text:
\begin{document}

\title[Noncommutative Fourier Series on $\mathbb{Z}^2\backslash SE(2)$]{Noncommutative Fourier Series on $\mathbb{Z}^2\backslash SE(2)$}

\author[A. Ghaani Farashahi]{Arash Ghaani Farashahi$^*$}

\address{Laboratory for Computational Sensing and Robotics (LCSR), Whiting School of Engineering, Johns Hopkins University, Baltimore, Maryland, United States.}
\email{arash.ghaanifarashahi@jhu.edu}
\email{ghaanifarashahi@outlook.com}

\author[G.S. Chirikjian]{Gregory S. Chirikjian}

\address{Laboratory for Computational Sensing and Robotics (LCSR), Whiting School of Engineering, Johns Hopkins University, Baltimore, Maryland, United States.}
\email{gregc@jhu.edu}
\email{gchirik@gmail.com}

\subjclass[2010]{Primary 43A30, 43A85, Secondary 20H15, 43A10, 43A15, 43A20, 68T40, 74E15, 82D25.}

\date{}

\keywords{Special Euclidean group, noncommutative Fourier series, right coset space, discrete (crystallographic) subgroup.}
\thanks{$^*$Corresponding author}
\thanks{E-mail addresses: arash.ghaanifarashahi@jhu.edu (Arash Ghaani Farashahi) and gregc@jhu.edu (Gregory S. Chirikjian)}

\begin{abstract}
This paper begins with a systematic study of noncommutative Fourier series on $\mathbb{Z}^2\backslash SE(2)$. Let $\mu$ be the finite $SE(2)$-invariant measure on the right coset space $\mathbb{Z}^2\backslash SE(2)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given (discrete) basis of the Hilbert function space $L^2(\mathbb{Z}^2\backslash SE(2),\mu)$. We then investigate the presented theory for the case of a canonical basis originated from a fundamental domain of $\mathbb{Z}^2$ in $SE(2)$. The paper is concluded by some convolution results.

\end{abstract}

\maketitle

\section{{\bf Introduction}}

The 2D special Euclidean group, usually denoted as $SE(2)$, is one of the simplest examples of a noncommutative, noncompact real finite dimensional Lie group. It is classically used for modeling of the group of planar rigid body motions, and plays significant roles in planar geometric analysis, quantum mechanics, and coherent state analysis \cite{Kisil.book, Pere.Coh}. Over the last decades, different applied and computational aspects of Fourier analysis on the unimodular group $SE(2)$ have achieved significant popularity in non-classical application areas, see \cite{PIb2, PIb4, PI5, PI6}.

The right coset space of the orthogonal lattice $\mathbb{Z}^2$ in $SE(2)$, that is $\mathbb{Z}^2\backslash SE(2)$, appears as the configuration space in many recent applications in computational science and engineering including computer vision, robotics, mathematical crystallography, computational biology, and material science \cite{PI1, PI3, PI6, PI7, PI8,  PI.Kya.2000, Kya.PI.2000, etal.PI.2006}.
Invoking the algebraic structure of $SE(2)$, the subgroup $\mathbb{Z}^2$ is not normal in $SE(2)$ and hence the right coset space $\mathbb{Z}^2\backslash SE(2)$ is not a group.
Therefore, the classical notion of non-commutative Fourier transform and thus Fourier type expansion on this compact 3-dimensional manifold are meaningless. However, there are still both algebraic and geometric structures on it, namely  the transitive right action of the Lie group $SE(2)$ on the manifold $\mathbb{Z}^2\backslash SE(2)$, which makes the right coset space into a homogeneous space.
In a nutshell, homogeneous spaces are group-like structures with many interesting applications in mathematical physics, differential geometry, geometric analysis, and coherent state (covariant) transforms, see \cite{FollH, Kisil.book, 50}.
Fourier expansions on homogeneous spaces of compact groups, coset spaces of compact groups, have been studied at depth in \cite{AGHF.GGD, AGHF.AMP, AGHF.CMB}.
This theory is strongly benefited from the compactness assumption about the group which is not as the case for $SE(2)$ and hence a different approach is required for the right coset space $\mathbb{Z}^2\backslash SE(2)$.

The present article focuses on constructive aspects of Fourier expansions on the right coset space $\mathbb{Z}^2\backslash SE(2)$. Our aim is to further develop a non-commutative Fourier-type reconstruction on the homogeneous space $\mathbb{Z}^2\backslash SE(2)$, based on the algebraic structure of the coset space, which has not been studied as extensively as the Fourier expansions on the group $SE(2)$.
We shall also address analytic aspects of
the proposed expansion as a constructive approximation, using tools
from abstract harmonic analysis and representation theory.

This article contains 4 sections. Section
2 is devoted to fixing notation and gives a
brief summary of noncommutative Fourier analysis on
the unimodular group $SE(2)$ and classical harmonic analysis on
the right coset space $\mathbb{Z}^2\backslash SE(2)$. In Section
3, we present the general theory of noncommutative Fourier series
for a class of $L^2$-functions defined on the right coset space $\mathbb{Z}^2\backslash SE(2)$,
for any given (discrete) basis of the Hilbert function space $L^2(\mathbb{Z}^2\backslash SE(2),\mu)$, where $\mu$ is the finite $SE(2)$-invariant measure on $\mathbb{Z}^2\backslash SE(2)$, normalized with respect to Weil's formula. We then investigate the presented theory for the case of a basis originated from a fundamental domain of $\mathbb{Z}^2$ in $SE(2)$. As the main result we present
a constructive series which can be viewed as reconstruction formula for a class of functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$, including matrix elements of the noncommutative Fourier transform
on $SE(2)$.
The main advantage of the presented noncommutative Fourier type series which is benefited from the algebraic structure of the right coset space
is the discrete sampling from the interval $[0,\infty)$.
We then present the notion of convolution of functions on $SE(2)$ by functions on $\mathbb{Z}^2\backslash SE(2)$, also called as the canonical module action of the Banach algebra $L^1(SE(2))$ on the Banach function space $L^1(\mathbb{Z}^2\backslash SE(2),\mu)$. We shall also study different analytic aspects of Fourier series for approximating the convolution functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$. As applications for convolution functions series we conclude the paper by some Plancherel type formulas for a class of $L^2$-functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$.

\newpage
\section{{\bf Preliminaries and Notation}}

Throughout this section we shall present preliminaries and the notation.
The 2D special Euclidean group, $SE(2)$, is the semidirect product of $\mathbb{R}^2$
with the 2D special orthogonal group $SO(2)$. That is,
\[
SE(2)=\mathbb{R}^2\rtimes SO(2)=SO(2)\ltimes\mathbb{R}^2.
\]
We denote elements $g\in SE(2)$ as $g=(\mathbf{x},\mathbf{R})$ where $\mathbf{x}\in\mathbb{R}^2$ and $\mathbf{R}\in SO(2)$. For any $g=(\mathbf{x},\mathbf{R})$ and
$g'=(\mathbf{x}',\mathbf{R}')\in SE(2)$ the group law is written as
\[
g\circ g'=(\mathbf{x}+\mathbf{R}\mathbf{x}',\mathbf{RR}'),
\]
and
\[
g^{-1}=(-\mathbf{R}^T\mathbf{x},\mathbf{R}^T)=(-\mathbf{R}^{-1}\mathbf{x},\mathbf{R}^{-1}).
\]
Alternatively, the special Euclidean group $SE(2)$ can be viewed as the set of matrices of the form
\begin{equation}
g(x_1,x_2,\theta):=\left(\begin{array}{ccc}
\cos\theta & -\sin\theta & x_1 \\
\sin\theta & \cos\theta & x_2 \\
0 & 0 & 1
\end{array}\right),
\end{equation}
with $\theta\in(0,2\pi]$ and $x_1,x_2\in\mathbb{R}$, or
\begin{equation}
g(a,\phi,\theta):=\left(\begin{array}{ccc}
\cos\theta & -\sin\theta & a\cos\phi \\
\sin\theta & \cos\theta & a\sin\phi \\
0 & 0 & 1
\end{array}\right).
\end{equation}
In semidirect product notation, we may denote it as $(\mathbf{x},\mathbf{R}_\theta)$, where $\mathbf{x}=(x_1,x_2)^T$ and
\[
\mathbf{R}_\theta=\left(\begin{array}{cc}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array}\right).
\]
The special Euclidean group $SE(2)$ is a unimodular group with the normalized Haar measure given by
\[
dg=\frac{1}{4\pi^2}dx_1dx_2d\theta=\frac{1}{4\pi^2}adad\phi d\theta.
\]
The convolution of functions $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$, is given by
\[
(f_1\star f_2)(h)=\int_{SE(2)}f_1(g)f_2(g^{-1}\circ h)dg,
\]
for $h\in SE(2)$.

The 2D special Euclidean group is a solvable Lie group, and hence classical methods for constructing unitary representations of solvable lie groups could be applied, see \cite{M.Sugiu}.
The set consists of all equivalence classes of continuous unitary irreducible representations of Euclidean group $SE(2)$, that is $\widehat{SE(2)}$, also called as spectrum of $SE(2)$, is given by
\begin{equation}
\widehat{SE(2)}:=\left\{U_p:p> 0\right\}\cup\left\{\chi_n:n\in\mathbb{Z}\right\}.
\end{equation}
For each $p>0$, the irreducible representation $U_p:SE(2)\to\mathcal{U}(L^2(\mathbb{S}^1))$ is given by $g\to U_p(g)=U(g,p)$ with
\begin{equation}
[U_p(g)\varphi](\mathbf{u})=[U(g,p)\varphi](\mathbf{u}):=e^{-\ii p\langle\mathbf{u},\mathbf{t}\rangle}\cdot\varphi(\mathbf{R}_\theta^T\mathbf{u}),
\end{equation}
for all $g=(\mathbf{t},\mathbf{R}_\theta)\in SE(2)$, $\varphi\in L^2(\mathbb{S}^1)$, and $\mathbf{u}\in\mathbb{S}^1$.\\
The Fourier transform of each $f\in L^1(SE(2))$ at $p>0$ is given by
\begin{equation}\label{hatp}
\widehat{f}(p):=\int_{SE(2)}f(g)U_p(g^{-1})dg.
\end{equation}
The corresponding convolution formula for functions $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ at $p>0$ is
\begin{equation}\label{hatconvp}
\widehat{(f_1\star f_2)}(p)=\widehat{f_2}(p)\widehat{f_1}(p).
\end{equation}
For each $n\in\mathbb{Z}$, the one-dimensional representation $\chi_n:SE(2)\to\mathbb{T}=\mathcal{U}(\mathbb{C})$ is given by
\begin{equation}
\chi_n(g):=e^{in\theta},
\end{equation}
for all $g=(\mathbf{t},\mathbf{R}_\theta)\in SE(2)$.
The Fourier transform of each $f\in L^1(SE(2))$ at $n\in\mathbb{Z}$ is given by
\begin{equation}
\widehat{f}[n]:=\int_{SE(2)}f(g)\overline{\chi_n(g)}dg.
\end{equation}
The corresponding convolution formula for functions $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ at $n\in\mathbb{Z}$ is
\begin{equation}
\widehat{(f_1\star f_2)}[n]=\widehat{f_1}[n]\widehat{f_2}[n].
\end{equation}
The standard orthonormal basis for the Hilbert function space $L^2(\mathbb{S}^1)$ is given by
the family $\mathcal{B}=\{\mathbf{e}_k:k\in\mathbb{Z}\}$, where for each $k\in\mathbb{Z}$, $\mathbf{e}_k:\mathbb{S}^1\to\mathbb{C}$
is given by
\[
\mathbf{e}_k(\mathbf{u}_\alpha):=\mathbf{e}_k(\cos\alpha,\sin\alpha)=e^{\ii k\alpha}.
\]
For each $g\in SE(2)$ and $p>0$, the matrix elements of the operator $U_p(g)$ with respect to the basis $\mathcal{B}$, are expressed as
\[
\mathrm{u}_{mn}(g,p)=\langle U_p(g)\mathbf{e}_m,\mathbf{e}_n\rangle_{L^2(\mathbb{S}^1)},
\]
for each $m,n\in\mathbb{Z}$, where
\[
\langle u,v\rangle=\frac{1}{2\pi}\int_0^{2\pi}u(\alpha)\overline{v(\alpha)}d\alpha,
\]
for all $u,v\in L^2(\mathbb{S}^1)$.
When expressed in this way, (\ref{hatp}) becomes an infinite-dimensional matrix and (\ref{hatconvp}) becomes a matrix product.

