probability domain in \eqref{eqn_Dkj} to the LLR domain in \eqref{eqn_Dkj_LLRs} is straightforward.

B) {\bf If} $j^*> k+1, \quad \text{where}\quad j^*\equiv arg \max\limits_{j>k} D_{kj}$, set the decision regions of the symbols $x_m$, where $k<m<j^*$, to empty, and set $k=j^*.$
\\
\hspace*{0.34cm} {\bf Else}, set $k=k+1.$
\\
{\bf End While}
\item Enumeration step over constellation points of $s_1$ and $s_2$.
\\
{\bf For} $k=1:M$
\\
a) Compute the following quantities for $\bar{s}_1(k),\bar{s}_2(k) \in \chi$
\begin{equation}\nonumber
Z_{(\bar{s}_1\hspace{-0.02cm})}\hspace{-0.1cm}=\hspace{-0.1cm}\frac{\tilde{\bh}_2^H}{\|\tilde{\bh}_2\|^2}\left(\hspace{-0.05cm}\tilde{\by}\hspace{-0.1cm}-\hspace{-0.1cm}\tilde{\bh}_1 \bar{s}_1(k)\hspace{-0.05cm}\right)\hspace{-0.1cm},
Z_{(\bar{s}_2\hspace{-0.02cm})}\hspace{-0.1cm}=\hspace{-0.1cm}\frac{\tilde{\bh}_1^H}{\|\tilde{\bh}_1\|^2}\left(\hspace{-0.05cm}\tilde{\by}\hspace{-0.1cm}-\hspace{-0.1cm}\tilde{\bh}_2 \bar{s}_2(k)\hspace{-0.05cm}\right)
\end{equation}
b) Slice the real and imaginary parts of $Z_{(\bar{s}_1)}$ and $Z_{(\bar{s}_2)}$ using the thresholds obtained in Step \ref{item_1} to obtain $\bar{s}^*_2(k)$ and $\bar{s}^*_1(k)$, respectively.
\\
c) Compute the following metrics
\vspace{-0.2cm}
\begin{align}
\eta_1(k)&=\sum_{n=1}^q \left(\bar{b}_{1,n}(k)L_a(c_{1n})+\bar{b}^*_{2,n}(k)L_a(c_{2n})\right)\nonumber\\
&-\|\tilde{\by}-\tilde{\bh}_1 \bar{s}_1(k) -\tilde{\bh}_2 \bar{s}^*_2(k) \|^2\label{eqn_metric_eta1}\\
\eta_2(k)&=\sum_{n=1}^q \left(\bar{b}^*_{1,n}(k)L_a(c_{1n})+\bar{b}_{2,n}(k)L_a(c_{2n})\right)\nonumber\\
&-\|\tilde{\by}-\tilde{\bh}_1 \bar{s}^*_1(k) -\tilde{\bh}_2 \bar{s}_2(k) \|^2
\end{align}
where $\{\bar{b}_{1,n}(k),\bar{b}_{2,n}(k),\bar{b}^*_{1,n}(k),\hspace{-0.05cm}\bar{b}^*_{2,n}(k)\hspace{-0.05cm}\}_{n=1}^{q}$ are the bit vectors of $\bar{s}_1(k), \bar{s}_2(k), \bar{s}_1^*(k), \bar{s}^*_2(k)$, respectively.
\\
{\bf End For}
\item Compute the detector LLRs for $i=1,2$ and $0\leq n\leq q$
\begin{align}
L(c_{in})=\max_{k\,:\,\bar{b}_{i,n}(k)=1} \eta_i(k) - \max_{k\,:\,\bar{b}_{i,n}(k)=0} \eta_i(k)
\end{align}
\end{enumerate}

\begin{figure}
\centering
\epsfig{file=SystemModel_Cropped.eps,height=1.1in,width=3.45in}
\caption{Receiver model with our proposed approach.} \label{fig_system_model}
\end{figure}

\section{Complexity analysis}\label{sec_complexity}
We count the number of required metrics computations to obtain the $2\text{log}_2(M)$ detector LLRs corresponding to $s_1$ and $s_2$. To get the new decision regions, we need to compute the probabilistic boundaries between every two symbols of the $L$ symbols (for both real and imaginary parts). Since $D_{jk}=D_{kj}$, the number of metrics (boundaries) to be computed is
\begin{equation}
N_{metrics}^{s_1} = 2\frac{L(L-1)}{2} = L^2-L = M-\sqrt{M} < M
\end{equation}
Note that these boundaries are computed only once and are not included inside the enumeration over $s_1$ in \eqref{eqn_LLR_s1_MaxLog}. Hence, the total number of metric computations to obtain the LLRs of the code bits corresponding to $s_1$ is $M+N_{metrics}^{s_1}=2M-\sqrt{M}$. To obtain the $2q$ LLRs corresponding to $\bs$ (i.e., $s_1$ and $s_2$), the number of metric computations per tone becomes
\begin{equation}
N_{metrics}^{Total}= 4M-2\sqrt{M} < 4M
\end{equation}

In Table \ref{Table_cmplx_comp}, we compare our algorithm with the Turbo-LORD (T-LORD) \cite{TLORD_TWC} and the brute-force algorithms in terms of the number of metrics to be computed, number of real multiplications (Muls), and number of real additions (Adds) per tone per iteration as function of the constellation size and the number of receive antennas. In Table \ref{Table_cmplx_comp_256QAM}, we compare these algorithms for 256-QAM and two receive antennas where we observe the significant computational complexity saving without any performance loss since our algorithm obtains the exact solution rather than the approximate solution in \cite{TLORD_TWC}. In T-LORD \cite{TLORD_TWC}, while enumerating over $s_1$, three candidates for $s_2$ are obtained for every possible value of the $M$ candidiases of $s_1$. Hence, we have $3M$ candidates for the $\left(s_1,s_2\right)$ pair, and the metric in \eqref{eqn_metric_eta1} is computed for each candidate. Doing the same for $s_2$, we have another $3M$ metrics summing up to $6M$ metrics to be computed.

\begin{table}
\caption{Complexity comparison between various ML detectors}
\centering
\begin{tabular}{|p{1.5cm}|p{0.8cm}|p{2cm}|p{2.6cm}|}
\hline
Detector & Metrics & Real Muls & Real Adds \\ \hline
Proposed & $4M-2\sqrt{M}$ & $\left(16N_r+18\right)M-2\sqrt{M}$  &  $\left(12N_r+3q+18\right)M-\left(q+2\right)\sqrt{M}-4$  \\ \hline
T-LORD & $6M$   & $\left(16N_r+48\right)M $  &  $\left(12N_r+2q+52\right)M-4$ \\ \hline
Brute force & $M^2$ & $8M^2$ & $12M^2-4M$ \\ \hline
\end{tabular}
\label{Table_cmplx_comp}
\end{table}

\begin{table}
\caption{Complexity of various ML detectors for $M=$256 and $N_r=2$}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Detector & Metrics & Real Muls & Real Adds \\ \hline
Proposed & 992 & 12768  &  16732 \\ \hline
T-LORD & 1536   & 20480  &  23548 \\ \hline
Brute force & 65536 & 524288 & 785408 \\ \hline
\end{tabular}
\label{Table_cmplx_comp_256QAM}
\end{table}

\section{Conclusion}\label{sec_Conclusion}
We developed the \textit{exact} max-log MAP detector for IDD in dual-layer MIMO schemes with computational complexity less than $4M$. The idea is to use the a priori LLRs in modifying the decision thresholds of the constellation symbols. We also showed that the a priori LLRs can lead to constellation symbols with empty decision regions, reducing the search space of the slicing block. Comparing the computational complexity with the Turbo-LORD approximate solution and the exact brute force solutions, we show that our algorithm achieves significant complexity reduction while achieving the exact max-log MAP solution. We have numerically verified that our method yields the same performance as the brute force solution for various simulation parameters but the simulation results are not shown here due to space limitations.

config: arxiv


TITLE: Phase Portrait of Symmetric Potential Well
QUESTION [0 upvotes]: I'm asked to sketch the phase portrait of a symmetric potential well and I had a few questions regarding the behavior of the phase portrait. Would the outer edges of the phase portrait behave similar to how is shown in my sketch as the solid line where they circle the two centers or would they deviate asymptotically along what would be the x-axis depicted by the dashed lines? Another question I had was if damping existed in the potential function and the centers were no longer neutrally stable, what would determine which unstable ends of the saddle point would be drawn to which center?

Any input is appreciated

REPLY [0 votes]: No, the dashed lines are not "physical", the solid lines are fine. Your solutions are bounded by their energy levels, that is, you can draw a horizontal line of the energy level in the graph of the potential energy, and the crossings define the extremal points between which the solution oscillates.


TITLE: "There are exactly two values of $x$ for which $P(x)$ is true" formula using logical symbols
QUESTION [4 upvotes]: Assuming $P(x)$ is true. The statement: "There are exactly two values of $x$ for which $P(x)$ is true" can be rewritten using logical symbols as follows:
$$\exists x \exists y[(P(x) \wedge P(y) \wedge  x\neq y)\wedge \forall w(P(w) \Rightarrow (w=x)\vee (w=y))]$$
Is this reasoning true, or there is any mistake.
Thanks.

REPLY [1 votes]: You're close. First, try to say there are two distinct elements that satisfy $P$. Your approach is pretty close but not right, it should be something like

There exist two elements, $x$ and $y$, such that $P(x)$ and $P(y)$ and $x$ is distinct from $y$

This phrase is easier to turn into a formula.
For the "uniqueness" part, you're right, but as @Git Gud notices, the existential quantifiers for $x$ and $y$ should take that part too.


TITLE: Limit of $\lim \limits_{x \rightarrow \infty}\frac{(x!)^n}{(ax)!}$
QUESTION [2 upvotes]: For a given $n \geq 1$ $\hspace{2mm}(n \in \mathbb{R})$, I know that
$$\lim \limits_{x \rightarrow \infty}\frac{(x!)^n}{(ax)!},$$ only exists and it is equal to zero if $a \geq n.$ However, I cannot prove that.
Could you give me some ideas? Thanks in advance!

REPLY [3 votes]: You can use Stirling's formula: $x! \sim \sqrt{2\pi x} (x/e)^x$. In your case, assuming $a > 0$ you get
$$
\frac{(x!)^n}{(ax)!} \sim (2\pi)^{(n-1)/2} \frac{x^{nx}}{e^{nx}} \frac{e^{ax}}{(ax)^{ax}} = (2\pi)^{(n-1)/2} \frac{(x/e)^{(n-a)x}}{a^{ax}}.
$$
If $a \geq n$ and $a > 1$ then this tends to zero. If $n > a$ then it tends to infinity. This leaves the corner case $a = n = 1$, in which the limit is $1$.


