# Convergence of fractional Fourier series on the torus and applications

Zunwei Fu<EMAIL_ADDRESS>Xianming Hou<EMAIL_ADDRESS>Qingyan Wu
<EMAIL_ADDRESS>School of Mathematics and Statistics, Linyi University,
Linyi 276000, China  College of Information Technology, The University of
Suwon, Hwaseong-si 18323, South Korea

###### Abstract

In this paper, we introduce the fractional Fourier series on the fractional
torus and study some basic facts of fractional Fourier series, such as
fractional convolution and fractional approximation. Meanwhile, fractional
Fourier inversion and Poisson summation formula are also given. We further
discuss the relationship between the decay of fractional Fourier coefficients
and the smoothness of a function. Using the properties of fractional Féjer
kernel, the pointwise convergence of fractional Fourier series can be
established. Finally, we present the applications of fractional Fourier series
to fractional partial differential equations with periodic boundary condition.
Moreover, we apply approximation methods on the fractional torus to recover
the non-stationary signals.

###### keywords:

fractional Fourier series, fractional approximate identity, fractional Fejér
kernel, fractional Fourier inversion, convergence

3.5cm3cm3cm3cm

††footnotetext: 2020 Mathematics Subject Classification. 42A20;
41A35.††footnotetext: This work was partially supported by the Natural Science
Foundation of Shandong Province (Nos. ZR2020QA006, ZR2021MA031) and the
National Natural Foundation of China (Nos. 12071197, 12171221, 12271232)

###### Contents

  1. 1 Introduction
  2. 2 Fractional Fourier series on the torus
  3. 3 Reproduction of Functions from Their Fractional Fourier Coefficients
    1. 3.1 Fractional Fourier inversion
    2. 3.2 Fractional Fourier series of square summable functions
    3. 3.3 The fractional Poisson summation formula
  4. 4 Decay of Fractional Fourier coefficient
    1. 4.1 Decay of fractional Fourier coefficients of integrable function
    2. 4.2 Decay of Fractional Fourier Coefficients of Smooth Functions
  5. 5 Pointwise Convergence of fractional Fourier series
    1. 5.1 Pointwise Convergence of the fractional Fejér Means
    2. 5.2 Almost Everywhere Convergence of the fractional Fejér Means
  6. 6 Applications to Partial Differential Equations
  7. 7 Applications to non-stationary signals
  8. 8 References

##  1 Introduction

It is well known that the Fourier series plays an important role in studying
boundary value problems, such as the vibrating string and the heat equation.
Fourier solved the problem of heat distribution in the cube $\mathbf{T}^{3}$
by using the triple sine series of three-variable functions. Dirichlet [4]
further studied the pointwise convergence of the Fourier series of piecewise
monotonic functions in terms of the certain kernel on the circle. Meanwhile,
motivated by periodic astronomical and geophysical occurrences, many scholars
are devoted to studying the expansions of periodic functions, such as [28].
Both theories and applications on the torus can be expressed by time and
frequency analysis, it is natural to study the properties of the Fourier
series.

The $n$-torus $\mathbf{T}^{n}$ is the cube $[0,1]^{n}$ with opposite sides
identified. Note that functions on $\mathbf{T}^{n}$ are $1$-periodic in every
coordinate. By Fourier transform, the $m$-th Fourier coefficient is defined as
follows:

$\displaystyle\mathcal{F}(f)(m)=\int_{\mathbf{T}^{n}}f(x)e^{-2\pi im\cdot
x}dx,$

where $f\in L^{1}(\mathbf{T}^{n})$, $m\in\mathbf{Z}^{n}$. The Fourier series
of $f$ is the series

$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}(f)(m)e^{2\pi im\cdot x}.$
(1.1)

Kolmogorov [12, 13] showed that there exists $f\in L^{1}(\mathbf{T}^{1})$
whose Fourier series diverges almost everywhere. For $1<p<\infty$, Carleson
and Hunt observed that the Fourier series of an $L^{p}(\mathbf{T}^{1})$
function is almost everywhere convergent, see [2, 9]. Fefferman [5] presented
an alternative proof for the above result. Lacey and Thiele [14], combining
ideas from [2] and [5], gave the third approach to Carleson’s theorem on the
pointwise convergence of the Fourier series. For other essential properties
and applications of the Fourier series on the torus, one can refer to [7, 8,
15, 16, 20, 27] and the references therein.