Using polar coordinates, the matrix elements satisfy the following closed form
\begin{equation}\label{upmat}
\mathrm{u}_{mn}(g(a,\phi,\theta),p)=\ii^{m-n}e^{-\ii(m\theta+(n-m)\phi)}J_{m-n}(pa),
\end{equation}
for $m,n\in\mathbb{Z}$, where $J_{q}$ is the $q$-th order Bessel function of the first kind, for each $q\in\mathbb{Z}$.

The Bessel functions satisfy the following orthogonality relation
\[
\int_0^\infty J_q(\alpha r)J_q(\beta r)rdr=\frac{\delta(\alpha-\beta)}{\alpha},
\]
for each $\alpha,\beta>0$ and $q\in\mathbb{Z}$, where $\delta$ is the Dirac delta function.

The classical Fourier Plancherel/Parseval formula on the group $SE(2)$ is given by
\[
\int_{SE(2)}|f(g)|^2dg=\int_0^\infty\|\widehat{f}(p)\|_{\rm HS}^2pdp,
\]
and also the noncommutative Fourier reconstruction formula is given by
\[
f(g)=\int_0^\infty\mathrm{tr}\left[\widehat{f}(p)U_p(g)\right]pdp,
\]
for $f\in L^1\cap L^2(SE(2))$ and $g\in SE(2)$, see \cite{M.Sugiu}.
It is worthwhile to mention that one-dimensional irreducible representations of $SE(2)$
play no part in the Fourier reconstruction integral on $SE(2)$.

Let $\Gamma$ be a discrete co-compact subgroup of $SE(2)$
with the counting measure as the Haar measure. Consequentially, the right coset space $\Gamma\backslash SE(2):=\{\Gamma g:g\in SE(2)\}$ is compact as a homogeneous space which the Lie group $SE(2)$ acts on from the right. Each discrete co-compact subgroup $\Gamma$ of $SE(2)$ is isomorphic to a group of the form $\mathbb{Z}^2\rtimes\mathbb{P}$, where $\mathbb{P}$ is a finite subgroup of $SO(2)$. Such a $\Gamma$ must necessarily belongs to one of the five handedness preserving wallpaper groups. Therefore, harmonic analysis on $\Gamma\backslash SE(2)$ is straightforward after $\mathbb{Z}^2\backslash SE(2)$.

The classical aspects of abstract harmonic analysis on locally compact homogeneous spaces are quite well studied by several authors, see \cite{FollH, AGHF.CJM, AGHF.IntJM, AGHF.JAusMS, AGHF.IJM, HR1, HR2, 50} and classical references therein.
The function space $\mathcal{C}(\Gamma\backslash SE(2))$, that is the set of all continuous functions on $\Gamma\backslash SE(2)$, consists of all functions
$\widetilde{f}$, where
$f\in\mathcal{C}_c(SE(2))$ and
\begin{equation}\label{5.1}
\widetilde{f}(\Gamma g):=\sum_{\gamma\in\Gamma}f(\gamma\circ g),
\end{equation}
for all $g\in SE(2)$.

Let $\mu$ be a Radon measure on the right coset space $\Gamma\backslash SE(2)$ and $h\in SE(2)$. The right translation $\mu_h$ of $\mu$ is defined by $\mu_h(E):=\mu(E\circ h)$, for all Borel subsets $E$ of $\Gamma\backslash SE(2)$, where
\[
E\circ h:=\{\Gamma g\circ h:\Gamma g\in E\}.
\]
The measure $\mu$ is called $SE(2)$-invariant if $\mu_h=\mu$, for all $h\in SE(2)$.
Since $SE(2)$ is unimodular, $\Gamma$ is discrete and $\Gamma\backslash SE(2)$ is compact, the right coset space $\Gamma\backslash SE(2)$ has a finite $SE(2)$-invariant measure $\mu$, which satisfies the following Weil's formula
\begin{equation}\label{TH.m}
\int_{\Gamma\backslash SE(2)}\widetilde{f}(\Gamma g)d\mu(\Gamma g)=\int_{SE(2)}f(g)dg,
\end{equation}
and hence the linear map $f\mapsto\widetilde{f}$ is norm-decreasing from $L^1(SE(2))$ into $L^1(\Gamma\backslash SE(2),\mu)$, that is
$$\|\widetilde{f}\|_{L^1(\Gamma\backslash SE(2),\mu)}\le \|f\|_{L^1(SE(2))},$$
for all $f\in L^1(SE(2))$, see \cite{FollH, HR1, 50}.

\newpage
\section{\bf Noncommutative Fourier Series on $\mathbb{Z}^2\backslash SE(2)$}

This section is dedicated to present the notion of noncommutative Fourier series for
square integrable functions on the right coset space of the orthogonal lattice $\Gamma:=\mathbb{Z}^2$, as the simplest wallpaper (crystallographic) group, in $SE(2)$. Throughout this section we assume that $\mu$ is the finite $SE(2)$-invariant measure on the right coset space $\Gamma\backslash SE(2)$ which is normalized with respect to Weil's formula (\ref{TH.m}).

First, we need some preliminary results concerning $L^2$-function space on the right coset space $\Gamma\backslash SE(2)$.

\begin{proposition}\label{TL2}
{\it Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
We then have $f\in L^2(SE(2))$ with
\[
\|f\|_{L^2(SE(2))}\le\|\widetilde{|f|}\|_{L^2(\Gamma\backslash SE(2),\mu)}.
\]
}\end{proposition}
\begin{proof}
Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$. We then claim that $f\in L^2(SE(2))$. To see this, we first note that
\[
\sum_{\gamma\in\Gamma}|f(\gamma\circ g)|^2\le\left(\sum_{\gamma\in\Gamma}|f(\gamma\circ g)|\right)^2,
\]
for $g\in SE(2)$. Then, using Weil's formula, we get
\begin{align*}
\|f\|_{L^2(SE(2))}^2&=\int_{SE(2)}|f(g)|^2dg=\int_{\Gamma\backslash SE(2)}\sum_{\gamma\in\Gamma}|f(\gamma\circ g)|^2d\mu(\Gamma g)\\&\le\int_{\Gamma\backslash SE(2)}\left(\sum_{\gamma\in\Gamma}|f(\gamma\circ g)|\right)^2d\mu(\Gamma g)
=\int_{\Gamma\backslash SE(2)}\left(\sum_{\gamma\in\Gamma}|f|(\gamma\circ g)\right)^2d\mu(\Gamma g)
\\&=\int_{\Gamma\backslash SE(2)}\widetilde{|f|}(\Gamma g)^2d\mu(\Gamma g)=\|\widetilde{|f|}\|^2_{L^2(\Gamma\backslash SE(2),\mu)},
\end{align*}
which implies that $f\in L^2(SE(2))$.
\end{proof}

\begin{corollary}
{\it Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
We then have $\widetilde{f}\in L^2(\Gamma\backslash SE(2),\mu)$ with
\[
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}\le\|\widetilde{|f|}\|_{L^2(\Gamma\backslash SE(2),\mu)}.
\]
}\end{corollary}
\begin{proof}
Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
Then, for each $g\in SE(2)$, we have
\begin{align*}
|\widetilde{f}(\Gamma g)|=\left|\sum_{\gamma\in\Gamma}f(\gamma\circ g)\right|
\le\sum_{\gamma\in\Gamma}|f(\gamma\circ g)|=\widetilde{|f|}(\Gamma g).
\end{align*}
Hence, we get
\begin{align*}
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2
&=\int_{\Gamma\backslash SE(2)}|\widetilde{f}(\Gamma g)|^2d\mu(\Gamma g)
\le\int_{\Gamma\backslash SE(2)}\widetilde{|f|}(\Gamma g)^2d\mu(\Gamma g)
=\|\widetilde{|f|}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2.
\end{align*}
\end{proof}

We then present the following constructive Fourier type approximation for $L^2$-functions on the right coset space $\Gamma\backslash SE(2)$.

\begin{theorem}\label{MainRecTh0}
Let $\Gamma:=\mathbb{Z}^2$ and $\mathcal{E}(\Gamma):=\{\psi_\ell:\Gamma\backslash SE(2)\to\mathbb{C}\ |\ \ell\in\mathbb{I}\}$ be a (discrete) orthonormal basis for the Hilbert function space $L^2(\Gamma\backslash SE(2),\mu)$. Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$. We then have
\begin{equation}\label{FourierInvQ0}
\widetilde{f}(\Gamma h)=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h),
\end{equation}
for $h\in SE(2)$, where
\[
Q^\ell(p)_{mn}:=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\ell(\Gamma g)}dg,\hspace{1cm}\widehat{f}(p)_{nm}=\langle\widehat{f}(p)\mathbf{e}_n,\mathbf{e}_m\rangle,
\]
for $p>0$, $\ell\in\mathbb{I}$, and $m,n\in\mathbb{Z}$.
\end{theorem}
\begin{proof}
Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
Then, $\psi:=\widetilde{f}$ is in $L^2(\Gamma\backslash SE(2),\mu)$ as well.
Since $\{\psi_\ell:\ell\in\mathbb{I}\}$ is an orthonormal basis, we get
\begin{equation}\label{cldecG}
\psi=\sum_{\ell\in\mathbb{I}}\langle\psi,\psi_\ell\rangle\psi_\ell,
\end{equation}
which implies that
\begin{equation}\label{cldec}
\psi(\Gamma h)=\sum_{\ell\in\mathbb{I}}\langle\psi,\psi_\ell\rangle\psi_\ell(\Gamma h),
\end{equation}
for $h\in SE(2)$. Invoking Proposition \ref{TL2}, we get $f\in L^2(SE(2))$. Thus, we have
\[
f(g)=\int_0^\infty\mathrm{tr}[\widehat{f}(p)U_p(g)]pdp,
\]
for $g\in SE(2)$.