TITLE: Polynomial vector space with linear operator
QUESTION [0 upvotes]: Suppose that $Ax(t) = x(t+1)$ for every $x$ in $P_n$; prove that if $D$ is the differentiation operator, then $$1+\frac{D}{1!}+\frac{D^2}{2!}+\cdots +\frac{D^{n-1}}{n-1!}=A$$
Please, explain to me how to start.

REPLY [1 votes]: After you have proven that $A$ is linear, it is enough to show equality of the operators on monomials. For that, try the binomial theorem.


TITLE: Is Wigner D matrix also a spherical tensor?
QUESTION [2 upvotes]: Can one think of Wigner D-matrix $\mathcal D_{mm''}^{(l)}$
as a spherical tensor even in case when $m''\neq0$ ?
In other words does relation like
\begin{equation}
  \tag{1}
  \mathcal D^\dagger(R_1)\mathcal D^{(l)}_{mm''}(R_2)\mathcal D(R_1)
  = \sum_{m'} D_{mm'}^{(l)*}(R_1)D^{(l)}_{m'm''}(R_2)
\end{equation}
hold? And if not, can one argue that this makes physically sense?
In case when $m''=0$ relation above holds, since
$\mathcal D_{m0}^{(l)}(\phi, \theta,\gamma)\sim Y_l^{m*}(\theta,\phi)$, and for spherical harmonics we have
\begin{equation}
  \tag{2}
  \mathcal D^\dagger(R)Y_l^{m}(\hat n)\mathcal D(R)
  = \sum_{m'} D_{mm'}^{(l)*}(R)Y_l^{m'}(\hat n).
\end{equation}
For a general case, I've tried using the product rule,
and I got close but not quite there:
\begin{equation}
  \tag{3}
  D_{mm''}^{(l)}(R_1R_2)
  =\sum_{m'} D_{mm'}^{(l)}(R_1) D_{m'm''}^{(l)}(R_2),
\end{equation}
but I'm not quite sure if I can interpret l.h.s here
as $D_{mm''}^{(l)}(R_1R_2)
= \mathcal D^\dagger(R_1)D_{mm''}^{(l)}(R_2)\mathcal D(R_2)$.
But, even then it is not quite right since I'd need a conjugate on the r.h.s, i.e. $D_{m''m'}^{(l)*}(R_1)$ instead of $D_{m''m'}^{(l)}(R_1)$.
Than again, in case my interpretation of $D_{mm''}^{(l)}(R_1R_2)$ is ok,
this has to reduce to spherical harmonic case again when $m''=0$.
So I'm left a bit confused... any help would be much appreciated.

REPLY [2 votes]: I can't think of what it might mean to say that the $\mathfrak{D}^{(\ell)}$ matrices are spherical tensors.  But they are certainly related.  The best source I know of for this kind of thing is Thorne's 1980 review, in which he shows how various presentations relate to each other.  For example, he shows that the various common vector and tensor harmonics can be written by "coupling" the scalar spherical harmonics to basis vectors.  For example, using the usual Clebsch-Gordan coefficients, we can write
\begin{equation}
  Y^{\ell', \ell m}(\theta, \phi) = \sum_{m', m''} (1\ell'm''m' | \ell m) \boldsymbol{\xi}^{m''} Y^{\ell'm'},
\end{equation}
where
\begin{equation}
  \boldsymbol{\xi}^0 = \mathbf{e}_z
  \quad \text{and} \quad
  \boldsymbol{\xi}^{\pm 1} = \mp (\mathbf{e}_x \pm i \mathbf{e}_y) / \sqrt{2}.
\end{equation}
He also shows how these can be written in terms of symmetric trace-free tensors, which are another useful presentation that shows up especially in older work.  Alternatively, related vector and tensor harmonics can be written in terms of derivatives of the $Y^{\ell m}$.
A related idea is the spin-weighted spherical harmonic (SWSH).  These generalize the formula that you gave above for standard spherical harmonics, and are sometimes defined as
\begin{equation}
  {}_{s}Y_{\ell m}(\theta, \phi) = (-1)^s \sqrt{ \frac{2\ell+1} {4\pi}} \mathfrak{D}^{(\ell)}_{-s, m}(\phi, \theta, 0).
\end{equation}
Unlike "scalar" spherical harmonics, SWSHs don't transform among themselves under rotation, because there's an extra factor of $e^{i s \gamma}$ floating around (where $\gamma$ is that third Euler angle, which was set to $0$ in the equation above).  I actually wrote a paper on essentially this topic, in which I showed that SWSHs really need to be understood as functions on the rotation group [or more generally on $SU(2)$], instead of the sphere.  Then, and only then, do they correctly transform among themselves under rotations.  (Except, of course, when $s=0$ — in which case the extra factor floating around is just $1$ and the SWSH is equal to the usual spherical harmonic.)
However, as Thorne shows, standard SWSHs can also be "coupled" to basis vectors, and do end up resulting in spherical tensors again.  Basically, if your basis vectors are also poorly behaved, so that they transform with a factor of $e^{-i s \gamma}$, you can cancel out the misbehavior of spin-weighted spherical harmonics and get something that does actually transform correctly as a tensor.

Transformation of the $\mathfrak{D}^{(\ell)}$ matrices
After the discussion in the comments, it seems that the question really is: How do the Wigner-$\mathfrak{D}$ matrices transform under rotation?  To be specific, suppose we have some $R_1$, we choose some $\ell$, and now we can write down the actual numerical values of $\mathfrak{D}^{(\ell)}_{m, m'}(R_1)$.  But those numerical values depend on the basis that we chose for the vector space that is implicitly present in the definition of $R_1$.  So now we wonder what would happen to all those values if we rotated the vector basis by some other rotation — say $R_2$.
So first we have to figure out how rotations themselves — in particular $R_1$ — transform under rotations.  This isn't hard: rotations form a group, where the group operation is composition.  If you think of rotations in terms of the rotation matrix, composition is just matrix multiplication.(*)  So the rotated version of $R_1$ is just $R_1 R_2$.
Now knowing this, the answer is already contained in the question above.  The numerical values of the rotated $\mathfrak{D}^{(\ell)}_{m, m'}(R_1)$ matrix are just the values of the matrix of the rotated rotation:
\begin{equation}
  \tag{A}
  \mathfrak{D}^{(\ell)}_{m, m'}(R_1\, R_2)
  =
  \sum_\mu \mathfrak{D}^{(\ell)}_{m, \mu}(R_1) \mathfrak{D}^{(\ell)}_{\mu, m'}(R_2).
\end{equation}
This is the transformation law for the Wigner-$\mathfrak{D}$ matrices.
You can also extend this as needed for multiple rotations.  For example,
\begin{equation}
  \mathfrak{D}^{(\ell)}_{m, m'}(R_1\, R_2\, R_3)
  =
  \sum_{\mu,\mu'} \mathfrak{D}^{(\ell)}_{m, \mu}(R_1) \mathfrak{D}^{(\ell)}_{\mu, \mu'}(R_2) \mathfrak{D}^{(\ell)}_{\mu', m'}(R_3).
\end{equation}
So if, for some reason, you wanted to conjugate your original rotation, you would have
\begin{equation}
  \mathfrak{D}^{(\ell)}_{m, m'}(R_2\, R_1\, R_2^{-1})
  =
  \sum_{\mu,\mu'} \mathfrak{D}^{(\ell)}_{m, \mu}(R_2) \mathfrak{D}^{(\ell)}_{\mu, \mu'}(R_1) \mathfrak{D}^{(\ell)}_{\mu', m'}(R_2^{-1}).
\end{equation}
It's also worth noting that equation (A) tells us that the $\mathfrak{D}$ matrix for an inverse rotation is just the matrix-inverse of the original $\mathfrak{D}$ matrix:
\begin{equation}
  \mathfrak{D}^{(\ell)}_{m, m'}(R^{-1})
  =
  \left( \mathfrak{D}^{(\ell)}_{m, m'}(R) \right)^{-1}.
\end{equation}

(*) There's a little bit of subtlety about whether you mean your rotations to be active or passive, etc.  Depending on what exactly your conventions are, the correct result might be $R_2\, R_1$.  You need to work this out for yourself by actually rotating, say, the basis vectors and deciding what you personally mean by all these things.  If there's one thing I've learned in years of working on this stuff, it's that there is no universal agreement on literally anything when it comes to rotations.  For example, even the words "active" and "passive" can refer to totally different concepts, and different people might mean opposite things even when they're referring to the same concept.  So the most important rule is to check the basics for yourself.

REPLY [0 votes]: I'm not sure about your notation but as I understand it I would start by writing
$$
D^\ell_{mm'}(R_2)=
\langle \ell m\vert R_2\vert \ell m'\rangle\, .
$$
Then
\begin{align}
(R_1)^{-1}D^\ell_{mm'}(R_2)R_1&:=
\langle \ell m\vert R_1^{-1}R_2R_1\vert  \ell m'\rangle\, , \\
&=\sum_{\mu\mu'}
\langle \ell m\vert R_1^{-1} \vert \ell \mu\rangle
\langle \ell \mu \vert R_2 \vert \ell \mu'\rangle
\langle \ell \mu'\vert R_1 \vert\ell m'\rangle\, , \\
&=\sum_{\mu\mu'}D^{\ell}_{m'\mu }(R^{-1}_1)D^{\ell}_{\mu \mu'}(R_2)
D^\ell_{\mu' m'}(R_1)
\end{align}
so it doesn't transform as a tensor because of the double sum.


text:
\begin{document}

\title[Inverse spectral problems]{Inverse spectral problems
    for Sturm-Liouville operators with singular potentials, IV.\\
    Potentials in the Sobolev space scale${}^{\dag}$}

\author[R.~O.~Hryniv, Ya.~V.~Mykytyuk]{Rostyslav O.~Hryniv and Yaroslav V.~Mykytyuk}
\address{Institute for Applied Problems of Mechanics and Mathematics,
3b~Naukova st., 79601 Lviv, Ukraine and Lviv National University,
1 Universytetska st.,
79602 Lviv, Ukraine}
\email{rhryniv@iapmm.lviv.ua, yamykytyuk@yahoo.com}

\address{\emph{Current address of R.H.:}
Institut f\"ur Angewandte Mathematik, Abteilung f\"ur
Wahr\-schein\-lich\-keitstheorie und Mathematische Statistik,
Wegelerstr.~6, D-53115 Bonn, Germany}
\email{rhryniv@wiener.iam.uni-bonn.de}

\thanks{${}^{\dag}$The work was partially supported by Ukrainian Foundation
for Basic Research, grant No.~01.07/00172.
R.~H. acknowledges the support of the Alexander von Humboldt foundation}

\subjclass[2000]{Primary 34A55, Secondary 34B24, 34L05, 34L20}
\keywords{Inverse spectral problems, Sturm-Liouville operators, singular
potentials, Sobolev spaces}