With the development of signal processing, the Fourier transform has gradually
exposed its limitations in processing non-stationary signals. In this
situation, the fractional Fourier transform (FRFT for short) was proposed to
overcome this problem. On the other hand, if signal $f$ is $t$-periodic with
$t<1$, then $f$ is not well defined on $\mathbf{T}^{n}$, thus we can not
obtain its frequency components. We need to introduce the matched fractional
torus to study such kind of signal (or function).

######  Definition 1.1.

Let $\alpha\in\mathbb{R}$, $\alpha\neq\pi\mathbf{Z}$. The fractional torus of
order $\alpha$, denoted by $\mathbf{T}^{n}_{\alpha}$, is the cube
$[0,|\sin\alpha|]^{n}$ with opposite sides identified.

Note that $\mathbf{T}^{n}_{\alpha}=\mathbf{T}^{n}$ for
$\alpha=\pi/2+2\pi\mathbf{Z}$, see Figure 1.1 (a) for two-dimensional torus.
The graphs of fractional torus $\mathbf{T}^{2}_{\alpha}$ for
$\alpha=\pi/2,\pi/3,\pi/6$ are shown in Figure 1.1 (b). Functions on
$\mathbf{T}^{n}_{\alpha}$ are functions $f$ on ${\mathbb{R}^{n}}$ that satisfy
$f(x+|\sin\alpha|m)=f(x)$ for any $x\in{\mathbb{R}^{n}}$ and
$m\in\mathbf{Z}^{n}$. Such functions are called $|\sin\alpha|$-periodic in
every coordinate for fixed $\alpha$.

(a) Torus $\mathbf{T}^{2}$

(b) fractional torus of order $\alpha$ $\mathbf{T}^{2}_{\alpha}$

Figure 1.1: The two-dimensional fractional torus of order $\alpha$
$\mathbf{T}^{2}_{\alpha}$

The idea of the Fourier operator’s fractional power appeared first in the work
of Wiener [29]. Namias [22] introduced the FRFT to solve certain types of
ordinary and partial differential equations arising in quantum mechanics. In
[21], McBride and Kerr provided a rigorous definition of the FRFT in integral
form on the Schwartz space $S(\mathbb{R})$ based on a modification of Namias’
fractional operators. Later, Kerr [11] discussed the $L^{2}(\mathbb{R})$
theory of FRFT. Zayed [30] introduced a new convolution structure for the FRFT
that preserves the convolution theorem for the Fourier transform. In [31],
Zayed presented a new class of fractional integral transforms that includes
the fractional Fourier and Hankel transforms. Zayed [32] gave two-dimensional
FRFT and studied its properties, such as the inversion theorem, convolution
theorem, and Poisson summation formula. Kamalakkannan and Roopkumar [10]
established the convolution theorem and product theorem for the
multidimensional fractional Fourier transform. Chen et al [3] established the
$L^{p}(\mathbb{R})$ theory of FRFT for $1\leq p<2$. The FRFT also has various
applications in many fields, such as optics [23], signal processing [24, 26],
image processing [17, 18], and so on. In this paper, we are devoted to
studying the convergence and applications of fractional Fourier series on the
torus.

This paper is organized as follows. In Section 2, we introduce fractional
Fourier coefficients and give some basic properties including fractional
convolution on $\mathbf{T}^{n}_{\alpha}$. In Section 3, we establish the
fractional Fourier inversion and Possion summation formula. Section 4 is
devoted to the relationship between decay of fractional Fourier coefficients
and the smoothness of a function. The pointwise convergences of fractional
Fejér means are given in Section 5. In Section 6, using fractional Fourier
series on $\mathbf{T}^{n}_{\alpha}$, we obtain the solutions of fractional
heat equation and fractional Dirichlet problem. Finally, we present a non-
stationary signal on $\mathbf{T}^{n}_{\alpha}$, which can be recovered by
approximating method.

##  2 Fractional Fourier series on the torus

In this section, we firstly introduce the definition of fractional Fourier
series in the setting $\mathbf{T}^{n}_{\alpha}$. Then we establish some basic
facts of fractional Fourier analysis. Moreover, we give the fractional
convolution and obtain fractional approximation in
$L^{p}(\mathbf{T}^{n}_{\alpha})$, $1\leq p\leq\infty$.