Let $\ell\in\mathbb{I}$ be given. Therefore, using Weil's formula, we get
\begin{align*}
\langle\psi,\psi_\ell\rangle&=\int_{\Gamma\backslash SE(2)}\psi(\Gamma g)
\overline{\psi_\ell(\Gamma g)}d\mu(\Gamma g)=\int_{\Gamma\backslash SE(2)}\widetilde{f}(\Gamma g)
\overline{\psi_\ell(\Gamma g)}d\mu(\Gamma g)
\\&=\int_{SE(2)}f(g)\overline{\psi_\ell(\Gamma g)}dg=\int_{SE(2)}\left(\int_0^\infty\mathrm{tr}[\widehat{f}(p)U_p(g)]pdp\right)\overline{\psi_\ell(\Gamma g)}dg
\\&=\int_0^\infty\left(\int_{SE(2)}\mathrm{tr}[\widehat{f}(p)U_p(g)]\overline{\psi_\ell(\Gamma g)}dg\right)pdp.
\end{align*}
Also, we have
\begin{align*}
\mathrm{tr}[\widehat{f}(p)U_p(g)]&=\sum_{m\in\mathbb{Z}}\langle\widehat{f}(p)U_p(g)\mathbf{e}_m,\mathbf{e}_m\rangle=\sum_{m\in\mathbb{Z}}\left\langle\widehat{f}(p)\left(\sum_{n\in\mathbb{Z}}\langle U_p(g)\mathbf{e}_m,\mathbf{e}_n\rangle\mathbf{e}_n\right),\mathbf{e}_m\right\rangle
\\&=\sum_{m\in\mathbb{Z}}\left\langle\left(\sum_{n\in\mathbb{Z}}\langle U_p(g)\mathbf{e}_m,\mathbf{e}_n\rangle\widehat{f}(p)\mathbf{e}_n\right),\mathbf{e}_m\right\rangle=\sum_{m\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\langle U_p(g)\mathbf{e}_m,\mathbf{e}_n\rangle\langle\widehat{f}(p)\mathbf{e}_n,\mathbf{e}_m\rangle
\\&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\widehat{f}(p)_{nm}\mathrm{u}_{mn}(g,p).
\end{align*}
We then have
\begin{align*}
\int_{SE(2)}\mathrm{tr}[\widehat{f}(p)U_p(g)]\overline{\psi_\ell(\Gamma g)}dg
&=\int_{SE(2)}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\widehat{f}(p)_{nm}\mathrm{u}_{mn}(g,p)\right)\overline{\psi_\ell(\Gamma g)}dg
\\&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\widehat{f}(p)_{nm}
\left(\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\ell(\Gamma g)}dg\right)
\\&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\widehat{f}(p)_{nm}Q^\ell(p)_{mn}.
\end{align*}
Therefore, we get
\begin{equation}\label{cl}
\langle\psi,\psi_\ell\rangle=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\left(\int_0^\infty\widehat{f}(p)_{nm}Q^\ell(p)_{mn}pdp\right).
\end{equation}
for all $\ell\in\mathbb{I}$. Applying (\ref{cl}) in (\ref{cldec}) we get
\begin{align*}
\widetilde{f}(\Gamma h)&=\psi(\Gamma h)
=\sum_{\ell\in\mathbb{I}}\langle\psi,\psi_\ell\rangle\psi_\ell(\Gamma h)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h).
\end{align*}
\end{proof}

\begin{corollary}
{\it  Let $\Gamma:=\mathbb{Z}^2$ and $\mathcal{E}(\Gamma):=\{\psi_\ell:\Gamma\backslash SE(2)\to\mathbb{C}\ |\ \ell\in\mathbb{I}\}$ be a (discrete) orthonormal basis for the Hilbert function space $L^2(\Gamma\backslash SE(2),\mu)$. Let $f:SE(2)\to\mathbb{C}$ be a continuous function supported in the fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$. We then have
\begin{equation}
f(h)=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h),
\end{equation}
for $h\in F\subseteq SE(2)$, where for $p>0$, $\ell\in\mathbb{I}$, and $m,n\in\mathbb{Z}$ we have
\[
Q^\ell(p)_{mn}:=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\ell(\Gamma g)}dg.
\]
}\end{corollary}

One of the classical ways to construct an orthogonal basis for the Hilbert function space $L^2(\mathbb{Z}^2\backslash SE(2),\mu)$ is by employing a fixed fundamental domain of $\mathbb{Z}^2$ in $SE(2)$. From now on, we shall apply the most canonical fundamental domain of $\mathbb{Z}^2$ in $SE(2)$.
Let $\Omega=[0,1)^2\times[0,2\pi)$. Then, $F_{\mathbb{Z}^2\backslash SE(2)}:=\Omega$ is a fundamental domain for $\mathbb{Z}^2$ in $SE(2)$\footnote{i.e. $SE(2)=\mathbb{Z}^2\circ F_{\mathbb{Z}^2\backslash SE(2)}$}.

For each integral vector $\mathbf{k}=(k_1,k_2,k_3)^T\in\mathbb{Z}^3$, define the function
$\phi_\mathbf{k}:\Omega\to\mathbb{C}$ by
\begin{equation}
\phi_\mathbf{k}(x,y,\theta):=\exp\left(2\pi \ii(k_1x+k_2y)\right)\exp(\ii k_3\theta),
\end{equation}
for all $(x,y,\theta)\in\Omega$.

\begin{proposition}
{\it Let $\Omega=[0,1)^2\times[0,2\pi)$. Then, $\mathcal{E}:=\{\phi_\mathbf{k}:\mathbf{k}\in\mathbb{Z}^3\}$ is an orthonormal basis for the Hilbert function space $L^2(\Omega,(2\pi)^{-1}dxdyd\theta)$.
}\end{proposition}
\begin{proof}
Let $\mathbf{n},\mathbf{m}\in\mathbb{Z}^3$ be integral vectors.
We then can write
\begin{align*}
\langle\phi_{\mathbf{n}},\phi_{\mathbf{m}}\rangle_{L^2(\Omega)}
&=\frac{1}{2\pi}\int_0^1\int_0^1\int_0^{2\pi}\phi_{\mathbf{n}}(x,y,\theta)\overline{\phi_{\mathbf{m}}(x,y,\theta)}dxdyd\theta
\\&=\frac{1}{2\pi}\int_0^1\int_0^1\int_0^{2\pi}\exp\left(2\pi \ii((n_1-m_1)x+(n_2-m_2)y)\right)\exp(\ii (n_3-m_3)\theta)dxdyd\theta
\\&=\left(\int_0^1\int_0^1\exp\left(2\pi \ii((n_1-m_1)x+(n_2-m_2)y)\right)dxdy\right)\left(\frac{1}{2\pi}\int_0^{2\pi}\exp(\ii (n_3-m_3)\theta)d\theta\right)
\\&=\left(\int_0^1\exp\left(2\pi \ii(n_1-m_1)x\right)dxdy\right)\left(\int_0^1\exp\left(2\pi \ii(n_2-m_2)y\right)dy\right)
\left(\frac{1}{2\pi}\int_0^{2\pi}\exp(\ii (n_3-m_3)\theta)d\theta\right)
\\&=\delta_{n_1,m_1}\cdot\delta_{n_2,m_2}\cdot\delta_{n_3,m_3}=\delta_{\mathbf{n},\mathbf{m}}.
\end{align*}
Also, completeness of the family $\mathcal{E}$ in $L^2(\Omega)$ is straightforward.
\end{proof}

For each $\mathbf{k}\in\mathbb{Z}^3$, one can consider the function $\phi_\mathbf{k}$
as a function on the right coset space $\Gamma\backslash SE(2)$. From, now on we shall denote these functions by $\psi_\mathbf{k}$. In details, we have
\begin{equation}\label{MainBasisFun}
\psi_\mathbf{k}(\Gamma g(x,y,\theta)):=\phi_\mathbf{k}(x,y,\theta)=\exp\left(2\pi \ii(k_1x+k_2y)\right)\exp(\ii k_3\theta),
\end{equation}
for all $(x,y,\theta)\in\Omega$ and $\mathbf{k}:=(k_1,k_2,k_3)^T\in\mathbb{Z}^3$.

We then conclude the following results.

\begin{theorem}
Let $\mu$ be the normalized $SE(2)$-invariant measure on the right coset space $\mathbb{Z}^2\backslash SE(2)$ and $\Omega:=[0,1)^2\times[0,2\pi)$. We then have
\begin{enumerate}
\item The Hilbert function spaces $L^2(\mathbb{Z}^2\backslash SE(2),\mu)$ and $L^2(\Omega,(2\pi)^{-1}dxdyd\theta)$ are isometric isomorphic.
\item The family $\mathcal{E}(\Gamma):=\{\psi_\mathbf{k}:\mathbf{k}\in\mathbb{Z}^3\}$ is an orthonormal basis for the Hilbert function space $L^2(\mathbb{Z}^2\backslash SE(2),\mu)$.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) The mapping $H:L^2(\mathbb{Z}^2\backslash SE(2),\mu)\to L^2(\Omega,(2\pi)^{-1}dxdyd\theta)$ given by $\psi\mapsto H\psi$, where
\[
H\psi(x,y,\theta):=\psi(\mathbb{Z}^2g(x,y,\theta)),
\]
for all $\psi\in L^2(\mathbb{Z}^2\backslash SE(2),\mu)$ and $(x,y,\theta)\in\Omega$, is a linear unitary operator, i.e. it is an isometric isomorphism of Hilbert spaces.

(2) It is straightforward from (1).
\end{proof}

We hereby continue by computing matrix elements $Q^\mathbf{k}(p)_{mn}$. To this end, we first need the following lemma.

\begin{lemma}\label{Qalt}
Let $j\in\mathbb{Z}$, $x,y\in\mathbb{R}$ and $r>0$. We then have
\begin{equation}
\int_{0}^{2\pi}e^{\ii j\phi}e^{-2\pi\ii r(x\cos\phi+y\sin\phi)}d\phi=(2\pi)\ii^{-j}J_{j}\left(2\pi r\sqrt{x^2+y^2}\right)e^{\ii j\Phi(x,y)},
\end{equation}
where $0\le\Phi(x,y)<2\pi$ is given by
\begin{equation}\label{polxy}
\frac{x}{\sqrt{x^2+y^2}}=\cos\Phi(x,y),\hspace{2cm}\frac{y}{\sqrt{x^2+y^2}}=\sin\Phi(x,y).
\end{equation}
\end{lemma}
\begin{proof}
Let $j\in\mathbb{Z}$, $x,y\in\mathbb{R}$ and $r>0$. Let $0\le\Phi(x,y)<2\pi$
be given by (\ref{polxy}). Hence, using Equation (4.31) of \cite{PI5}, we get
\[
e^{-2\pi\ii r(x\cos\phi+y\sin\phi)}=\sum_{\ell\in\mathbb{Z}}\ii^{-\ell}
J_\ell\left(2\pi r\sqrt{x^2+y^2}\right)e^{-\ii\ell\phi}e^{i\ell\Phi(x,y)}.
\]
We then can write
\begin{align*}
\int_{0}^{2\pi}e^{\ii j\phi}e^{-2\pi\ii r(x\cos\phi+y\sin\phi)}d\phi
&=\int_{0}^{2\pi}e^{\ii j\phi}\left(\sum_{\ell\in\mathbb{Z}}\ii^{-\ell}
J_\ell\left(2\pi r\sqrt{x^2+y^2}\right)e^{-\ii\ell\phi}e^{i\ell\Phi(x,y)}\right)d\phi
\\&=\sum_{\ell\in\mathbb{Z}}\ii^{-\ell}
J_\ell\left(2\pi r\sqrt{x^2+y^2}\right)\cdot e^{i\ell\Phi(x,y)}\cdot\left(\int_{0}^{2\pi}e^{\ii j\phi}e^{-\ii\ell\phi}d\phi\right)
\\&=(2\pi)\sum_{\ell\in\mathbb{Z}}\ii^{-\ell}
J_\ell\left(2\pi r\sqrt{x^2+y^2}\right)e^{\ii\ell\Phi(x,y)}\delta_{j,\ell}
=(2\pi)\ii^{-j}J_{j}\left(2\pi r\sqrt{x^2+y^2}\right)e^{\ii j\Phi(x,y)}.
\end{align*}
\end{proof}

Next results present explicit formulas for matrix elements $Q^\mathbf{k}(p)_{mn}$.