\date{\today}

\begin{abstract}
We solve the inverse spectral problems for the class of
Sturm--Liouville operators with singular real-valued potentials
from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The
potential is recovered from two spectra or from the spectrum and
norming constants. Necessary and sufficient conditions on the
spectral data to correspond to the potential in~$W^{s-1}_2(0,1)$
are established.
\end{abstract}
\maketitle

\section{Introduction}\label{sec:intr}

Suppose that $q$ is a real-valued distribution from~$W^{-1}_2(0,1)$. We
denote by $(\la_n^2)$ and $(\mu_n^2)$, $n\in\bN$, eigenvalues of
Sturm--Liouville operators $T_{\mathrm D}$ and $T_{\mathrm N}$
generated by the differential expression $-\frac{d^2}{dx^2} +q$ subject
to the Dirichlet and Neumann--Dirichlet boundary conditions
respectively (see Section~\ref{sec:pre} for precise definitions). The
operators $T_{\mathrm D}$ and $T_{\mathrm N}$ are bounded
below~\cite{SS} and thus become positive after addition of a suitable
constant to the potential~$q$. Henceforth there is no loss of
generality in assuming that the operators $T_{\mathrm D}$ and
$T_{\mathrm N}$ (and the numbers $\la_n$ and $\mu_n$) are positive. The
eigenvalues $\la_n^2$ and $\mu_n^2$ are simple; we arrange them in
increasing order and recall the following their
properties~\cite{An,HMtwo,SS1}:
\begin{itemize}
\item [(A1)] the sequences $(\la_n)$ and $(\mu_n)$ interlace, i.e.,
    $\mu_n<\la_n<\mu_{n+1}$ for all $n\in\bN$;
\item [(A2)] the numbers $\la_n$ and $\mu_n$ obey the asymptotics
    \begin{equation}\label{eq:lamu}
        \la_n = \pi n + \wt\la_n, \qquad \mu_n=\pi(n-1/2) + \wt\mu_n
    \end{equation}
 with some $\ell_2$-sequences $(\wt\la_n)$ and $(\wt\mu_n)$.
\end{itemize}

Conversely, it was shown in~\cite{HMtwo} that if two sequences
$(\la_n^2)$ and $(\mu_n^2)$ of positive numbers satisfy properties
(A1) and (A2), then there exists a unique $q\in W^{-1}_2(0,1)$
such that $(\la_n^2)$ and $(\mu_n^2)$ are eigenvalues of the
Sturm--Liouville operators $T_{\mathrm D}$ and $T_{\mathrm N}$
with potential~$q$. In other words, the inverse spectral problem
of recovering the potential by two spectra is uniquely soluble in
the class of Sturm--Liouville operators with singular potentials
from~$W^{-1}_2(0,1)$. The papers~\cite{HMinv,HMtwo} give the
corresponding reconstruction algorithm and thus extend the
classical inverse spectral theory for Sturm--Liouville operators
developed by Gelfand, Levitan, Marchenko, and Krein in
1950-ies~\cite{GL,K,M}.

The aim of the present work is to show that the above inverse spectral
problem is completely soluble in the class of Sturm--Liouville
operators with potentials from~$W^{s-1}_2(0,1)$ for every $s\in[0,1]$.
More precisely, we shall formulate necessary and sufficient conditions
on sequences $(\la_n^2)$ and $(\mu_n^2)$ in order that they are
Dirichlet and Neumann--Dirichlet eigenvalues of a Sturm--Liouville
operator with singular potential~$q$ from~$W^{s-1}_2(0,1)$.

The particular case $s=1$ corresponds to potentials in $L_2(0,1)$;
the classical theorem by Marchenko~\cite[Theorem 3.4.1]{Ma} states
that necessary and sufficient conditions on the eigenvalues
$(\la_n^2)$ and $(\mu_n^2)$ are (A1) and (A2) with the
specification that $\wt\la_n$ and $\wt\mu_n$ have the form
\[
    \wt\la_n = \frac{c}n + \frac{\wh \la_n}{n}, \qquad
    \wt\mu_n = \frac{c}n + \frac{\wh \mu_n}{n}
\]
with $c\in\bR$ and some $\ell_2$-sequences $(\wh\la_n)$ and
$(\wh\mu_n)$.

For an arbitrary intermediate value $s\in(0,1)$, the direct spectral
problem was studied in~\cite{HMas,KM,SS1}. For instance, it was proved
in~\cite{HMas} that $\wt\la_n$ and $\wt\mu_n$ are respectively even and odd
sine Fourier coefficients of some function from~$W^{s}_2(0,1)$ (cf. the
above-mentioned cases $s=0$ and $s=1$). More exactly, the main result
from~\cite{HMas} reads as follows.

\begin{theorem*}{Theorem~A. }
Assume that $q\in W^{s-1}_2(0,1)$ for some $s\in[0,1]$ and that $\wt\la_n$,
$\wt\mu_n$ are defined through~\eqref{eq:lamu}. Then the function $\si^*$
given by
\begin{equation}\label{eq:si*}
    \si^*(x):=  2 \sum_{n=1}^\infty \wt\mu_n\sin[(2n-1)\pi x] -
                2 \sum_{n=1}^\infty \wt\la_n\sin (2\pi nx)
\end{equation}
belongs to~$W^s_2(0,1)$. Moreover, $\si^*-\si \in W^{2s}_2(0,1)$, where
$\si$ is any of the distributional primitives of~$q$.
\end{theorem*}

In the present paper we show that the condition $\si^*\in
W^s_2(0,1)$ is a sufficient addendum to (A1) and (A2) guaranteeing
that the corresponding potential belongs to~$W^{s-1}_2(0,1)$. Our
main result is as follows.

\begin{theorem}\label{thm:main1}
In order that two sequences $(\la_n^2)$ and $(\mu_n^2)$ be
eigenvalues of (positive) Sturm--Liouville operators $T_{\mathrm
D}$ and $T_{\mathrm N}$ with potential from $W^{s-1}_2(0,1)$,
$s\in[0,1]$, it is necessary and sufficient that assumptions (A1),
(A2) hold and that the function~$\si^*$ of~\eqref{eq:si*} belongs
to $W^s_2(0,1)$.
\end{theorem}

As an intermediate step we solve the inverse spectral problem of
recovering the potential of a Sturm--Liouville expression by its
Dirichlet spectrum $(\la_n^2)$ and the so-called norming
constants~$(\al_n)$. We recall that
\[
    \al_n:= \Bigl(2\int_0^1|u_n(x)|^2\,dx\Bigr)^{-1},
\]
where $u_n$ is an eigenfunction of the operator $T_{\mathrm D}$
that corresponds to the eigenvalue $\la_n^2$ and satisfies the
initial condition $u_n^{[1]}(0)=\la_n$, with $u^{[1]}$ denoting
the quasi-derivative of a function~$u$, see Section~\ref{sec:pre}.
Alternatively, we can reduce the inverse spectral problem by two
spectra to recovering the potential by the spectrum $(\mu_n^2)$ of
the operator $T_{\mathrm N}$ and the norming constants $(\be_n)$;
the latter are defined as
\[
    \be_n:= \Bigl(2\int_0^1 |v_n(x)|^2\,dx\Bigr)^{-1},
\]
where $v_n$ is an eigenfunction of the operator $T_{\mathrm N}$
that corresponds to the eigenvalue~$\mu_n^2$ and satisfies the
initial condition $v_n(0)=1$.

In the case $q\in W^{-1}_2(0,1)$ (i.e., for $s=0$) the norming constants
$\al_n$ and $\be_n$ have the asymptotics $\al_n = 1 + \wt\al_n$,
$\be_n=1+\wt\be_n$ with $\ell_2$-sequences $(\wt\al_n)$ and $(\wt\be_n)$,
see~\cite{HMinv}. For $s\in(0,1]$ this asymptotics refines as follows. We
introduce the function $\ga$ via the formula
\begin{equation}\label{eq:ga}
    \ga(x):=    2 \sum_{n=1}^\infty \wt\be_n \cos[(2n-1)\pi x] -
                2 \sum_{n=1}^\infty \wt\al_n \cos (2\pi nx)
\end{equation}
and also put
\[
    \ga^*(x) := -2x \si^*(1-x).
\]

\begin{theorem}\label{thm:albe}
Assume that $\si^*\in W^s_2(0,1)$ for some $s\in[0,1]$. Then the function
$\ga$ also belongs to $W^s_2(0,1)$; moreover, $\ga-\ga^*\in W^{2s}_2(0,1)$.
\end{theorem}

In particular, we see that if a primitive $\si$ of $q$ belongs to
$W^s_2(0,1)$, then the sequences $(-\wt\al_n)$ and $(\wt\be_n)$ are even
and odd cosine Fourier coefficients respectively of the function $\ga\in
W^s_2(0,1)$ given by~\eqref{eq:ga}. In the reverse direction the claim is
that if the even and odd parts of the functions $\si^*$ and $\ga$ belong to
$W^s_2(0,1)$, then $\si$ is an $W^s_2$-function (and hence the potential
$q$ belongs to~$W^{s-1}_2(0,1)$). More exactly, the following two
statements hold true.

\begin{theorem}\label{thm:main2}
Sequences $(\la_n^2)$ and $(\al_n)$ of positive numbers are
eigenvalues and norming constants for some Sturm--Liouville
operator $T_{\mathrm D}$ with real-valued potential $q\in
W^{s-1}_2(0,1)$, $s\in[0,1]$, if and only if the following
conditions are satisfied:
\begin{itemize}
\item [(B1)] the numbers $\la_1<\la_2<\dots$ obey the asymptotics
    $\la_n=\pi n + \wt\la_n$, where $\wt\la_n$ are even sine Fourier
    coefficients of some function from $W^s_2(0,1)$;
\item [(B2)] the numbers $\wt\al_n:=\al_n-1$ are even cosine Fourier
    coefficients of some function from $W^s_2(0,1)$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{thm:main3}
Sequences $(\mu_n^2)$ and $(\be_n)$ of positive numbers are
eigenvalues and norming constants for some Sturm--Liouville
operator $T_{\mathrm N}$ with real-valued potential $q\in
W^{s-1}_2(0,1)$, $s\in[0,1]$ if and only if the following
conditions are satisfied:
\begin{itemize}
\item [(C1)] the numbers $\mu_1<\mu_2<\dots$ obey the asymptotics
    $\mu_n=\pi (n-1/2) + \wt\mu_n$, where $\wt\mu_n$ are odd sine Fourier
    coefficients of some function from $W^s_2(0,1)$;
\item [(C2)] the numbers $\wt\be_n:=\be_n-1$ are odd cosine Fourier
    coefficients of some function from $W^s_2(0,1)$.
\end{itemize}
\end{theorem}

The organization of the paper is as follows. In
Section~\ref{sec:pre} we give the precise definitions of the
operators $T_{\mathrm D}$ and $T_{\mathrm N}$. In
Section~\ref{sec:norm} asymptotics of the norming constants is
established, based on which Theorem~\ref{thm:albe} is proved. The
algorithm of solution of the inverse spectral problems under
consideration and the proofs of Theorems~\ref{thm:main1},
\ref{thm:main2}, and \ref{thm:main3} are given in
Section~\ref{sec:inverse}. Finally, in Appendix~\ref{sec:sobolev}
some necessary facts about Sobolev spaces $W_2^s(0,1)$ and Fourier
series therein are gathered.