\begin{theorem}\label{Qnm}
Let $p>0$, $n,m\in\mathbb{Z}$, and $\mathbf{k}=(k_1,k_2,k_3)^T\in\mathbb{Z}^3$ with $k_1,k_2\not=0$. We then have
\[
Q^\mathbf{k}(p)_{mn}=\frac{\delta_{k_3,-m}\cdot e^{\ii(m-n)\Phi(k_1,k_2)}}{2\pi\sqrt{k_1^2+k_2^2}}\cdot\delta\left(p-2\pi\sqrt{k_1^2+k_2^2}\right),
\]
where $0\le\Phi(k_1,k_2)<2\pi$ is given by $(k_1,k_2)=\sqrt{k_1^2+k_2^2}(\cos\Phi(k_1,k_2),\sin\Phi(k_1,k_2))$.
\end{theorem}
\begin{proof}
Let $n,m\in\mathbb{Z}$. Let $\mathbf{k}=(k_1,k_2,k_3)^T\in\mathbb{Z}^3$ be an integral vector with $k_1,k_2\not=0$ and $p>0$. Using Lemma \ref{Qalt} with $j:=m-n$, we can write
\begin{align*}
Q^\mathbf{k}(p)_{mn}
&=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\mathbf{k}(\Gamma g)}dg
\\&=\frac{1}{(2\pi)^2}\int_0^{2\pi}\int_0^\infty\int_0^{2\pi}\mathrm{u}_{mn}(g(r,\phi,\theta),p)\overline{\psi_\mathbf{k}(\Gamma g(r,\phi,\theta))} d\theta rdrd\phi
\\&=\frac{\ii^{m-n}}{(2\pi)^2}\int_0^{2\pi}\int_0^\infty\int_0^{2\pi}e^{-\ii(m\theta+(n-m)\phi)}J_{m-n}(pr)e^{-2\pi\ii(k_1r\cos\phi+k_2r\sin\phi)}e^{-\ii k_3\theta}d\theta rdrd\phi
\\&=\frac{\ii^{m-n}}{(2\pi)^2}\left(\int_0^{2\pi}e^{-\ii(k_3+m)\theta}d\theta\right)\left(\int_0^\infty\int_0^{2\pi}e^{-\ii(n-m)\phi}J_{m-n}(pr)e^{-2\pi\ii(k_1r\cos\phi+k_2r\sin\phi)}rdrd\phi\right)
\\&=\frac{\ii^{m-n}\delta_{k_3,-m}}{2\pi}\left(\int_0^\infty\int_0^{2\pi}e^{-\ii(n-m)\phi}J_{m-n}(pr)e^{-2\pi\ii(k_1r\cos\phi+k_2r\sin\phi)}rdrd\phi\right)
\\&=\frac{\ii^{m-n}\delta_{k_3,-m}}{2\pi}\left(\int_0^\infty J_{m-n}(pr)\left(\int_0^{2\pi}e^{\ii(m-n)\phi}e^{-2\pi\ii(k_1r\cos\phi+k_2r\sin\phi)}d\phi\right)rdr\right)
\\&=\delta_{k_3,-m}e^{\ii(m-n)\Phi(k_1,k_2)}\cdot\left(\int_0^\infty J_{m-n}(pr)J_{m-n}\left(2\pi r\sqrt{k_1^2+k_2^2}\right)rdr\right)
\\&=\frac{\delta_{k_3,-m}e^{\ii(m-n)\Phi(k_1,k_2)}}{2\pi\sqrt{k_1^2+k_2^2}}\delta\left(p-2\pi\sqrt{k_1^2+k_2^2}\right).
\end{align*}
\end{proof}

\begin{lemma}
Let $0\not=\alpha\in\mathbb{R}$. We then have
\begin{equation}\label{J0}
\int_0^\infty J_0(\alpha r)rdr=\frac{\delta(\alpha)}{\alpha}.
\end{equation}
\end{lemma}
\begin{proof}
Let $0\not=\alpha\in\mathbb{R}$.
Let $\mathbf{x}\in\mathbb{R}^2$ with $\mathbf{x}=(\alpha\cos\phi,\alpha\sin\phi)$ where $0<\phi\le2\pi$. Then, using Equation (4.31) of \cite{PI5}, we have
\begin{align*}
\delta(\mathbf{x})&=\frac{1}{(2\pi)^2}\int_{\mathbb{R}^2}e^{\ii\langle\bom,\mathbf{x}\rangle}d\bom
\\&=\frac{1}{(2\pi)^2}\int_{0}^{2\pi}\int_0^{\infty}\left(\sum_{k\in\mathbb{Z}}\ii^{k}J_k(\alpha r)e^{\ii k\theta}e^{-\ii k\phi}\right)rdrd\theta
\\&=\frac{1}{(2\pi)^2}\sum_{k\in\mathbb{Z}}\ii^{k}\left(\int_0^{\infty}J_k(\alpha r)rdr\right)\left(\int_{0}^{2\pi}e^{\ii k\theta}d\theta\right)e^{-\ii k\phi}
\\&=\frac{1}{2\pi}\sum_{k\in\mathbb{Z}}\ii^{k}\left(\int_0^{\infty}J_k(\alpha r)rdr\right)\delta_{0,k}e^{-\ii k\phi}=\frac{1}{2\pi}\int_0^\infty J_0(\alpha r)rdr.
\end{align*}
Since we have $\delta(\mathbf{x})=\frac{1}{2\pi}\frac{\delta(\alpha)}{\alpha}$,
we conclude (\ref{J0}).
\end{proof}

\begin{theorem}\label{Qnm00}
Let $p>0$, $n,m,k_3\in\mathbb{Z}$, and $\mathbf{k}:=(0,0,k_3)^T\in\mathbb{Z}^3$. We then have
\[
Q^\mathbf{k}(p)_{mn}=\delta_{k_3,-m}\delta_{n,m}
\frac{\delta\left(p\right)}{p}.
\]
\end{theorem}
\begin{proof}
Let $p>0$, $n,m,k_3\in\mathbb{Z}$, and $\mathbf{k}:=(0,0,k_3)^T\in\mathbb{Z}^3$.
Then, using (\ref{J0}), we have
\begin{align*}
Q^\mathbf{k}(p)_{mn}&=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\mathbf{k}(\Gamma g)}dg
\\&=\frac{1}{(2\pi)^2}\int_0^{2\pi}\int_0^\infty\int_0^{2\pi}\mathrm{u}_{mn}(g(r,\phi,\theta),p)\overline{\psi_\mathbf{k}(\Gamma g(r,\phi,\theta))} d\theta rdrd\phi
\\&=\frac{\ii^{m-n}}{(2\pi)^2}\int_0^{2\pi}\int_0^\infty\int_0^{2\pi}e^{-\ii(m\theta+(n-m)\phi)}J_{m-n}(pr)e^{-\ii k_3\theta}d\theta rdrd\phi
\\&=\frac{\ii^{m-n}}{(2\pi)^2}\left(\int_0^{2\pi}e^{-\ii(k_3+m)\theta}d\theta\right)\left(\int_0^\infty\int_0^{2\pi}e^{-\ii(n-m)\phi}J_{m-n}(pr)rdrd\phi\right)
\\&=\frac{\ii^{m-n}\delta_{k_3,-m}}{2\pi}\left(\int_0^{2\pi}e^{-\ii(n-m)\phi}d\phi\right)\left(\int_0^\infty J_{m-n}(pr)rdr\right)
\\&=\ii^{m-n}\delta_{k_3,-m}\delta_{n,m}\left(\int_0^\infty J_{m-n}(pr)rdrd\phi\right)=\ii^{m-n}\delta_{k_3,-m}\delta_{n,m}
\frac{\delta\left(p\right)}{p}=\delta_{k_3,-m}\delta_{n,m}
\frac{\delta\left(p\right)}{p}.
\end{align*}
\end{proof}

\begin{proposition}\label{1D}
{\it Let $f\in L^1(SE(2))$ and $n,m\in\mathbb{Z}$. We then have
\begin{equation}
\lim_{p\to 0^+}\widehat{f}(p)_{nm}=\delta_{n,m}\widehat{f}[-m]=\delta_{n,m}\widehat{f}[-n].
\end{equation}
}\end{proposition}
\begin{proof}
Let $f\in L^1(SE(2))$ and $n,m\in\mathbb{Z}$. Then, for each $p>0$, we have
\begin{align*}
\widehat{f}(p)_{nm}&=\langle\widehat{f}(p)\mathbf{e}_n,\mathbf{e}_m\rangle
=\int_{SE(2)}f(g)\langle U_p(g^{-1})\mathbf{e}_n,\mathbf{e}_m\rangle dg
\\&=\int_{SE(2)}f(g)\langle U_p(g)^*\mathbf{e}_n,\mathbf{e}_m\rangle dg
=\int_{SE(2)}f(g)\langle \mathbf{e}_n,U_p(g)\mathbf{e}_m\rangle dg
\\&=\int_{SE(2)}f(g)\overline{\langle U_p(g)\mathbf{e}_m,\mathbf{e}_n\rangle} dg
=\int_{SE(2)}f(g)\overline{\mathrm{u}_{mn}(g)} dg
\\&=\int_{SE(2)}f(g(a,\phi,\theta))\overline{\mathrm{u}_{mn}(g(a,\phi,\theta))}dg(a,\phi,\theta)
\\&=\ii^{n-m}\int_{SE(2)}f(g(a,\phi,\theta))e^{\ii(m\theta+(n-m)\phi)}J_{m-n}(pa)dg(a,\phi,\theta).
\end{align*}
We then have
\begin{align*}
\lim_{p\to 0^+}\widehat{f}(p)_{nm}&=\ii^{n-m}\cdot\lim_{p\to 0^+}\int_{SE(2)}f(g(a,\phi,\theta))e^{\ii(m\theta+(n-m)\phi)}J_{m-n}(pa)dg(a,\phi,\theta)
\\&=\ii^{n-m}\int_{SE(2)}f(g(a,\phi,\theta))e^{\ii(m\theta+(n-m)\phi)}\left(\lim_{p\to 0^+}J_{m-n}(pa)\right)dg(a,\phi,\theta)
\\&=\ii^{n-m}J_{m-n}(0)\int_{SE(2)}f(g(a,\phi,\theta))e^{\ii(m\theta+(n-m)\phi)}dg(a,\phi,\theta)
\\&=\delta_{n,m}\int_{SE(2)}f(g(a,\phi,\theta))e^{\ii m\theta}dg(a,\phi,\theta)
\\&=\delta_{n,m}\int_{SE(2)}f(g)\overline{\chi_{-m}(g)}dg=\delta_{n,m}\widehat{f}[-m]=\delta_{n,m}\widehat{f}[-n].
\end{align*}
\end{proof}

For each $k_1,k_2\in\mathbb{Z}$, let $\rho(k_1,k_2):=\sqrt{k_1^2+k_2^2}$ and
$0\le \Phi(k_1,k_2)<2\pi$ be given by
\[
k_1=\rho(k_1,k_2)\cos\Phi(k_1,k_2),\hspace{1cm}k_2=\rho(k_1,k_2)\sin\Phi(k_1,k_2).
\]

We then present the following closed form characterization for coefficients, if the basis functions are given by (\ref{MainBasisFun}).

\begin{theorem}\label{mainCoef}
 Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$, and $\psi:=\widetilde{f}$.
\begin{enumerate}
\item For any integral vector $\mathbf{k}:=(k_1,k_2,k_3)^T\in\mathbb{Z}^3$ with $k_1,k_2\not=0$, we have
\begin{equation}\label{kkk}
\langle\psi,\psi_\mathbf{k}\rangle=\sum_{n\in\mathbb{Z}}e^{-\ii(n+k_3)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{n,-k_3}.
\end{equation}
\item For any integral vector $\mathbf{k}:=(0,0,k_3)^T\in\mathbb{Z}^3$ with $k_3\in\mathbb{Z}$, we have
\begin{equation}\label{00k}
\langle\psi,\psi_\mathbf{k}\rangle=\widehat{f}[k_3].
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
(1) Let $\mathbf{k}=(k_1,k_2,k_3)^T\in\mathbb{Z}^3$ be an integral vector with $k_1,k_2\not=0$, and $n,m\in\mathbb{Z}$. Using Theorem \ref{Qnm}, we have
\[
Q^\mathbf{k}(p)_{mn}=\frac{\delta_{k_3,-m}e^{\ii(m-n)\Phi(k_1,k_2)}}{2\pi\rho(k_1,k_2)}\delta\left(p-2\pi\rho(k_1,k_2)\right).
\]
Thus, we get
\begin{align*}
\int_0^\infty\widehat{f}(p)_{nm}Q^\mathbf{k}(p)_{mn}pdp&=\frac{\delta_{k_3,-m}e^{\ii(m-n)\Phi(k_1,k_2)}}{2\pi\rho(k_1,k_2)}\int_0^\infty\widehat{f}(p)_{nm}\delta\left(p-2\pi\rho(k_1,k_2)\right)pdp
\\&=\delta_{k_3,-m}e^{\ii(m-n)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{nm}.
\end{align*}
Hence, we have
\begin{align*}
\langle\psi,\psi_\mathbf{k}\rangle&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\mathbf{k}(p)_{mn}pdp
\\&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\delta_{k_3,-m}e^{\ii(m-n)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{nm}
\\&=\sum_{n\in\mathbb{Z}}e^{-\ii(n+k_3)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{n,-k_3}.
\end{align*}
(2) Let $\mathbf{k}=(0,0,k_3)^T\in\mathbb{Z}^3$ be an integral vector with $k_3\in\mathbb{Z}$, and $n,m\in\mathbb{Z}$. Using Theorem \ref{Qnm00}, we have
\[
Q^{(0,0,k_3)}(p)_{mn}=\delta_{k_3,-m}\delta_{n,m}\frac{\delta(p)}{p}.
\]
Thus, using Proposition \ref{1D}, we get
\begin{align*}
\int_0^\infty\widehat{f}(p)_{nm}Q^\mathbf{k}(p)_{mn}pdp
&=\delta_{k_3,-m}\delta_{n,m}\int_0^\infty\widehat{f}(p)_{nm}\delta(p)dp
\\&=\delta_{k_3,-m}\delta_{n,m}\left(\lim_{p\to 0^+}\widehat{f}(p)_{nm}\right)
=\delta_{k_3,-m}\delta_{n,m}\widehat{f}[-n].
\end{align*}
Hence, we have
\begin{align*}
\langle\psi,\psi_\mathbf{k}\rangle
&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\mathbf{k}(p)_{mn}pdp
\\&=\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\delta_{k_3,-m}\delta_{n,m}\widehat{f}[-n]
\\&=\sum_{n\in\mathbb{Z}}\delta_{n,-k_3}\widehat{f}[-n]=\widehat{f}[k_3].
\end{align*}
\end{proof}

We then deduce the following version of the reconstruction formula (\ref{FourierInvQ0}) in terms of matrix elements, if the basis functions are given by (\ref{MainBasisFun}).

\begin{theorem}
 Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
We then have
\begin{equation}\label{MainRecFourierGammaSE2}
\widetilde{f}(\Gamma h)=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{(k_1,k_2)\in\mathbb{Z}^2_\times}\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{\ii(k_3-n)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{nk_3}e^{2\pi\ii a(k_1\cos\phi+k_2\sin\phi)}e^{-\ii k_3\alpha},
\end{equation}
for $h=g(a,\phi,\alpha)\in SE(2)$.
\end{theorem}
\begin{proof}
Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
Let $h=g(a,\phi,\alpha)\in SE(2)$ be given. Using Equation (\ref{FourierInvQ0}), we have
\begin{equation}\label{FourierInvQ00}
\widetilde{f}(\Gamma h)=\sum_{\mathbf{k}\in\mathbb{Z}^3}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\mathbf{k}(p)_{mn}pdp\right)\psi_\mathbf{k}(\Gamma h),
\end{equation}
for $h\in SE(2)$, where
\[
Q^\mathbf{k}(p)_{mn}:=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\mathbf{k}(\Gamma g)}dg,
\]
for $p>0$, $\mathbf{k}\in\mathbb{Z}^3$, and $m,n\in\mathbb{Z}$.
Therefore, using Equations (\ref{kkk}) and (\ref{00k}), for $h=g(a,\phi,\alpha)$, we conclude that
\begin{align*}
\widetilde{f}(\Gamma h)&=\sum_{\mathbf{k}\in\mathbb{Z}^3}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\mathbf{k}(p)_{mn}pdp\right)\psi_\mathbf{k}(\Gamma h)
\\&=\sum_{k_1\in\mathbb{Z}}\sum_{k_2\in\mathbb{Z}}\sum_{k_3\in\mathbb{Z}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^{(k_1,k_2,k_3)}(p)_{mn}pdp\right)\psi_{(k_1,k_2,k_3)}(\Gamma h)
\\&=\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\left(\int_0^\infty \widehat{f}(p)_{nm}Q^{(0,0,k_3)}(p)_{mn}pdp\right)\psi_{(0,0,k_3)}(\Gamma h)
\\&\ \ \ +\sum_{(k_1,k_2)\in\mathbb{Z}^2_\times}\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\left(\int_0^\infty\widehat{f}(p)_{nm}Q^{(k_1,k_2,k_3)}(p)_{mn}pdp\right)\psi_{(k_1,k_2,k_3)}(\Gamma h)
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{(k_1,k_2)\in\mathbb{Z}^2_\times}\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{-\ii(n+k_3)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{n,-k_3}e^{2\pi\ii a(k_1\cos\phi+k_2\sin\phi)}e^{\ii k_3\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{(k_1,k_2)\in\mathbb{Z}^2_\times}\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{-\ii(n-k_3)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{nk_3}e^{2\pi\ii a(k_1\cos\phi+k_2\sin\phi)}e^{-\ii k_3\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{(k_1,k_2)\in\mathbb{Z}^2_\times}\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{\ii(k_3-n)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{nk_3}e^{2\pi\ii a(k_1\cos\phi+k_2\sin\phi)}e^{-\ii k_3\alpha}.
\end{align*}
\end{proof}

Let $\mathcal{R}:=\left\{\rho(k_1,k_2):k_1,k_2\in\mathbb{Z}\right\}$.
For each $r\in\mathcal{R}$, let
\[
\Theta_r:=\left\{\Phi(i,j):r=\rho(i,j),~~i,j\in\mathbb{Z}\right\}.
\]

\begin{proposition}
{\it With above assumptions we have
\begin{enumerate}
\item $\mathbb{N}\cup\{0\}\subseteq\mathcal{R}\subseteq\sqrt{\mathbb{N}}:=\{\sqrt{n}:n\in\mathbb{N}\cup\{0\}\}$.
\item $\mathcal{R}$ is a discrete subset of $[0,\infty)$.
\item For each $r\in\mathcal{R}$, the set $\Theta_r$ is a finite subset of $[0,2\pi)$.
\item $\mathbb{Z}^2=\bigcup_{r\in\mathcal{R}}\{(r\cos\theta,r\sin\theta):\theta\in\Theta_r\}$.
\end{enumerate}
}\end{proposition}
\begin{proof}
(1)-(3) are straightforward.

(4) Let $\mathbf{x}\in\bigcup_{r\in\mathcal{R}}\{(r\cos\theta,r\sin\theta):\theta\in\Theta_r\}$. Suppose $r\in\mathcal{R}$ and $\theta\in\Theta_r$ with $\mathbf{x}=(r\cos\theta,r\sin\theta)$. Hence, $\theta=\Phi(i,j)$ with $\rho(i,j)=r$, for some $i,j\in\mathbb{Z}$. We then have
\[
r\cos\theta=\rho(i,j)\cos\Phi(i,j)=\sqrt{i^2+j^2}\frac{i}{\sqrt{i^2+j^2}}=i\in\mathbb{Z},
\hspace{1cm}
r\sin\theta=\rho(i,j)\sin\Phi(i,j)=\sqrt{i^2+j^2}\frac{j}{\sqrt{i^2+j^2}}=j\in\mathbb{Z}.
\]
Thus, we deduce that $\mathbf{x}=(r\cos\theta,r\sin\theta)\in\mathbb{Z}^2$.
Therefore, we get $\bigcup_{r\in\mathcal{R}}\{(r\cos\theta,r\sin\theta):\theta\in\Theta_r\}\subseteq\mathbb{Z}^2$. Conversely, let $\mathbf{x}=(k_1,k_2)\in\mathbb{Z}^2$ be given.
We then have $k_1,k_2\in\mathbb{Z}$ and hence we get
$k_1=\rho(k_1,k_2)\cos\Phi(k_1,k_2)$, and $k_2=\rho(k_1,k_2)\sin\Phi(k_1,k_2)$.
Then, we conclude that $\mathbf{x}=(r\cos\theta,r\sin\theta)$,
with $r:=\rho(k_1,k_2)$ and $\theta:=\Phi(k_1,k_2)$. This implies that $\mathbf{x}\in\bigcup_{r\in\mathcal{R}}\{(r\cos\theta,r\sin\theta):\theta\in\Theta_r\}$ and hence
$\mathbb{Z}^2\subseteq\bigcup_{r\in\mathcal{R}}\{(r\cos\theta,r\sin\theta):\theta\in\Theta_r\}$.
\end{proof}

For each $\rho\in\mathcal{R}$ and $q\in\mathbb{Z}$, let $K_\rho^q:\mathbb{R}^2\to\mathbb{C}$ be given by
\[
K_\rho^q(a,\phi):=\sum_{\theta\in\Theta_\rho}e^{2\pi\ii a\rho\cos(\phi-\theta)}e^{\ii q\theta},
\]
for all $(a,\phi)\in\mathbb{R}^2$.

\begin{theorem}
 Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$.
We then have
\begin{equation}\label{mainR}
\widetilde{f}(\Gamma h)=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{f}(2\pi\rho)_{nk}e^{-\ii k\alpha},
\end{equation}
for $h=g(a,\phi,\alpha)\in SE(2)$.
\end{theorem}
\begin{proof}
Let $f\in L^1(SE(2))$ such that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$,
and $h=g(a,\phi,\alpha)\in SE(2)$.
Then, using (\ref{MainRecFourierGammaSE2}), we have
\begin{align*}
\widetilde{f}(\Gamma h)&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{(k_1,k_2)\in\mathbb{Z}^2_\times}\sum_{k_3\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{\ii(k_3-n)\Phi(k_1,k_2)}\widehat{f}(2\pi\rho(k_1,k_2))_{nk_3}e^{2\pi\ii a(k_1\cos\phi+k_2\sin\phi)}e^{-\ii k_3\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{\theta\in\Phi_\rho}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{\ii(k-n)\theta}\widehat{f}(2\pi\rho)_{nk}e^{2\pi\ii a\rho\cos(\phi-\theta)}e^{-\ii k\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\left(\sum_{\theta\in\Phi_\rho}e^{2\pi\ii a\rho\cos(\phi-\theta)}e^{\ii(k-n)\theta}\right)\widehat{f}(2\pi\rho)_{nk}e^{-\ii k\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{f}(2\pi\rho)_{nk}e^{-\ii k\alpha},
\end{align*}
\end{proof}

\begin{corollary}
{\it  Let $f:SE(2)\to\mathbb{C}$ be a continuous function supported in the fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$. We then have
\begin{equation}
f(h)=\sum_{k\in\mathbb{Z}}\widehat{f}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{f}(2\pi\rho)_{nk}e^{-\ii k\alpha},
\end{equation}
for $h=g(a,\phi,\alpha)\in F\subseteq SE(2)$.
}\end{corollary}

\section{{\bf Noncommutative Fourier Series of Convolutions on $\Gamma\backslash SE(2)$}}
Throughout this section we still assume that $\Gamma:=\mathbb{Z}^2$ and $\mu$ is the finite $SE(2)$-invariant measure on the right coset space $\Gamma\backslash SE(2)$ which is normalized with respect to Weil's formula. We then introduce the notion of convolution of functions on $SE(2)$ by functions on $\Gamma\backslash SE(2)$.
Also, we shall study different aspects of noncommutative Fourier series for approximating the convolution functions on the right coset space $\Gamma\backslash SE(2)$. As applications for noncommutative Fourier series of convolution functions, we conclude the paper by some noncommutative Plancherel type formulas for functions on the right coset space $\Gamma\backslash SE(2)$.