\section{Preliminaries}\label{sec:pre}

Suppose that $q\in W^{-1}_2(0,1)$ is real-valued. We fix an arbitrary
real-valued distributional primitive $\si\in L_2(0,1)$ of~$q$ (so that
$q=\si'$ in the sense of distributions) and consider the differential
expression
\[
    l_\si(u) := -(u'-\si u)' -\si u'
\]
on its ``maximal'' domain in~$L_2(0,1)$,
\[
    \fD(l_\si) = \{ u \in W^1_1(0,1) \mid u^{[1]} \in W^1_1(0,1),\
        l_\si(u) \in L_2(0,1)\}.
\]
Here and hereafter, $u^{[1]}$ stands for the \emph{quasi-derivative}
$u'-\si u$ of a function $u$. It is easily seen that $l_\si(u) = -u'' + q
u$ in the sense of distributions, so that $l_\si$ is a
\emph{regularization} of the differential expression $-\frac{d^2}{dx^2}+q$.

The operators $T_{\mathrm D}$ and $T_{\mathrm N}$ are defined as
the restrictions of $l_\si$ imposing the corresponding boundary
conditions
\begin{align*}
    \fD(T_{\mathrm D}) &= \{ u \in \fD(l_\si) \mid u(0)=u(1)=0\},\\
    \fD(T_{\mathrm N}) &= \{ u \in \fD(l_\si) \mid u^{[1]}(0)=u(1)=0\}.
\end{align*}
It is known~\cite{SS} that the operators $T_{\mathrm D}$ and
$T_{\mathrm N}$ are selfadjoint, bounded below, and have discrete
spectra. In some sense they are the most natural Dirichlet and
Neumann--Dirichlet Sturm--Liouville operators associated with the
differential expression~$-\frac{d^2}{dx^2} + q$ with singular~$q$; see
the discussion in~\cite{SS1}.

\section{Asymptotics of the norming constants}\label{sec:norm}

Suppose that $\si\in L_2(0,1)$ is real-valued. We denote by
$u_\pm(\cdot,\la)$ and $v_-(\cdot,\la)$ solutions of the equation
 \(
    l_\si (u) = \la^2 u
 \)
that satisfy the initial conditions
\[
    u_-(0,\la) = u_+(1,\la) = v_-^{[1]}(0,\la) = 0, \qquad
    u_-^{[1]}(0,\la) = u_+^{[1]}(1,\la) = v_-(0,\la) = 1.
\]
Observe that according to the definition of~$l_\si$ the equation
$l_\si(u) = \la^2 u$ is to be regarded as the first-order system
\[
    \frac{d}{dx}\binom{u}{u^{[1]}} =
        \begin{pmatrix} \si & 1\\ -\si^2-\la^2 & -\si
        \end{pmatrix} \binom{u}{u^{[1]}}.
\]
Since the entries of the above matrix are integrable, this system enjoys
the standard existence and uniqueness properties; in particular, the
solutions $u_\pm(\cdot,\la)$ and $v_-(\cdot,\la)$ are well defined for all
$\la\in\bC$.

Set $\Phi(\la):=\la u_-(1,\la)$ and $\Psi(\la):=u_+^{[1]}(0,\la)$;
then the numbers $\pm\la_n$ and $\pm\mu_n$ are zeros of $\Phi$ and
$\Psi$ respectively. $\Phi$ and $\Psi$ are respectively odd and
even entire functions of order~$1$ and hence can be represented by
their canonical Hadamard products~\cite{HMtwo}, namely,
\[
    \Phi(\la) = \la\prod_{n=1}^\infty \frac{\la_n^2-\la^2}{\pi^2 n^2}, \qquad
    \Psi(\la) = \prod_{n=1}^\infty \frac{\mu_n^2-\la^2}{\pi^2 (n-1/2)^2}.
\]
Moreover, it turns out that the norming constants
$\al_n=(\sqrt2\la_n\|u_-(\cdot,\la_n)\|)^{-2}$ and
$\be_n=(\sqrt2\|v_-(\cdot,\mu_n)\|)^{-2}$ can be expressed via the
functions~$\Phi$ and $\Psi$ only (cf.~\cite{GS,HMtwo}).

\begin{lemma}\label{lem:aln}
The norming constants~$\al_n$ and $\be_n$ satisfy the following equalities:
\begin{equation}\label{eq:alben}
    \al_n = \frac{\Psi(\la_n)}{\dot{\Phi}(\la_n)},\qquad
    \be_n =-\frac{\Phi(\mu_n)}{\dot{\Psi}(\mu_n)}.
\end{equation}
\end{lemma}

\begin{proof}
The Green function $G_{\mathrm D}(x,y,\la^2)$ of the
operator~$T_{\mathrm D}$ (i.e., the kernel of the resolvent
$(T_{\mathrm D}-\la^2)^{-1}$) equals
\[
    G_{\mathrm D}(x,y,\la^2) = \sum_{n=1}^{\infty}
        \frac{2\al_n\la_n^2 u_-(x,\la_n) u_-(y,\la_n)}{\la^2_n - \la^2}.
\]
On the other hand, we have
\[
    G_{\mathrm D}(x,y,\la^2) = \frac1{W(\la)} \left\{
        \begin{aligned}{u_-(x,\la) {u_+(y,\la)}},& \qquad
                        0\le x\le y\le1,\\
                    {u_-(y,\la) {u_+(x,\la)}},& \qquad
                        0\le y\le x\le1,
        \end{aligned} \right.
\]
where
     $W(\la):=u_+(x,\la)u^{[1]}_-(x,\la)
                -u_-(x,\la)u^{[1]}_+(x,\la)$
is the Wronskian of $u_+$ and $u_-$. The value of $W(\la)$ is
independent of $x\in[0,1]$; in particular, taking $x=1$ and $x=0$
we find that
\begin{equation}\label{eq:u+u-}
W(\la)\equiv -u_-(1,\la)\equiv u_+(0,\la).
\end{equation}

Equating the two expressions and comparing the residues at the poles
$\la=\la_n$, we find that
\[
    \al_n\la_n u_-(y,\la_n) =  \frac{u_+(y,\la_n)}{\dot{u}_-(1,\la_n)}.
\]
Observe that $u_-(\cdot,\la_n)$ and $u_+(\cdot,\la_n)$ are collinear and
hence
\[
    u_+(\cdot,\la_n)/u_-(\cdot,\la_n) =
    u_+^{[1]}(\cdot,\la_n)/u_-^{[1]}(\cdot,\la_n) =
    u_+^{[1]}(0,\la_n)=\Psi(\la_n);
\]
combining the above relations, we conclude that
\[
    \al_n = \frac{u_+(y,\la_n)}{u_-(y,\la_n)}
            \frac{1}{\la_n\dot{u}_-(1,\la_n)}
          = \frac{\Psi(\la_n)}{\dot{\Phi}(\la_n)}
\]
as claimed.

In a similar fashion we equate two expressions for the Green's
function~$G_{\mathrm N}(x,y,\la^2)$ of the operator $T_{\mathrm
N}$, namely,
\[
    \sum_{n=1}^\infty \frac{2\be_n v_-(x,\mu_n) v_-(y,\mu_n)}{\mu_n^2-\la^2}
    \equiv
    \frac1{W_1(\la)} \left\{
        \begin{aligned}{v_-(x,\la) {u_+(y,\la)}},& \qquad
                        0\le x\le y\le1,\\
                    {v_-(y,\la) {u_+(x,\la)}},& \qquad
                        0\le y\le x\le1.
        \end{aligned} \right.
\]
Here $W_1$ is the Wronskian of $u_+$ and $v_-$ and it is identically equal
to $-\Psi$, as follows from the equalities
\[
    W_1(\la):=u_+(x,\la)v^{[1]}_-(x,\la)-v_-(x,\la)u^{[1]}_+(x,\la)
        = -u^{[1]}_+(0,\la) = -\Psi(\la).
\]
Therefore we find that
\[
    \be_n =\frac{\mu_n u_+(y,\mu_n)}{v_-(y,\mu_n)\dot{\Psi}(\mu_n)}
        = \frac{\mu_n u_+(0,\mu_n)}{\dot{\Psi}(\mu_n)}
        =-\frac{\mu_n u_-(1,\mu_n)}{\dot{\Psi}(\mu_n)}
        =-\frac{\Phi(\mu_n)}{\dot{\Psi}(\mu_n)},
\]
where the second equality is obtained by taking $y=0$, while the third one
follows from~\eqref{eq:u+u-}. The lemma is proved.
\end{proof}

In what follows, we shall say that a function $f\in L_2(0,1)$ is \emph{odd}
(respectively \emph{even\/}) if $f(1-x)\equiv -f(x)$ (respectively, if
$f(1-x)\equiv f(x)$). Denote by $L_{2,\mathrm{o}}(0,1)$ and
$L_{2,\mathrm{e}}(0,1)$ the subspaces of $L_2(0,1)$ consisting of odd and
even functions respectively. We shall denote by $f_{\mathrm{o}}$ and
$f_{\mathrm{e}}$ respectively the odd and even parts of a function $f$;
obviously,
\[
    f_{\mathrm{o}}(x) = \tfrac12 \bigl[f(x)-f(1-x)\bigr], \qquad
    f_{\mathrm{e}}(x) = \tfrac12 \bigl[f(x)+f(1-x)\bigr].
\]

\begin{lemma}\label{lem:int}
The functions $\Phi$ and $\Psi$ admit the integral representations
\begin{equation}\label{eq:int}
\begin{aligned}
    \Phi(\la) &= \sin\la + \int_0^1 \phi(x) \sin [\la (1-2x)]\,dx,\\
    \Psi(\la) &= \cos\la + \int_0^1 \psi(x) \cos [\la (1-2x)]\,dx,
\end{aligned}
\end{equation}
in which $\phi\in L_{2,\mathrm{o}}(0,1)$ and $\psi\in
L_{2,\mathrm{e}}(0,1)$.
\end{lemma}

\begin{proof}
Using the technique of the transformation operators~\cite{HMtr}, $\Phi$ and
$\Psi$ can be shown to admit the integral representations of the form
\begin{align*}
    \Phi(\la) &= \sin\la + \int_0^1 \wt\phi(x) \sin \la x\,dx,\\
    \Psi(\la) &= \cos\la + \int_0^1 \wt\psi(x) \cos \la x\,dx
\end{align*}
with some $L_2$-functions $\wt\phi$ and $\wt\psi$; see detailed
derivation in~\cite{HMtwo}. Now we put
\begin{align*}
    \phi(x) &:= \begin{cases}
                 \wt\phi(1-2x) \quad &\text{if}\quad x\in[0,1/2]\\
                -\wt\phi(2x-1) \quad &\text{if}\quad x\in(1/2,1]
                \end{cases},\\
    \psi(x) &:= \begin{cases}
                 \wt\psi(1-2x) \qquad &\text{if}\quad x\in[0,1/2]\\
                 \wt\psi(2x-1) \qquad &\text{if}\quad x\in(1/2,1].
                \end{cases}
\end{align*}
It is easily seen that $\phi\in L_{2,\mathrm{o}}(0,1)$, $\psi\in
L_{2,\mathrm{e}}(0,1)$, and that equalities~\eqref{eq:int} hold. The lemma
is proved.
\end{proof}

The next lemma tells us that the values of $\Phi$ and $\Psi$ at the points
$\la_n$ and $\mu_n$ are expressed through sine and cosine Fourier
coefficients of some related functions. In the following $s_n(f)$ and
$c_n(f)$ will stand for respectively $n$-th sine and $n$-th cosine Fourier
coefficients of a function $f\in L_2(0,1)$; see~\eqref{eq:frtr} for exact
formulae. We also denote by $S$ the operator of multiplication by $1-2x$,
i.e., $(Sf)(x) = (1-2x)f(x)$.

\begin{lemma}\label{lem:sncn}
For an arbitrary $f\in L_2(0,1)$, the following equalities hold:
\begin{enumerate}
\item $\int_0^1 f(x) \sin[\la_n(1-2x)]\,dx =
            (-1)^{n+1}\bigl[s_{2n}(f) - \wt\la_n c_{2n}(Sf) +
            \wt\la_n^2 s_{2n}(f_1)\bigr]$;
\item $\int_0^1 f(x) \cos[\la_n(1-2x)]\,dx =
            (-1)^n\bigl[c_{2n}(f) + \wt\la_n s_{2n}(Sf) +
            \wt\la_n^2 c_{2n}(f_2)\bigr]$;
\item $\int_0^1 f(x) \sin[\mu_n(1-2x)]\,dx =
            (-1)^{n+1}\bigl[c_{2n-1}(f) + \wt\mu_n s_{2n-1}(Sf) +
            \wt\mu_n^2 c_{2n-1}(f_3)\bigr]$;
\item $\int_0^1 f(x) \cos[\mu_n(1-2x)]\,dx =
            (-1)^{n+1}\bigl[s_{2n-1}(f) - \wt\mu_n c_{2n-1}(Sf) +
            \wt\mu_n^2 s_{2n-1}(f_4)\bigr]$,
\end{enumerate}
where $f_j$, $j=1,2,3,4$, are some functions from $L_2(0,1)$.
\end{lemma}

\begin{proof}
We shall prove only part (1) as the other parts are established
analogously. Using the equality
\begin{align*}
    \sin [\la_n (1-2x)] &= (-1)^{n+1} \cos[\wt\la_n(1-2x)]\sin (2\pi nx)\\
        &\quad + (-1)^n  \sin[\wt\la_n(1-2x)]\cos (2\pi nx),
\end{align*}
the asymptotic relations
\[
    \sin t = t + O(t^3), \qquad \cos t = 1- t^2/2 + O(t^4), \quad t\to0,
\]
and the fact that $(\wt\la_n)\in\ell_2$, we find that
\[
    \int_0^1 f(x)\sin [\la_n (1-2x)]\,dx =
        (-1)^{n+1} \bigl[ s_{2n}(f) - \wt\la_n c_{2n}(Sf)
            + \wt\la_n^2 a_n\bigr]
\]
for some $\ell_2$-sequence $(a_n)$. Clearly, there exists a function
$f_1\in L_2(0,1)$ such that $a_n = s_{2n}(f_1)$ for all $n\in\bN$ and the
proof of part (1) is complete.
\end{proof}

\begin{remark}\label{rem:sncn}
Put
\begin{alignat*}{2}
    g_1&:= \si^* \tast Sf  + \si^* \tast(\si^* \hast f_1),&\qquad
    g_2&:= -\si^* \hast Sf + \si^* \hast(\si^* \tast f_2),\\
    g_3&:= \si^* \hast Sf  + \si^* \hast(\si^* \tast f_3),&\qquad
    g_4&:= -\si^* \tast Sf + \si^* \tast(\si^* \hast f_4) ,
\end{alignat*}
where the operations $\hast$ and $\tast$ are introduced in
Appendix~\ref{sec:sobolev}. By virtue of Lemma~\ref{lem:conv} we can
restate equalities (1)--(4) of the previous lemma as follows:
\begin{itemize}
\item[(1$'$)] $\int_0^1 f(x) \sin[\la_n(1-2x)]\,dx =
            (-1)^{n+1} s_{2n}(f+g_1) $;
\item[(2$'$)] $\int_0^1 f(x) \cos[\la_n(1-2x)]\,dx =
            (-1)^n c_{2n}(f+g_2)$;
\item[(3$'$)] $\int_0^1 f(x) \sin[\mu_n(1-2x)]\,dx =
            (-1)^{n+1} c_{2n-1}(f+g_3)$;
\item[(4$'$)] $\int_0^1 f(x) \cos[\mu_n(1-2x)]\,dx =
            (-1)^{n+1} s_{2n-1}(f+g_4)$.
\end{itemize}
If the functions $\si^*$ and $f$ belong to $W^s_2(0,1)$ for some
$s\in[0,1]$, then $Sf\in W^s_2(0,1)$ by Proposition~\ref{pro:interp}, and
thus Corollary~\ref{cor:conv} implies that the above functions $g_j$,
$j=1,2,3,4$, belong to $W^{2s}_2(0,1)$.
\end{remark}

Using~\eqref{eq:alben}, integral representations for $\Phi$ and $\Psi$, and
asymptotics of $\la_n$ and $\mu_n$, we can show that the norming constants
$\al_n$ and $\be_n$ obey the asymptotics $\al_n = 1 + \wt\al_n$ and $\be_n
= 1 +\wt\be_n$ with $\ell_2$-sequences $(\wt\al_n)$ and $(\wt\be_n)$. It
turns out that if the spectral data $(\la_n^2)$ and $(\mu_n^2)$ have better
asymptotics, then the functions $\phi$ and $\psi$ in~\eqref{eq:int} become
smoother, and the asymptotics of $\al_n$ and $\be_n$ refine.

\begin{lemma}\label{lem:phipsi} Assume that the numbers
$\wt\la_n:=\la_n-\pi n$ and $\wt\mu_n =\mu_n-\pi (n-1/2)$ are such that the
function $\si^*$ of~\eqref{eq:si*} belongs to $W^s_2(0,1)$ for some
$s\in[0,1]$. Then the functions $\phi$ and $\psi$ in integral
representation~\eqref{eq:int} of~$\Phi$ and~$\Psi$ have the form
\[
    \phi =-\si^*_{\mathrm{o}} + \phi_1, \qquad
    \psi = \si^*_{\mathrm{e}} + \psi_1,
\]
where $\phi_1$ and $\psi_1$ are respectively some odd and even functions
from $W^{2s}_2(0,1)$.
\end{lemma}

\begin{proof}
In virtue of Lemma~\ref{lem:int} the equality $\Phi(\la_n)=0$ can be recast
as
\[
    \sin\la_n + \int_0^1 \phi(x) \sin[\la_n(1-2x)]\,dx = 0.
\]
Observe that $\sin\la_n=(-1)^n\sin\wt\la_n$ and that $\sin\wt\la_n =
\wt\la_n + \wt\la_n^2 b_n$ for some $\ell_2$-sequence~$(b_n)$. Combining
this observation with Lemma~\ref{lem:sncn}, we arrive at the relation
\[
     \wt\la_n - s_{2n}(\phi) + \wt\la_n c_{2n}(S\phi)
                       +\wt\la_n^2 s_{2n}(\wh\phi)=0
\]
for some odd function $\wh\phi\in L_{2,\mathrm{o}}(0,1)$. Using
Lemma~\ref{lem:conv} and recalling that $\wt\la_n=-s_{2n}(\si^*) =
-s_{2n}(\si^*_{\mathrm{o}})$, we conclude that
\[
    \phi = -\si^*_{\mathrm{o}} - \si^*_{\mathrm{o}} \tast
    \bigl[(S\phi) - \si^*_{\mathrm{o}} \hast \wh\phi \bigr].
\]
In particular, $\phi\in W^s_2(0,1)$ by Corollary~\ref{cor:conv}, so that
the function $(S\phi) - \si^*_{\mathrm{o}} \hast \wh\phi$ belongs to
$W^s_2(0,1)$, and again by Corollary~\ref{cor:conv} we get
$\phi_1:=\phi+\si^*_{\mathrm{o}} = \si^*_{\mathrm{o}} \tast
    \bigl[\si^*_{\mathrm{o}} \hast \wh\phi -(S\phi)\bigr]\in W^{2s}_2(0,1)$.
The fact that $\phi_1\in L_{2,\mathrm{o}}(0,1)$ is obvious.

In a similar manner, using the relations $\Psi(\mu_n)=0$ and
$\mu_n = \pi (n-\tfrac12) + \wt\mu_n$ and Lemmata~\ref{lem:int}
and \ref{lem:sncn} we find that
\[
    \wt\mu_n - s_{2n-1}(\psi) + \wt\mu_n c_{2n-1}(S\psi) +
                \wt\mu_n^2 s_{2n-1}(\wh\psi) =0
\]
for some function $\wh\psi\in L_{2,\mathrm{e}}(0,1)$. Replicating the
above reasoning, we conclude that $\psi = \si^*_{\mathrm{e}} + \psi_1$
for some even function $\psi_1$ from $W^{2s}_2(0,1)$ as claimed. The
proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:albe}]
Formula~\eqref{eq:alben} implies that
\[
    \wt\al_n=\al_n-1 =
        \frac{\Psi(\la_n)-\dot{\Phi}(\la_n)}{\dot{\Phi}(\la_n)}, \qquad
    \wt\be_n=\be_n-1 =
        \frac{-\Phi(\mu_n)-\dot{\Psi}(\mu_n)}{\dot{\Psi}(\mu_n)}
\]
for all $n\in\bN$. According to Lemma~\ref{lem:int} we have
\begin{align*}
    \Psi(\la_n)-\dot{\Phi}(\la_n) &
            = \int_0^1 \th_1(x)\cos[\la_n(1-2x)]\,dx,\\
    -\Phi(\mu_n)-\dot{\Psi}(\mu_n) &
            = \int_0^1 \th_2(x)\sin[\mu_n(1-2x)]\,dx
\end{align*}
with $\th_1:= \psi - S\phi$ and $\th_2:=-\phi + S\psi$. Further, in view of
Lemma~\ref{lem:sncn} and Remark~\ref{rem:sncn},
\begin{align*}
    \int_0^1 \th_1(x) \cos[\la_n(1-2x)]\,dx
            &= (-1)^n c_{2n}(\th_1 + \wt\th_1),\\
    \int_0^1 \th_2(x) \sin[\mu_n(1-2x)]\,dx
            &= (-1)^{n+1} c_{2n-1}(\th_2 + \wt\th_2)
\end{align*}
with some functions $\wt\th_1$ and $\wt\th_2$ from $W^{2s}_2(0,1)$.