\subsection{Convolution Functions}
Let $f\in L^1(SE(2))$ and $\psi\in L^1(\Gamma\backslash SE(2),\mu)$. We then define the convolution of $f$ with $\psi$ as the function $\psi\oslash f:\Gamma\backslash SE(2)\to\mathbb{C}$ via
\begin{equation}\label{oslash}
(\psi\oslash f)(\Gamma g):=\int_{SE(2)} \psi(\Gamma h)f(h^{-1}\circ g)dh,
\end{equation}
for $g\in SE(2)$.

For each $\gamma\in\Gamma$ and $g\in SE(2)$, we can write
\begin{align*}
\int_{SE(2)}\psi(\Gamma h)f(h^{-1}\circ\gamma\circ g)dh
&=\int_{SE(2)}\psi(\Gamma \gamma\circ h)f((\gamma\circ h)^{-1}\circ\gamma\circ g)d(\gamma\circ h)
\\&=\int_{SE(2)}\psi(\Gamma h)f((\gamma\circ h)^{-1}\circ\gamma\circ g)dh
\\&=\int_{SE(2)}\psi(\Gamma h)f(h^{-1}\circ\gamma^{-1}\circ\gamma\circ g)dh
=\int_{SE(2)}\psi(\Gamma h)f(h^{-1}\circ g)dh,
\end{align*}
hence we deduce that
\[
\Gamma g\mapsto \int_{SE(2)}\psi(\Gamma h)f(h^{-1}\circ g)dh,
\]
is well-defined as a function on the right coset space $\Gamma\backslash SE(2)$.

Next result shows that $L^1(\Gamma\backslash SE(2),\mu)$ with respect to the module action of $L^1(SE(2))$ given by (\ref{oslash}) is a Banach module.

\begin{theorem}
Let $f\in L^1(SE(2))$ and $\psi\in L^1(\Gamma\backslash SE(2),\mu)$. We then have
$\psi\oslash f\in L^1(\Gamma\backslash SE(2),\mu)$ with
\[
\|\psi\oslash f\|_{L^1(\Gamma\backslash SE(2),\mu)}\le \|f\|_{L^1(SE(2))}\|\psi\|_{L^1(\Gamma\backslash SE(2),\mu)}.
\]
\end{theorem}
\begin{proof}
Let $f\in L^1(SE(2))$ and $\psi\in L^1(\Gamma\backslash SE(2),\mu)$. We then have
\begin{align*}
\|\psi\oslash f\|_{L^1(\Gamma\backslash SE(2),\mu)}
&=\int_{\Gamma\backslash SE(2)}\left|\int_{SE(2)} \psi(\Gamma h)f(h^{-1}\circ g)dh\right|d\mu(\Gamma g)
\\&\le\int_{\Gamma\backslash SE(2)}\int_{SE(2)} \left|\psi(\Gamma h)f(h^{-1}\circ g)\right|dhd\mu(\Gamma g)
\\&=\int_{\Gamma\backslash SE(2)}\int_{SE(2)}|\psi(\Gamma h)||f(h^{-1}\circ g)|dhd\mu(\Gamma g)
\\&=\int_{\Gamma\backslash SE(2)}\int_{SE(2)}|\psi(\Gamma g\circ h)||f((g\circ h)^{-1}\circ g)|d(g\circ h)d\mu(\Gamma g)
\\&=\int_{\Gamma\backslash SE(2)}\int_{SE(2)}|\psi(\Gamma g\circ h)||f(h^{-1})|dhd\mu(\Gamma g)
\\&=\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}|\psi(\Gamma g\circ h)|d\mu(\Gamma g)\right)|f(h^{-1})|dh
\\&=\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}|\psi(\Gamma g)|d\mu(\Gamma g\circ h^{-1})\right)|f(h^{-1})|dh
\\&=\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}|\psi(\Gamma g)|d\mu(\Gamma g)\right)|f(h^{-1})|dh
\\&=\|\psi\|_{L^1(\Gamma\backslash SE(2),\mu)}\left(\int_{SE(2)}|f(h^{-1})|dh\right)=\|f\|_{L^1(SE(2))}\|\psi\|_{L^1(\Gamma\backslash SE(2),\mu)}.
\end{align*}
\end{proof}

\begin{proposition}
{\it Let $f\in L^1(SE(2))$ and $\psi\in L^1(\Gamma\backslash SE(2),\mu)$. We then have
\begin{equation}
(\psi\oslash f)(\Gamma g)=\int_{\Gamma\backslash SE(2)}\psi(\Gamma h)\left(\sum_{\gamma\in\Gamma} f(h^{-1}\circ\gamma^{-1}\circ g)\right)d\mu(\Gamma h).
\end{equation}
}\end{proposition}
\begin{proof}
Let $f\in L^1(SE(2))$ and $\psi\in L^1(\Gamma\backslash SE(2),\mu)$. Let $g\in SE(2)$. Using Weil's formula, we can write
\begin{align*}
(\psi\oslash f)(\Gamma g)
&=\int_{SE(2)} \psi(\Gamma h)f(h^{-1}g)dh
\\&=\int_{\Gamma\backslash SE(2)}\left(\sum_{\gamma\in\Gamma}\psi(\Gamma \gamma\circ h) f((\gamma\circ h)^{-1}\circ g)\right)d\mu(\Gamma h)
\\&=\int_{\Gamma\backslash SE(2)}\left(\sum_{\gamma\in\Gamma}\psi(\Gamma h)f((\gamma\circ h)^{-1}\circ g)\right)d\mu(\Gamma h)
=\int_{\Gamma\backslash SE(2)}\psi(\Gamma h)\left(\sum_{\gamma\in\Gamma} f(h^{-1}\circ\gamma^{-1}\circ g)\right)d\mu(\Gamma h).
\end{align*}
\end{proof}

\begin{theorem}
Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ and $g\in SE(2)$. We then have
\begin{equation}
\widetilde{(f_1\star f_2)}(\Gamma g)=(\widetilde{f_1}\oslash f_2)(\Gamma g).
\end{equation}
\end{theorem}
\begin{proof}
Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ and $g\in SE(2)$. We then have
\begin{align*}
\widetilde{(f_1\star f_2)}(\Gamma g)
&=\sum_{\gamma\in\Gamma}(f_1\star f_2)(\gamma\circ g)
=\sum_{\gamma\in\Gamma}\left(\int_{SE(2)}f_1(h)f_2(h^{-1}\circ\gamma\circ g)dh\right)
\\&=\sum_{\gamma\in\Gamma}\left(\int_{SE(2)}f_1(\gamma\circ h)f_2(h^{-1}\circ g)dh\right)
=\int_{SE(2)}\left(\sum_{\gamma\in\Gamma}f_1(\gamma\circ h)\right)f_2(h^{-1}\circ g)dh
\\&=\int_{SE(2)}\widetilde{f_1}(\Gamma h)f_2(h^{-1}\circ g)dh=(\widetilde{f_1}\oslash f_2)(\Gamma g).
\end{align*}
\end{proof}

\begin{corollary}
{\it Let $f_k:SE(2)\to\mathbb{C}$ with $k\in\{1,2\}$
be continuous functions such that $f_1\star f_2$ is supported in a fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$ and $g\in F$. We then have
\begin{align}
(f_1\star f_2)(g)=(\widetilde{f_1}\oslash f_2)(\Gamma g).
\end{align}
}\end{corollary}

\begin{proposition}
{\it Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$, $f:=f_1\star f_2$, and $g\in SE(2)$. We then have
\begin{equation}\label{oslash.abs}
\widetilde{|f|}(\Gamma g)\le(|\widetilde{f_1}|\oslash |f_2|)(\Gamma g).
\end{equation}
}\end{proposition}
\begin{proof}
Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$, $f:=f_1\star f_2$, and $g\in SE(2)$.
We then have
\begin{align*}
\widetilde{|f|}(\Gamma g)
&=\sum_{\gamma\in\Gamma}|(f_1\star f_2)|(\gamma\circ g)=\sum_{\gamma\in\Gamma}|(f_1\star f_2)(\gamma\circ g)|
=\sum_{\gamma\in\Gamma}\left|\int_{SE(2)}f_1(h)f_2(h^{-1}\circ\gamma\circ g)dh\right|
\\&\le\sum_{\gamma\in\Gamma}\int_{SE(2)}|f_1(\gamma\circ h)||f_2(h^{-1}\circ g)|dh
=\int_{SE(2)}\left(\sum_{\gamma\in\Gamma}|f_1(\gamma\circ h)|\right)|f_2(h^{-1}\circ g)|dh
\\&=\int_{SE(2)}\widetilde{|f_1|}(\Gamma h)|f_2(h^{-1}\circ g)|dh=\int_{SE(2)}\widetilde{|f_1|}(\Gamma h)|f_2|(h^{-1}\circ g)dh=(\widetilde{|f_1|}\oslash |f_2|)(\Gamma g).
\end{align*}
\end{proof}

Next result presents a constructive expansion for the convolution of
$L^1(SE(2))$ on $L^1(\Gamma\backslash SE(2),\mu)$.

\begin{theorem}
Let $\Gamma:=\mathbb{Z}^2$ and $\mathcal{E}(\Gamma):=\{\psi_\ell:\Gamma\backslash SE(2)\to\mathbb{C}\ |\ \ell\in\mathbb{I}\}$ be a (discrete) orthonormal basis for the Hilbert function space $L^2(\Gamma\backslash SE(2),\mu)$. Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ such that $\widetilde{|f_1|}\in L^2(\Gamma\backslash SE(2),\mu)$.  We then have
\begin{equation}\label{FourierInvQ0Conv}
(\widetilde{f_1}\oslash f_2)(\Gamma h)=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\int_0^\infty\widehat{f_1}(p)_{nk}\widehat{f_2}(p)_{km}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h),
\end{equation}
for $h\in SE(2)$, where for $p>0$, $\ell\in\mathbb{I}$, and $m,n\in\mathbb{Z}$ we have
\[
Q^\ell(p)_{mn}:=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\ell(\Gamma g)}dg.
\]
\end{theorem}
\begin{proof}
Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ such that $\widetilde{|f_1|}\in L^2(\Gamma\backslash SE(2),\mu)$. Let $f:=f_1\star f_2$. Then we have $f\in L^1(SE(2))$.
We then claim that $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$ as well.
To this end, first note that using (\ref{oslash.abs}), for each $g\in SE(2)$ we have
$$\widetilde{|f|}(\Gamma g)\le \widetilde{|f_1|}\oslash|f_2|(\Gamma g).$$
We then have
\begin{align*}
\|\widetilde{|f|}\|_{L^2(\Gamma\backslash SE(2),\mu)}