It follows from Lemma~\ref{lem:int} that
\begin{align*}
    \dot{\Phi}(\la_n) &= \cos\la_n
        + \int_0^1 (1-2x)\phi(x) \cos [\la_n (1-2x)]\,dx,\\
   -\dot{\Psi}(\mu_n) &= \sin\mu_n
        + \int_0^1 (1-2x)\psi(x) \sin [\mu_n (1-2x)]\,dx.
\end{align*}
We have $\cos\la_n = (-1)^n\cos\wt\la_n = (-1)^n (1 + \wt\la_n d_n)$ and
$\sin\mu_n = (-1)^{n+1} \cos\wt\mu_n =(-1)^{n+1} (1+ \wt\mu_n e_n)$ for
some $\ell_2$-sequences $(d_n)$ and $(e_n)$. Using Lemma~\ref{lem:sncn} and
Remark~\ref{rem:sncn}, we now conclude that
\[
    (-1)^n \dot{\Phi}(\la_n) = 1 + c_{2n}(g_1), \qquad
    (-1)^{n+1} \dot{\Psi}(\mu_n) = 1 + c_{2n-1}(g_2)
\]
for some functions $g_1$ and $g_2$ from $W^s_2(0,1)$. Since
$\dot{\Phi}(\la_n)\ne0$ and $\dot{\Psi}(\mu_n)\ne0$ for all $n\in\bN$,
Lemma~\ref{lem:wiener} implies that
\[
    \frac{(-1)^n}{\dot{\Phi}(\la_n)}     = 1 + c_{2n}(h_1), \qquad
    \frac{(-1)^{n+1}}{\dot{\Psi}(\mu_n)} = 1 + c_{2n-1}(h_2)
\]
for some functions $h_1$ and $h_2$ from $W^s_2(0,1)$.

We now combine the above relations to conclude that
\[
    \wt\al_n = c_{2n}(\th_1 + \wt\th_1)\bigl(1 + c_{2n}(h_1)\bigr), \qquad
    \wt\be_n = c_{2n-1}(\th_2 + \wt\th_2)\bigl(1 + c_{2n-1}(h_2)\bigr).
\]
It follows that $\ga = -\th_1+\th_2 + \wt\th$ for some $\wt\th\in
W^{2s}_2(0,1)$. Since by Lemma~\ref{lem:phipsi}
\begin{align*}
    -\th_1 + \th_2 = (S-I)(\phi+\psi)
        &= (S-I)(\si^*_{\mathrm{e}} -
            \si^*_{\mathrm{o}}) + (S-I)(\phi_1 + \psi_1)\\
        &= \ga^* + (S-I)(\phi_1 + \psi_1)
\end{align*}
and $(S-I)(\phi_1 + \psi_1)\in W^{2s}_2(0,1)$ by
Proposition~\ref{pro:interp}, we conclude that the function $\ga-\ga^*$ is
in $W^{2s}_2(0,1)$ as required. The theorem is proved.
\end{proof}

Observe that Theorem~\ref{thm:albe} gives necessary parts of
Theorems~\ref{thm:main2} and \ref{thm:main3}. Sufficient parts of these
theorems constitute the inverse spectral problem and are treated in the
next section.

\section{The inverse problem}\label{sec:inverse}

We start by recalling briefly the standard method of recovering
the potential of a Sturm--Liouville operator from the spectral
data---sequences of eigenvalues and the corresponding norming
constants. This method was suggested by Gelfand and Levitan
in~\cite{GL} for the case of regular (i.e., locally integrable)
potentials and was further developed in~\cite{HMinv} to cover
singular potentials from $W^{-1}_2(0,1)$.

Consider the functions
\begin{alignat*}{2}
    \om_1(x) &:= \sum_{n=1}^\infty
        \bigl[ \al_n \cos \la_n x - \cos(\pi nx)\bigr],& \qquad
                &x\in[0,2];\\
    \om_2(x) &:= \sum_{n=1}^\infty
        \bigl\{\be_n \cos \mu_n x - \cos[\pi (n-\tfrac12)x]\bigr\},& \qquad
                &x\in[0,2],
\end{alignat*}
which belong to $L_2(0,2)$ as soon as the functions $\si^*$ and $\ga$ are
in $L_2(0,1)$ (cf.~Lemmata~\ref{lem:om1} and \ref{lem:om2} below) and put
for $x,y\in[0,1]$
\begin{equation}\label{eq:f}
    f_j(x,y) := \om_j(|x-y|) + (-1)^j \om_j(x+y), \qquad j=1,2.
\end{equation}
We introduce an integral operator $F_j$ with kernel $f_j$; namely, $F_j$
acts in $L_2(0,1)$ according to the formula
\begin{equation}\label{eq:F}
    (F_j u)(x) := \int_0^1 f_j(x,y) u(y)\,dy.
\end{equation}

Let also $K_1$ and $K_2$ be the transformation operators for
$T_{\mathrm D}$ and $T_{\mathrm N}$ respectively. Recall that
$K_j$, $j=1,2$, is an integral operator with lower-triangular
kernel $k_j$, i.e., $k_j(x,y)=0$ a.e.~on the set $\{(x,y) \mid 0
\le x < t \le1\}$ and thus
\[
    (K_j u)(x) = \int_0^x k_j(x,y) u(y)\,dy, \qquad j=1,2.
\]
The operator $I+K_1$ transforms solutions of the unperturbed
equation $l_{0}(u)=\la^2 u$ (i.e., corresponding to $\si\equiv0$)
subject to the Dirichlet initial condition $u(0)=0$ into the
solutions of the equation $l_\si(u)=\la^2u$ subject to the
Dirichlet initial condition; the operator $I+K_2$ does the same
for the Neumann boundary condition at $x=0$.

Moreover, $f_j$ and $k_j$ are related through the so-called
Gelfand--Levitan--Marchenko (GLM) equation
\begin{equation}\label{eq:GLM}
    f_j(x,y) + k_j(x,y) + \int_0^x k_j(x,t)f_j(t,y)\,dt =0, \qquad
        0 \le y\le x \le 1.
\end{equation}
It is known~\cite{GKvolt,HMinv} that this GLM equation is naturally
related to the problem of factorisation of the operator $I+F_j$ in a
special manner and that~\eqref{eq:GLM} is uniquely soluble for~$k_j$ as
soon as the operator $I+F_j$ is (uniformly) positive. We note that
under properties (B1) and (B2) with $s=0$ the operator $I+F_1$ is
uniformly positive in $L_2(0,1)$, and the same is true of $I+F_2$ if
(C1) and (C2) hold with $s=0$ (see the details in~\cite{HMinv}).
Henceforth in both cases of interest the GLM equation possesses a
unique solution. Moreover, up to an additive constant $C_j$, the
primitive $\si$ of the potential $q$ equals
\begin{equation}\label{eq:si}
    \si(x) = (-1)^{j-1}2 \om_j(2x) -
        2\int_0^x k_j(x,t) f_j(t,x)\,dt + C_j.
\end{equation}
In what follows, we shall restrict ourselves to the case of the
operator $T_{\mathrm D}$; the proofs for the operator $T_{\mathrm
N}$ require only minor modifications.

In order to prove sufficiency part of Theorem~\ref{thm:main2} we need to
show first that the function $\om_1$ belongs to $W^s_2(0,1)$, then
establish some properties of the kernel $k_1$, and finally use
formula~\eqref{eq:si} to prove the inclusion $\si\in W^s_2(0,1)$.

\begin{lemma}\label{lem:om1}
Assume that the numbers $\wt\la_n$ and $\wt\al_n>-1$ are such that
there exist functions $g$ and $h$ in $W^s_2(0,1)$ with the property
that $\wt\la_n = s_{2n}(g)$ and $\wt\al_n=c_{2n}(h)$. Then the
function~$\om_1$ belongs to $W^s_2(0,2)$.
\end{lemma}

\begin{proof}
Observe first that, by the construction of $\si^*$ and $\ga$, we have
$\si^*_{\mathrm{o}}= g_{\mathrm{o}}$ and
$\ga_{\mathrm{e}}=h_{\mathrm{e}}$, whence $\si^*_{\mathrm{o}}\in
W^s_2(0,1)$ and $\ga_{\mathrm{e}}\in W^s_2(0,1)$. We write
\begin{align*}
    2\om_1(2x) &= 2\sum_{n=1}^\infty
        \bigl[(1+\wt\al_n)\cos(2\pi nx+2\wt\la_nx) - \cos(2\pi nx)\bigr]\\
    &= 2\sum_{n=1}^\infty \wt\al_n \cos(2\pi nx)
       + 2\sum_{n=1}^\infty (1+\wt\al_n)
                [\cos(2\wt\la_nx)-1] \cos(2\pi nx)\\
       & \quad -2\sum_{n=1}^\infty (1+\wt\al_n)
                \sin(2\wt\la_nx) \sin(2\pi nx)\\
        &=: -\ga_{\mathrm{e}}(x) + g_1(x) -  g_2(x),
\end{align*}
so that it remains to prove that the functions $g_1$ and $g_2$ belong
to $W^{s}_2(0,1)$.

Justification of the inclusions $g_1\in W^{s}_2(0,1)$ and $g_2\in
W^{s}_2(0,1)$ is similar, and we shall give it in detail only for the
function $g_1$. We have
\[
    g_1(x)=2\sum_{n=1}^\infty (1+\wt\al_n) \cos(2\pi nx)
        \sum_{k=1}^\infty (-1)^k\frac{(2\wt\la_nx)^{2k}}{(2k)!}.
\]
For $x\in[0,1]$, the estimate
\[
    \sum_{k=1}^\infty \frac{(2\wt\la_nx)^{2k}}{(2k)!}
        \le \cosh(2\wt\la_n)- 1 = \mathrm{O}(\wt\la_n^2)
\]
and the inclusion $(\wt\la_n)\in\ell_2$ imply that the above double
series for $g_1$ converges uniformly and absolutely. Changing the
summation order, we find that
\begin{equation}\label{eq:g1}
    g_1(x) = \sum_{k=1}^\infty \frac{(-1)^k2^{2k}}{(2k)!}x^{2k} h_k(x),
\end{equation}
where
\[
    h_k(x) := 2\sum_{n=1}^\infty (1+\wt\al_n)\,\wt\la_n^{2k}\cos(2\pi nx).
\]

Put $\tau:= \si^*_{\mathrm{o}}\tast \si^*_{\mathrm{o}}$; then by virtue
of Lemma~\ref{lem:conv} we find that
\[
    h_k = (\underbrace{\tau \oast \tau \oast \cdots
                    \oast \tau}_{k\ \text{times}}) +
        \ga_{\mathrm{e}} \oast (\underbrace{\tau \oast \tau \oast \cdots
                    \oast \tau}_{k\ \text{times}});
\]
see the definition of $\oast$ and $\tast$ in
Appendix~\ref{sec:sobolev}. Corollary~\ref{cor:conv} yields the
inclusion $h_k\in W^{s}_2(0,1)$; moreover, there exists a number
$C_1>0$ such that
\[
    \|h_k\|_{s} \le C_1 \bigl(1+\|\ga_\mathrm{e}\|_{0}\bigr)
        \ \|\si^*_{\mathrm{o}}\|^{2k}_{s}.
\]

Denote by $V$ the operator of multiplication by $x$; then $V^{2k}$ is
bounded both in $L_2(0,1)$ and in $W^1_2(0,1)$ and there exists
$C_2\ge1$ such that, for all $f\in W^1_2(0,1)$,
\[
    \|V^{2k}f\|_{0}\le \|f\|_{0}, \qquad
    \|V^{2k}f\|_{1} \le C_2 k \|f\|_{1}
\]
Interpolating between $W^1_2(0,1)$ and $L_2(0,1)$, we conclude that
$V^{2k}$ is bounded in every intermediate space $W^{s}_2(0,1)$ and
$\|V^{2k}f\|_{s} \le C_2 k^{s}\|f\|_{s}$ for all $f\in W^{s}_2(0,1)$.
Combining the above relations, we conclude that the series
in~\eqref{eq:g1} converges in~$W^{s}_2(0,1)$. Henceforth $g_1\in
W^{s}_2(0,1)$, which completes the proof.
\end{proof}

Denote by $\fA_s$ the set of all integral operators $K$ over~$(0,1)$,
whose kernels $k$ possess the following properties:
\begin{itemize}
\item [(1)] for every $x\in [0,1]$ the functions $k(x,\cdot)$ and
    $k(\cdot,x)$ belong to $W^s_2(0,1)$;
\item [(2)] the mappings
\[
    [0,1] \ni x \mapsto k(x,\cdot) \in W^s_2(0,1), \qquad
    [0,1] \ni x \mapsto k(\cdot,x) \in W^s_2(0,1)
\]
are continuous.
\end{itemize}

The results of~\cite{My} imply the following statement.

\begin{proposition}\label{pro:fact}
Assume that $F$ is an integral operator with kernel $f$ such that
$F\in\fA_s$, $s\in[0,\frac12)$, and $I+F>0$. Let $k$ be equal to a
solution of the corresponding GLM equation~\eqref{eq:GLM} in the domain
$0\le y\le x\le1$ and be zero in the domain $0\le x<y\le1$; then an
integral operator~$K$ with kernel $k$ also belongs to~$\fA_s$.
\end{proposition}

Let the assumptions of Lemma~\ref{lem:om1} hold for some $s\in
[0,\frac12)$, so that $\om_1\in W^s_2(0,1)$. Then the operator $F_1$ given
by~\eqref{eq:f}--\eqref{eq:F} belongs to $\fA_s$. Indeed, properties (1)
and (2) of the definition of $\fA_s$ for the kernel $f_1$ follow from the
fact that
\begin{itemize}
\item[(a)] the operator $P$ restricting a function on $\bR$
 onto $(0,1)$ is a bounded mapping from $W^s_2(\bR)$ into
 $W^s_2(0,1)$~\cite[Theorem~1.9.1]{LM};
\item[(b)] the translations $T_t f (\cdot) := f(\cdot+t)$,
    $t\in\bR$, form a $C_0$-group in $W^s_2(\bR)$ \cite{My}.
\end{itemize}

With these preliminaries in hand, we can complete the inverse spectral
analysis of Theorems~\ref{thm:main1}, \ref{thm:main2}, and \ref{thm:main3}.

\begin{proof}[Proof of Theorem~\ref{thm:main1}.]
The necessity part of the theorem follows from Theorem~A, hence we need to
prove only the sufficiency part, i.e., that properties (A1) and (A2) and
the inclusion $\si^*\in W^s_2(0,1)$ imply that $\si\in W^s_2(0,1)$.

Assume therefore that the sequences $(\la_n^2)$ and $(\mu_n^2)$ satisfy
properties (A1) and (A2). Applying the reconstruction procedure
explained above (and developed in detail for the case of singular
potentials from~$W^{-1}_2(0,1)$ in~\cite{HMtwo}), we find a unique
function $\si\in L_2(0,1)$ such that $\la_n^2$ and $\mu_n^2$ are
eigenvalues of the corresponding operators $T_{\mathrm D}$ and
$T_{\mathrm N}$. It remains to prove that the inclusion $\si^*\in
W^s_2(0,1)$ yields $\si\in W^s_2(0,1)$. We shall consider separately
the cases $s\in[0,\frac12)$, $s\in[\frac12,1)$, and $s=1$.

\textbf{Case 1: $s\in[0,\frac12)$.} By Theorem~\ref{thm:albe} the
function~$\ga$ belongs to~$W^s_2(0,1)$; hence by Lemma~\ref{lem:om1}
the function $\om_1$ is in $W^s_2(0,1)$ and, as explained above, the
integral operator~$F_1$ falls into the set~$\fA_s$. It follows from
Proposition~\ref{pro:fact} that the solution~$k_1$ of the GLM
equation~\eqref{eq:GLM} generates an integral operator $K_1$ that also
belongs to~$\fA_s$.

In view of formula~\eqref{eq:si} and the inclusion~$\om_1\in
W^s_2(0,1)$, the theorem will be proved as soon as we show that the
integral
 \(
    \int_0^x k_1(x,t) f_1(t,x)\,dt
 \)
defines a function from $W^s_2(0,1)$. Observe that $k_1(x,t)=0$
a.e.~for $0\le x < t\le1$, so that we can extend the range of
integration to $[0,1]$. Hence we put
\[
    \eta(x) := \int_0^1 k_1(x,t) f_1(t,x)\,dt
\]
and shall prove that $\eta \in W^s_2(0,1)$.

Recall \cite[Theorem~1.10.2]{LM} that one of the equivalent norms
in the space $W^s_2(0,1)$ is given by
\[
    \|\eta\|_{s} = \biggl( \|\eta\|^2_0
        + 2\int_0^1 \int_x^1
           \frac{|\eta(x)-\eta(y)|^2}{(x-y)^{1+2s}} \,dy\,dx\biggr)^{1/2}.
\]
Now we use the fact that the operators $K_1$ and $F_1$ belong to $\fA_s$.
In particular, there exists a constant $C_1>0$ such that
\[
    \max_{x\in [0,1]} \bigl( \|k_1(x,\cdot)\|^2_{s}+
        \|k_1(\cdot,x)\|^2_{s}+ \|f_1(x,\cdot)\|^2_{s}+
        \|f_1(\cdot,x)\|^2_{s}\bigr) \le C_1,
\]
and we use this inequality to derive the estimates
\[
    \|\eta\|^2_{0} \le \int_0^1 \Bigl(
        \int_0^1 |k_1(x,y)|^2\,dy \int_0^1 |f_1(y,x)|^2\,dy
            \Bigr) \,dx \le C_1^2
\]
and
\begin{align*}
    |\eta(x)-\eta(y)|^2 &\le
             2 \Bigl| \int_0^1 |k_1(x,t)-k_1(y,t)||f_1(t,x)|\,dt\Bigr|^2\\
    &\quad + 2 \Bigl| \int_0^1 |k_1(y,t)||f_1(t,x)-f_1(t,y)|\,dt\Bigr|^2\\
    &\le 2C_1 \Bigl( \int_0^1 |k_1(x,t)-k_1(y,t)|^2\,dt
        + \int_0^1 |f_1(t,x)-f_1(t,y)|^2\,dt\Bigr).
\end{align*}
It follows that
\[
    \|\eta\|^2_{s} \le C_1^2 +
        4C_1 \int_0^1 \Bigl[\|k_1(\cdot,t)\|^2_{s} +
            \|f_1(t,\cdot)\|^2_{s}\Bigr]\,dt \le 5C_1^2,
\]
so that $\eta\in W^s_2(0,1)$, and the proof for the case~$s\in[0,\frac12)$
is complete.

\textbf{Case 2: $s\in[\frac12,1)$.} Case 1 applied to the exponent
$\tfrac{s}2$ gives $\si\in W^{s/2}_2(0,1)$, so that $\si-\si^*\in
W^s_2(0,1)$ by Theorem~A. Since by assumption $\si^*\in W^s_2(0,1)$, we
have $\si\in W^s_2(0,1)$ as required.

\textbf{Case 3: $s=1$.} We again use the bootstrap method: first, by
Case~2, $\si\in W^t_2(0,1)$ for any $t\in[0,1)$, e.g., $t=\frac12$; then,
by Theorem~A, $\si-\si^*\in W^1_2(0,1)$. This inclusion yields $\si\in
W^1_2(0,1)$, and the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main2}.]
If $\si\in W^s_2(0,1)$, then $\si^*\in W^s_2(0,1)$ by Theorem~A, hence
$\ga\in W^s_2(0,1)$ by Theorem~\ref{thm:albe}. Properties (B1) and (B2)
then obviously hold as, by the construction of $\si^*$ and $\ga$, $\wt\la_n
= s_{2n}(-\si^*)$ and $\wt\al_n = c_{2n}(-\ga)$.

Conversely, assume (B1) and (B2). According to the results
of~\cite{HMinv}, there exists a unique Sturm--Liouville operator
$T_{\mathrm D}$ with potential $q$ from $W^{-1}_2(0,1)$ such that
$\la_n^2$ and $\al_n$ are eigenvalues and norming constants of
$T_{\mathrm D}$. We need to prove that validity of (B1) and (B2)
implies that the recovered potential $q$ belongs in fact to
$W^{s-1}_2(0,1)$ (i.e., that any primitive $\si$ of $q$ belongs to
$W^s_2(0,1)$). In fact, under (B1) and (B2) Lemma~\ref{lem:om1} yields
$\om_1\in W^s_2(0,2)$, and it remains to observe that the proof of
Theorem~\ref{thm:main1} derives from this the inclusion $\si\in
W^s_2(0,1)$. The proof is complete.
\end{proof}

Proof of Theorem~\ref{thm:main3} is completely analogous; the only
essential reservation is that Lemma~\ref{lem:om1} should be replaced with
the following its counterpart (we leave both proofs to the reader):

\begin{lemma}\label{lem:om2}
Assume that the numbers $\wt\mu_n$ and $\wt\be_n>-1$ are such that
there exist functions $g$ and $h$ in $W^s_2(0,1)$ with the property
that $\wt\mu_n = s_{2n-1}(g)$ and $\wt\be_n=c_{2n-1}(h)$. Then the
function~$\om_2$ belongs to $W^s_2(0,1)$.
\end{lemma}

\medskip
\textbf{Acknowledgements.} {R.~H. gratefully acknowledges the
financial support of the Alexander von Humboldt Foundation and
thanks the Institute for Applied Mathematics of Bonn University
for the warm hospitality.}

\appendix

\section{Sobolev spaces $W^s_2(0,1)$ and all that}\label{sec:sobolev}

We recall here some facts about the Sobolev spaces $W^s_2(0,1)$
and Fourier coefficients of functions from these spaces. For
details, we refer the reader to~\cite[Ch.~1]{LM}.