&=\left(\int_{\Gamma\backslash SE(2)}\widetilde{|f|}(\Gamma g)^2d\mu(\Gamma g)\right)^{1/2}
\\&\le\left(\int_{\Gamma\backslash SE(2)}\widetilde{|f_1|}\oslash|f_2|(\Gamma g)^2d\mu(\Gamma g)\right)^{1/2}
\\&=\left(\int_{\Gamma\backslash SE(2)}\left(\int_{SE(2)}\widetilde{|f_1|}(\Gamma h) |f_2|(h^{-1}\circ g)dh\right)^2d\mu(\Gamma g)\right)^{1/2}
\\&=\left(\int_{\Gamma\backslash SE(2)}\left(\int_{SE(2)}\widetilde{|f_1|}(\Gamma g\circ h) |f_2|((g\circ h)^{-1}\circ g)d(g\circ h)\right)^2d\mu(\Gamma g)\right)^{1/2}
\\&=\left(\int_{\Gamma\backslash SE(2)}\left(\int_{SE(2)}\widetilde{|f_1|}(\Gamma g\circ h)|f_2|(h^{-1})dh\right)^2d\mu(\Gamma g)\right)^{1/2}.
\end{align*}
Then, using Minkowski's Inequality for Integrals, we can write
\begin{align*}
\|\widetilde{|f|}\|_{L^2(\Gamma\backslash SE(2),\mu)}&\le\left(\int_{\Gamma\backslash SE(2)}\left(\int_{SE(2)}\widetilde{|f_1|}(\Gamma g\circ h) |f_2|(h^{-1})dh\right)^2d\mu(\Gamma g)\right)^{1/2}
\\&\le\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}\widetilde{|f_1|}(\Gamma g\circ h)^2 f_2(h^{-1})|^2d\mu(\Gamma g)\right)^{1/2}dh
\\&=\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}\widetilde{|f_1|}(\Gamma g\circ h)^2d\mu(\Gamma g)\right)^{1/2}| f_2(h^{-1})|dh
\\&=\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}\widetilde{|f_1|}(\Gamma g)^2d\mu(\Gamma g\circ h^{-1})\right)^{1/2}| f_2(h^{-1})|dh
\\&=\int_{SE(2)}\left(\int_{\Gamma\backslash SE(2)}\widetilde{|f_1|}(\Gamma g)^2d\mu(\Gamma g)\right)^{1/2}| f_2(h^{-1})|dh
=\|\widetilde{|f_1|}\|_{L^2(\Gamma\backslash SE(2),\mu)}\|f_2\|_{L^1(SE(2))}.
\end{align*}
Therefore, applying Equation (\ref{FourierInvQ0}) for $f$, we get
\begin{align*}
\widetilde{f_1}\oslash f_2(\Gamma h)&=\widetilde{f_1\star f_2}(\Gamma h)
=\widetilde{f}(\Gamma h)
=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\widehat{f_1\star f_2}(p)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\left(\widehat{f_2}(p)\widehat{f_1}(p)\right)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h).
\end{align*}
For each $n,m\in\mathbb{Z}$, we have
\begin{align*}
\left(\widehat{f_2}(p)\widehat{f_1}(p)\right)_{nm}
&=\langle\widehat{f_2}(p)\widehat{f_1}(p)\mathbf{e}_n,\mathbf{e}_m\rangle=\langle\widehat{f_1}(p)\mathbf{e}_n,\widehat{f_2}(p)^*\mathbf{e}_m\rangle
\\&=\sum_{k\in\mathbb{Z}}\langle\widehat{f_1}(p)\mathbf{e}_n,\mathbf{e}_k\rangle\langle\mathbf{e}_k,\widehat{f_2}(p)^*\mathbf{e}_m\rangle
\\&=\sum_{k\in\mathbb{Z}}\langle\widehat{f_1}(p)\mathbf{e}_n,\mathbf{e}_k\rangle\langle\widehat{f_2}(p)\mathbf{e}_k,\mathbf{e}_m\rangle
=\sum_{k\in\mathbb{Z}}\widehat{f_1}(p)_{nk}\widehat{f_2}(p)_{km}.
\end{align*}
Hence, we get
\begin{align*}
(\widetilde{f_1}\oslash f_2)(\Gamma h)&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\left(\widehat{f_2}(p)\widehat{f_1}(p)\right)_{nm}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\int_0^\infty\widehat{f_1}(p)_{nk}\widehat{f_2}(p)_{km}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h),
\end{align*}
for $h\in SE(2)$.
\end{proof}

\begin{corollary}
{\it Let $f_k:SE(2)\to\mathbb{C}$ with $k\in\{1,2\}$
be continuous functions such that $f_1\star f_2$ is supported in a fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$ and $h=g(a,\phi,\theta)\in F$. We then have
\begin{equation}
(f_1\star f_2)(h)=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\int_0^\infty\widehat{f_1}(p)_{nk}\widehat{f_2}(p)_{km}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma h),
\end{equation}
where for $p>0$, $\ell\in\mathbb{I}$, and $m,n\in\mathbb{Z}$ we have
\[
Q^\ell(p)_{mn}:=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\ell(\Gamma g)}dg,
\]

}\end{corollary}

We then deduce the following version of convolution theorem in terms of matrix elements, if the basis functions are given by (\ref{MainBasisFun}).

\begin{theorem}
Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ such that $\widetilde{|f_1|}\in L^2(\Gamma\backslash SE(2),\mu)$. We then have
\begin{equation}\label{3SumSE(2)convR}
(\widetilde{f_1}\oslash f_2)(\Gamma h)=\sum_{k\in\mathbb{Z}}\widehat{f_1}[k]\widehat{f_2}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{f_1}(2\pi\rho)_{nm}\widehat{f_2}(2\pi\rho)_{mk}e^{-\ii k\alpha},
\end{equation}
for $h=g(a,\phi,\alpha)\in SE(2)$.
\end{theorem}
\begin{proof}
Let $f_k\in L^1(SE(2))$ with $k\in\{1,2\}$ such that $\widetilde{|f_1|}\in L^2(\Gamma\backslash SE(2),\mu)$. Then, using Equation (\ref{mainR}) for $f_1\star f_2$, we have
\begin{align*}
(\widetilde{f_1}\oslash f_2)(\Gamma h)&=\widetilde{(f_1\star f_2)}(\Gamma h)
=\sum_{k\in\mathbb{Z}}\widehat{(f_1\star f_2)}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{(f_1\star f_2)}(2\pi\rho)_{nk}e^{-\ii k\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f_1}[k]\widehat{f_2}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\left(\widehat{f_2}(2\pi\rho)\widehat{f_1}(2\pi\rho)\right)_{nk}e^{-\ii k\alpha}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f_1}[k]\widehat{f_2}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{f_1}(2\pi\rho)_{nm}\widehat{f_2}(2\pi\rho)_{mk}e^{-\ii k\alpha}.
\end{align*}
\end{proof}

\begin{corollary}
{\it Let $f_k:SE(2)\to\mathbb{C}$ with $k\in\{1,2\}$
be continuous functions such that $f_1\star f_2$ is supported in a fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$ and $h=g(a,\phi,\theta)\in F$. We then have
\begin{equation}\label{3SumSE(2)onv.fun}
(f_1\star f_2)(h)=\sum_{k\in\mathbb{Z}}\widehat{f_1}[k]\widehat{f_2}[k]e^{\ii k\alpha}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(a,\phi)\widehat{f_1}(2\pi\rho)_{nm}\widehat{f_2}(2\pi\rho)_{mk}e^{-\ii k\alpha}.
\end{equation}
}\end{corollary}

\subsection{{\bf Plancherel Formulas}}

We shall continue by presenting Plancherel type formulas associated to the noncommutative Fourier series (\ref{FourierInvQ0}).

The following result presents the canonical connection of $L^2$-norms on the right coset space $\Gamma\backslash SE(2)$ with convolution of functions.

\begin{proposition}\label{plan.alt}
{\it Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$ be given. We then have
\[
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2=(\widetilde{f}\oslash f^*)(\Gamma).
\]
}\end{proposition}
\begin{proof}
Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$ be given. We then have $\widetilde{f}\in L^2(\Gamma\backslash SE(2),\mu)$. Then, using Weils formula, we can write
\begin{align*}
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2&=\int_{\Gamma\backslash SE(2)}|\widetilde{f}(\Gamma h)|^2d\mu(\Gamma h)=\int_{\Gamma\backslash SE(2)}\widetilde{f}(\Gamma h)\overline{\widetilde{f}(\Gamma h)}d\mu(\Gamma h)
\\&=\int_{\Gamma\backslash SE(2)}\sum_{\gamma\in\Gamma}\widetilde{f}(\Gamma\gamma\circ h)\overline{f(\gamma\circ h)}d\mu(\Gamma h)
=\int_{SE(2)}\widetilde{f}(\Gamma h)\overline{f(h)}dh
\\&=\int_{SE(2)}\widetilde{f}(\Gamma h)f^*(h^{-1})dh=(\widetilde{f}\oslash f^*)(\Gamma).
\end{align*}
\end{proof}

We then have the following general form of Plancherel formula associated to the noncommutative Fourier series (\ref{FourierInvQ0}).

\begin{theorem}\label{plan.general}
Let $\Gamma:=\mathbb{Z}^2$ and $\mathcal{E}(\Gamma):=\{\psi_\ell:\Gamma\backslash SE(2)\to\mathbb{C}\ |\ \ell\in\mathbb{I}\}$ be a (discrete) orthogonal basis for the Hilbert function space $L^2(\Gamma\backslash SE(2),\mu)$. Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$ be given. We then have
\begin{equation}\label{FourierInvQ0plan}
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\langle\widehat{f}(p)\mathbf{e}_n,\widehat{f}(p)\mathbf{e}_m\rangle Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma),
\end{equation}
where for $p>0$, $\ell\in\mathbb{I}$, and $m,n\in\mathbb{Z}$ we have
\[
Q^\ell(p)_{mn}:=\int_{SE(2)}\mathrm{u}_{mn}(g,p)\overline{\psi_\ell(\Gamma g)}dg,
\]
\end{theorem}
\begin{proof}
Let $f\in L^1(SE(2))$ with $\widetilde{f}\in L^2(\Gamma\backslash SE(2),\mu)$ be given. Using Equation (\ref{FourierInvQ0}) and Proposition \ref{plan.alt}, we get
\begin{align*}
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2
&=\widetilde{f}\oslash f^*(\Gamma)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\int_0^\infty\widehat{f^*}(p)_{km}\widehat{f}(p)_{nk}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nk}(\widehat{f}(p)^*)_{km}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\int_0^\infty\widehat{f}(p)_{nk}\overline{\widehat{f}(p)_{mk}}Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma)
\\&=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\langle\widehat{f}(p)\mathbf{e}_n,\widehat{f}(p)\mathbf{e}_m\rangle Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma).
\end{align*}
\end{proof}

We then conclude the following consequence, with assumptions in Theorem \ref{plan.general}.

\begin{corollary}
{\it Let $f:SE(2)\to\mathbb{C}$ be a continuous function supported in the fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$. We then have
\[
\|f\|_{L^2(SE(2))}^2=\sum_{\ell\in\mathbb{I}}\left(\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\int_0^\infty\langle\widehat{f}(p)\mathbf{e}_n,\widehat{f}(p)\mathbf{e}_m\rangle Q^\ell(p)_{mn}pdp\right)\psi_\ell(\Gamma).
\]
}\end{corollary}

Next we present the following version of Plancherel formula, when basis functions are given by (\ref{MainBasisFun}).