By definition, the space $W^0_2(0,1)$ coincides with $L_2(0,1)$ and the
norm~$\|\cdot\|_0$ in $W^0_2(0,1)$ is just the $L_2(0,1)$-norm. The
Sobolev space $W^2_2(0,1)$ consists of all functions $f$ in $L_2(0,1)$,
whose distributional derivatives $f'$ and $f''$ also fall
into~$L_2(0,1)$. Being endowed with the norm
\[
    \|f\|_2 := \bigl(\|f\|_0^2 + \|f'\|_0^2 + \|f''\|_0^2\bigr)^{1/2},
\]
$W_2^2(0,1)$ becomes a Hilbert space.

Now we interpolate~\cite[CH.~1.2.1]{LM} between $W^2_2(0,1)$ and
$W^0_2(0,1)$ to get the intermediate spaces $W^s_2(0,1)$ with
norms $\|\cdot\|_s$ for $s\in(0,2)$; namely,
\[
    W^{2s}_2(0,1) :=
        [W^2_2(0,1),W^0_2(0,1)]_{1-s}.
\]
The norms $\|\cdot\|_s$ are nondecreasing with $s\in[0,2]$, i.e.,
if $s<t$ and $f\in W^t_2(0,1)$, then $\|f\|_s\le\|f\|_t$. Since by
construction the spaces $W_2^s(0,1)$ form an interpolation scale,
the general interpolation theorem~\cite[Theorem~1.5.1]{LM} implies
the following interpolation property for operators in these
spaces.

\begin{proposition}\label{pro:interp}
Assume that an operator $T$ acts boundedly in $W^s_2(0,1)$ and
$W^r_2(0,1)$, $s<r$. Then $T$ is a bounded operator in
$W^{ts+(1-t)r}_2(0,1)$ for every $t\in[0,1]$; moreover,
$\|T\|_{ts+(1-t)r}\le \|T\|_s^t \|T\|_r^{1-t}$.
\end{proposition}

Proposition~\ref{pro:interp} yields boundedness in every $W^s_2(0,1)$,
$s\in [0,2]$, of the operators $R$ and $V$ given by $Rf(x)=f(1-x)$ and
$Vf(x)=xf(x)$.

For an arbitrary $f\in L_2(0,1)$ and an arbitrary $\la\in\bC$, we put
\begin{equation}\label{eq:frtr}
    s_\la(f):= \int_0^1 f(x) \sin(\pi\la x)\,dx, \qquad
    c_\la(f):= \int_0^1 f(x) \cos(\pi\la x)\,dx.
\end{equation}
As usual, $\ast$ denotes the convolution operation on~$(0,1)$,
i.e.,
\[
    (f\ast g)(x) := \int_0^x f(x-t) g(t)\,dt.
\]
We shall also introduce the following shorthand notations:
\begin{align*}
    (f \oast g)(x) &:= \tfrac12 \bigl[ R\, (R f\ast g + f\ast Rg)
                + f\ast g + Rf \ast Rg\bigr],\\
    (f \hast g)(x) &:= \tfrac12 \bigl[ R\, (R f\ast g + f\ast Rg)
                - f\ast g - Rf \ast Rg\bigr],\\
    (f \tast g)(x) &:= \tfrac12 \bigl[ R\, (R f\ast g - f\ast Rg)
                + f\ast g - Rf \ast Rg\bigr]
\end{align*}
(where, as earlier, $R$ stands for the reflection operator, $(Rf)(x) =
f(1-x)$). The operations $\oast, \hast$, and $\tast$ play the same role for
the sine and cosine Fourier transform on $(0,1)$ as the usual convolution
for the Fourier transform on the whole line. Namely, these operations have
the following properties.

\begin{lemma}\label{lem:conv}
For arbitrary $f,g\in L_2(0,1)$ and $\la\in\bC$ the following equalities
hold:
\[
    c_\la(f)c_\la(g) = c_\la(f\oast g), \qquad
    s_\la(f)s_\la(g) = c_\la(f\hast g), \qquad
    s_\la(f)c_\la(g) = s_\la(f\tast g).
\]
\end{lemma}

\begin{proof}
We shall prove only the first equality since the other two can be treated
analogously. We have
\[
    2 c_\la(f)c_\la(g) = \int_0^1 \int_0^1 f(x) g(t)
            \{\cos[\pi \la(x-t)] + \cos[\pi \la(x+t)]\}\,dx dt,
\]
and simple calculations lead to
\begin{align*}
    \int_0^1\int_0^1 &f(x) g(t)\cos\pi \la(x-t)\, dx dt \\
        &= \int_0^1
        \Bigl( \int_0^{1-s} f(s+t) g(t)\,dt
                + \int_0^{1-s} f(t) g(s+t)\,dt
        \Bigr)\cos(\pi \la s)\,ds, \\
    \int_0^1\int_0^1 &f(x) g(t)\cos[\pi \la(x+t)]\, dx dt \\
        &= \int_0^1
        \Bigl( \int_0^{s} f(s-t) g(t)\,dt
                + \int_0^{s} f(1-t) g(1-s+t)\,dt
        \Bigr)\cos\pi \la s\,ds.
\end{align*}
Taking into account the relations
\[
    \int_0^{1-s} f(s+t) g(t)\,dt = R(Rf\ast g)(s), \qquad
    \int_0^{s} f(1-t) g(1-s+t)\,dt = Rf \ast Rg,
\]
we get $c_\la(f) c_\la(g) = c_\la(f\oast g)$ as stated. The lemma is
proved.
\end{proof}

It is well known that convolution accumulates smoothness; the
precise meaning of this statement is as follows.

\begin{proposition}\label{pro:conv}
Assume that $s,t\in[0,1]$ and that $f\in W^s_2(0,1)$ and $g\in
W^t_2(0,1)$ are arbitrary. Then the function $h:= f\ast g$ belongs
to $W^{s+t}_2(0,1)$ and, moreover, there exists $C>0$ independent
of $f$ and $g$ such that $\|h\|_{s+t}\le C \|f\|_s\|g\|_t$.
\end{proposition}

Proof of this proposition is based on interpolation between the
extreme cases $s,t=0,1$, which are handled with directly.

Combining Proposition~\ref{pro:conv} with the fact that the
operator $R$ is bounded in the spaces~$W^s_2(0,1)$ for all
$s\in[0,1]$, we arrive at the following conclusion.

\begin{corollary}\label{cor:conv}
Assume that $s,t\in[0,1]$ and that $f\in W^s_2(0,1)$, $g\in W^t_2(0,1)$.
Then the functions $f\oast g, f\hast g$, and $f\tast g$ belong to
$W^{s+t}_2(0,1)$ and, moreover, there exists a number $C>0$ independent of
$f$ and $g$ such that
\[
    \max\bigl\{\|f\oast g\|_{s+t}, \|f\hast g\|_{s+t},
    \|f\tast g\|_{s+t} \bigr\} \le C \|f\|_s
    \|g\|_t.
\]
\end{corollary}

The following lemma is an analogue of the well-known Wiener lemma.

\begin{lemma}\label{lem:wiener}
Assume that $f\in W^s_2(0,1)$, where $s\in[0,1]$. If $1+ c_n(f)\ne0$ for
all $n\in\bN$, then there exists a function $g\in W^s_2(0,1)$ such that
\[
    \bigl(1 + c_n(f)\bigr)^{-1} = 1 + c_n(g), \qquad n\in\bN.
\]
\end{lemma}

\begin{proof}
We start the proof with some auxiliary constructions. Denote by $C$ the
operator that acts from $L_2(0,1)/\mathbb{C}$ into $\ell_2$ according
to the formula
\[
    Cf :=   \bigl( c_n(f)\bigr)_{n\in\bN}.
\]
This operator is isomorphic; we put
\[
    {\cW}^s:= \{ Cf \mid f\in W^s_2(0,1)\}
\]
and endow the linear space $\cW^s$ with the norm
\[
    \|x\|_{\cW^s}:= \|C^{-1}\,x\|_{s}.
\]
In view of Lemma~\ref{lem:conv} and Corollary~\ref{cor:conv}, the
elementwise multiplication $(xy)_n:=x_ny_n$ is a continuous
operation in~$\cW^s$. We adjoin to $\cW^s$ the unit element $e$
(with components $e_n$ equal to $1$) and denote the resulting
unital algebra by $\wh\cW^s$. By a well-known
result~\cite[Theorem~10.2]{Ru} one can introduce an equivalent
norm in $\wh\cW^s$ under which $\wh\cW^s$ becomes a commutative
Banach algebra.

Assume now that the assumptions of the lemma hold and denote by $x$ an
element of $\wh\cW^s$ with components $x_n:= 1+c_n(f)$. We shall prove
below that $x$ is invertible in $\wh\cW^s$; it then follows that $x^{-1} =
e + y$ for some $y\in \cW^s$ as required.

It is well known~\cite[Theorem~11.5]{Ru} that the element $x$ is
invertible in the unital Banach algebra~$\wh\cW^s$ if and only if
$x$ does not belong to any maximal ideal of $\wh\cW^s$. Assume, on
the contrary, that there exists a maximal ideal $\fm$ of
$\wh\cW^s$ containing $x$. Since $\wh\cW^s$ contains all finite
sequences and none of $x_n$ vanishes, $\fm$ also contains all
finite sequences. Finite sequences form a dense subset of $\cW^s$
because the set of all trigonometric polynomials in $\cos\pi nx$
is dense in $W^s_2(0,1)$. Recalling that maximal ideals are
closed, we conclude that $\cW^s\subset\fm$. Next we observe that
$\cW^s$ is a proper subset of $\fm$ (e.g., $x$ belongs to
$\fm\setminus\cW^s$) and that $\cW^s$ has codimension~$1$ in
$\wh\cW^s$. Henceforth $\fm = \wh\cW^s$, which contradicts our
assumption that $\fm$ is a maximal ideal of $\wh\cW^s$. As a
result, $x$ is not contained in any maximal ideal of $\wh\cW^s$
and thus is invertible in~$\wh\cW^s$. The lemma is proved.
\end{proof}