\begin{theorem}
Let $f\in L^1(SE(2))$ with $\widetilde{|f|}\in L^2(\Gamma\backslash SE(2),\mu)$ be given.
We then have
\begin{equation}\label{3SumSE(2)plan}
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2=\sum_{k\in\mathbb{Z}}|\widehat{f}[k]|^2+
\sum_{\rho\in\mathcal{R}}\sum_{\theta\in\Phi_\rho}\sum_{m\in\mathbb{Z}}\left|\sum_{n\in\mathbb{Z}}e^{-\ii n\theta}\widehat{f}(2\pi\rho)_{nm}\right|^2.
\end{equation}
\end{theorem}
\begin{proof}
Let $f\in L^1(SE(2))$ with $\widetilde{f}\in L^2(\Gamma\backslash SE(2),\mu)$ be given. Using Equation (\ref{3SumSE(2)convR}) for $f_1:=f$ and $f_2:=f^*$, we get
\begin{align*}
\|\widetilde{f}\|_{L^2(\Gamma\backslash SE(2),\mu)}^2&=\widetilde{f}\oslash f^*(\Gamma)
=\widetilde{f}\oslash f^*(\Gamma g(0,0,0))
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]\widehat{f^*}[k]+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(0,0)\widehat{f}(2\pi\rho)_{nm}\widehat{f^*}(2\pi\rho)_{mk}
\\&=\sum_{k\in\mathbb{Z}}\widehat{f}[k]\overline{\widehat{f}[k]}+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(0,0)\widehat{f}(2\pi\rho)_{nm}\left(\widehat{f}(2\pi\rho)^*\right)_{mk}
\\&=\sum_{k\in\mathbb{Z}}|\widehat{f}[k]|^2+\sum_{\rho\in\mathcal{R}}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(0,0)\widehat{f}(2\pi\rho)_{nm}\overline{\widehat{f}(2\pi\rho)_{km}}.
\end{align*}
Also, for each $\rho\in\mathcal{R}$, we have
\begin{align*}
\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}K_\rho^{k-n}(0,0)\widehat{f}(2\pi\rho)_{nm}\overline{\widehat{f}(2\pi\rho)_{km}}
&=\sum_{\theta\in\Theta_\rho}\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}e^{-\ii n\theta}e^{\ii k\theta}\widehat{f}(2\pi\rho)_{nm}\overline{\widehat{f}(2\pi\rho)_{km}}
\\&=\sum_{\theta\in\Theta_\rho}\sum_{m\in\mathbb{Z}}\left|\sum_{n\in\mathbb{Z}}e^{-\ii n\theta}\widehat{f}(2\pi\rho)_{nm}\right|^2,
\end{align*}
which completes the proof.
\end{proof}

\begin{corollary}
{\it Let $f:SE(2)\to\mathbb{C}$ be a continuous function supported in the fundamental domain $F$ of $\mathbb{Z}^2$ in $SE(2)$. We then have
\[
\|f\|_{L^2(SE(2))}^2=\sum_{k\in\mathbb{Z}}|\widehat{f}[k]|^2+
\sum_{\rho\in\mathcal{R}}\sum_{\theta\in\Theta_\rho}\sum_{m\in\mathbb{Z}}\left|\sum_{n\in\mathbb{Z}}e^{-\ii n\theta}\widehat{f}(2\pi\rho)_{nm}\right|^2.
\]
}\end{corollary}

{\bf Concluding remarks.}
Whereas the focus of this paper has been Fourier series of functions in $L^2(\mathbb{Z}^2\backslash SE(2))$, the extension to nonorthogonal lattices follows naturally. This is because any nonorthogonal lattice can be
expressed as $\mathbf{T}=\mathbf{A}\mathbb{Z}^2$ where $\mathbf{A}\in GL(2,\mathbb{R})$, and conjugation of both $\mathbb{Z}^2$ and
$SE(2)$ (with elements expressed as $3\times 3$ matrices) by $\mathbf{A}\oplus 1$ extends the current analysis trivially to
$L^2(\mathbf{A}\mathbb{Z}^2\backslash \mathbf{A}\,SE(2)\, \mathbf{A}^{-1})$ where $\mathbf{A}\,SE(2)\, \mathbf{A}^{-1}$ is $SE(2)$ expressed in nonorthogonal
coordinates.

Moreover, the results easily generalize to $L^2(\Gamma\backslash SE(2),\mu)$ where $\Gamma$ is
any wallpaper group. The reason for this is that due to the semidirect product structure of $\Gamma$,
every $\gamma \in \Gamma$ can be decomposed as $\gamma =\mathbf{t}\circ p$ where $\mathbf{t}\in \mathbf{T}= \mathbf{A}\mathbb{Z}^2$ and
$p \in \mathbb{P}$. Consequently,
$$\widetilde{f}(\Gamma g) = \sum_{\mathbf{t}\in\mathbf{T}} \left(\sum_{p \in \mathbb{P}}  f(\mathbf{t}\circ (p \circ g))\right)\,. $$
Hence for any $f \in L^2(SE(2))$
$$ \underline{f}(g) \,:=\, \frac{1}{|\mathbb{P}|} \sum_{p \in \mathbb{P}}  f(p \circ g) $$
(which is $f(g)$ averaged over $\mathbb{P}$), is also in  $L^2(SE(2))$.
Then, since
$$ \widetilde{f}(\Gamma g) = |\mathbb{P}| \sum_{\mathbf{t}\in\mathbf{T}}\underline{f}(\mathbf{t} \circ g) $$
and since $|\mathbb{P}|$ is finite, the theory in this paper can be used directly to characterize
$L^2(\Gamma\backslash SE(2),\mu)$ by restricting the discussion to the subspace of $L^2(SE(2))$ with the symmetry
$$ \underline{f}(g) \,=\, \underline{f}(p \circ g) \,\,\,{\rm for}\,\,\, p \in \mathbb{P}\,. $$

Future work will focus on noncommutative Fourier series on $\Gamma\backslash SE(3)$ where $\Gamma$ is
handedness-preserving crystallographic space group. This is more challenging than the planar case for several
reasons. First, the 3D crystallographic space groups are not all semi-direct products of $\mathbb{Z}^3$ with
a finite subgroup of $SO(3)$. Second, unlike $SE(2)$, the group $SE(3)$ is not solvable and the matrix elements
of irreducible unitary representations are significantly more complicated.

\

{\bf Acknowledgments.}
This work has been supported by the National Institute of General Medical Sciences of the NIH under award number R01GM113240, by the US National Science Foundation under grant NSF CCF-1640970, and by Office of Naval Research Award N00014-17-1-2142. The authors gratefully acknowledge the supporting agencies. The findings and opinions expressed here are only those of the authors, and not of the funding agencies.

\bibliographystyle{amsplain}

config: arxiv


TITLE: What is the best/most effective way to gain experince in using set theory to understand math, calculus in particular?
QUESTION [4 upvotes]: Note: While this question as phrased in the title is somewhat subjective, what I'm looking for as an answer should be specific enough to still have a clear/valid answer(s) to it. Also, apologies in advance if this is somewhat long winded but I feel the full context of my question is needed for clarity.
First let me provide some context. When it comes to learning new areas of math (or in this case, arguably, more fundamental/abstract areas), one problem all students face is 'vision'. Taking new concepts and adding them to old ones is often difficult since to do that requires seeing how things interrelate. However your typical education in math usually leaves you vastly unprepared to do this.
In a sense, most modern math education teaches students backwards. Most of the maths that students learn is gutted of the more important (and abstract) details. Details such as the 'real' proofs of certain theorems being replaced with a shorthand version that, while proves the part of the concept being focused on, ignores the parts that give it its foundation. "Proofs in a vacuum" as I call them since they usually don't bother proving (sometimes not even mentioning) other concepts that the main idea depends on.
This leads me to my problem. I have a rather strong foundation in multiple areas of math (particularly those often relevant to physics, such as calculus), but since my education has focused on results/applications rather then math as a whole I'm left without the 'glue' that ties it all together.
The language of set theory (as well as basic forms of logic) is still new to me. While I have a strong (enough) foundation in the basics after recently completing a class in discrete math, taking the concepts (and building onto them where needed) and applying them so as to add to my understanding of calculus for example is still difficult.
My question is this: What are some methods/pet projects I should try an implement to 'break out' of this novice phase as far set theory is concerned? To bridge the gap between knowing basic ideas concerning sets and relations between them and concepts from calculus for example? I don't want to look at an equation and see an equation. I wanna be able to see it as a more abstract entity.
Keep in mind what I'm looking for in an answer is:

Specific topics to research that show how sets and relations are used to construct other others of math
Go to examples for me to play around with so as to see how set theory is applied in practice (for example, how relations can be used to construct basic operations like addition, or differential operators).
Examples of a common (but not obvious) mistakes people tend to make applying set theory to describe common concepts in other area of math.

REPLY [2 votes]: If I understand your question correctly, although you have embedded a critique of how abstract math is taught, your question is really about how you can shore up your own understanding of abstract math via an enhanced understanding of set theory.
I would suggest that you learn about how the real number system is built up in steps from Peano's axioms for the natural numbers. There are three steps:

From Peano's axioms for the natural numbers to the integers.
From the integers to the rational numbers.
From the rational numbers to the real numbers.

There is one still deeper step:

From the ZFC axioms of set theory to Peano's axioms for the natural numbers (via the construction of the ordinal numbers).

I cannot recommend a single source to you for all four of these steps, although steps 2, 3, 4 at least are in many books at the advanced undergraduate or early graduate level, with steps 2 and 3 more likely to be in number theory books and steps 3 and 4 more likely to be in advanced calculus books. For Step 2, I like the closing two chapters of the book "Theory of Numbers" by B. M. Stewart. For Step 1, I like the early sections of Cohen's slim little atom bomb of a book "Set theory and the continuum hypothesis".


TITLE: Propositional logic "equivalent to" using union, intersection and negation
QUESTION [1 upvotes]: In the Maths book, "implies to" is described as
$A\rightarrow$B equals to $\lnot\ A \lor  B $
How can I represent $A \Leftrightarrow B$ in the same way?

REPLY [1 votes]: $A \iff B$ is equivalent to $( \neg A \lor B ) \land ( \neg B \lor A )$
which in turn is equivalent to $( \neg A \land \neg B ) \lor ( A \land B )$


TITLE: Are the following proofs correct(2)?
QUESTION [1 upvotes]: A list of some questions from class and my attempt at them are given below. Would appreciate any advice on what I did wrong or on how to attempt some of the questions.

Let A: X → Y be a linear operator from a linear space X to a linear space Y. Show that the image of a linear operator A: X → Y is a linear subspace of Y.

My attempt:
1.Showing non empty, I don't know any examples..
2.Showing additive, where $y_1, y_2 \in \ker(A):$

$Ax_1 = y_1$ and $Ax_2 = y_2$
$A(x_1 + x_2) = y_1 + y_2$

3.Showing scalar multiplication for $a \in F$

$a(Ax_1)=ay_1$
$ay_1 = ay_1$

_

4.(a) The differentiation operator $D : C^∞ → C^∞$ is a linear operator. Find the kernel and
  image of D. (b) The operator $D^n
: C^∞ → C^∞$ is a linear operator (differentiation n times).
  Find the kernel and image of $D^n$
  .

My attempt:
(a) ker(D) = c, c ∈ F and im(D) = $C^∞$ , I don't know how to prove that this is the case it just seems like it is such.
(b) ker($D^n$) = $x^n$, x ∈ $C^∞$ and im($D^n$) = $C^∞$ , I don't know how to prove that this is the case it just seems like it is such.

REPLY [1 votes]: 3.1:  See problem 1.
4(a):  It is a standard application of the mean value theorem that $Df = 0$ implies $f$ is constant.
4(b):  Your kernel is incorrect.  What can you say about all polynomials of degree $<n$?  For the image, what is the definition of $C^\infty$?