Let $\alpha\in\mathbb{R}$, $\alpha\neq\pi\mathbf{Z}$. Set
$e_{\alpha}(x):=e^{\pi i|x|^{2}\cot\alpha}$ and
$e_{\alpha}f(x)=e_{\alpha}(x)f(x)$ for a function $f$ on
$\mathbf{T}^{n}_{\alpha}$.

######  Definition 2.1.

Let $1\leq p\leq\infty$. We say that a function $f$ on
$\mathbf{T}^{n}_{\alpha}$ lies in the space
$e_{-\alpha}L^{p}(\mathbf{T}^{n}_{\alpha})$ if

$\displaystyle f(x)=e_{-\alpha}g(x),\quad g\in L^{p}(\mathbf{T}^{n}_{\alpha})$

and satisfies $\|f\|_{L^{p}(\mathbf{T}^{n}_{\alpha})}<\infty$.

######  Definition 2.2.

For a complex-valued function $f\in
e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$, $\alpha\in\mathbb{R}$ and
$m\in\mathbf{Z}^{n}$, we define

$\mathcal{F}_{\alpha}(f)(m)=\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}f(x)K_{\alpha}(m,x)dx,&\alpha\neq\pi\mathbf{Z},\\\
f(m),&\alpha=2\pi\mathbf{Z},\\\
f(-m),&\alpha=2\pi\mathbf{Z}+\pi,\end{array}\right.$

where

$\displaystyle
K_{\alpha}(m,x):=A_{\alpha}^{n}e_{\alpha}(x)e_{\alpha}(m,x)e_{\alpha}(m),$

here $A_{\alpha}=\sqrt{1-i\cot\alpha}$ and $e_{\alpha}(m,x)=e^{-2\pi i(m\cdot
x)\csc\alpha}$. We call $\mathcal{F}_{\alpha}(f)(m)$ the $m$-th fractional
Fourier coefficient of order $\alpha$ of $f$.

In order to state our results, we recall some notations. The spaces
$C^{k}(\mathbf{T}^{n}_{\alpha})$, $k\in\mathbf{Z}^{+}$, are defined as the
sets of functions $\phi$ for which $\partial^{\beta}\phi$ exist and are
continuous for all $|\beta|\leq k$. When $k=0$ we set
$C^{0}(\mathbf{T}^{n}_{\alpha})=C(\mathbf{T}^{n}_{\alpha})$ to be the space of
continuous functions on $\mathbf{T}^{n}_{\alpha}$. Let

$\displaystyle
C^{\infty}(\mathbf{T}^{n}_{\alpha}):=\bigcap_{k=0}^{\infty}C^{k}(\mathbf{T}^{n}_{\alpha}).$

######  Definition 2.3.

For $0\leq k\leq\infty$. The space $e_{-\alpha}C^{k}(\mathbf{T}^{n}_{\alpha})$
is defined to be the space of functions $f$ on $\mathbf{T}^{n}_{\alpha}$ such
that

$\displaystyle f(x)=e_{-\alpha}g(x),\quad g\in
C^{k}(\mathbf{T}^{n}_{\alpha}).$

Notice that the spaces $e_{-\alpha}C^{k}(\mathbf{T}^{n}_{\alpha})$ are
contained in $e_{-\alpha}L^{p}(\mathbf{T}^{n}_{\alpha})$ for all $1\leq
p\leq\infty$.

We denote by $\overline{f}$ the complex conjugate of the function $f$, by
$\tilde{f}$ the function $\tilde{f}(x)=f(-x)$, and by $\tau_{y}(f)(x)$ the
function $\tau_{y}(f)(x)=f(x-y)$ for all $y\in\mathbf{T}^{n}_{\alpha}$. Since
$\alpha=\pi\mathbf{Z}$, we get $\mathbf{T}^{n}_{\alpha}=\\{0\\}$. Hence,
throughout this paper, for $\alpha\neq\pi\mathbf{Z}$, we always assume
$\mathbf{T}^{n}_{\alpha}=[-|\sin\alpha|/2,|\sin\alpha|/2]^{n}$. Next, we give
some elementary properties of fractional Fourier coefficients.

######  Proposition 2.4.

Let $f,g$ be in $e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$. Then for all
$m,k\in\mathbf{Z}^{n}$, $\lambda\in\mathbb{C}$, $y\in\mathbf{T}^{n}_{\alpha}$,
we have

  1. 1.

$\mathcal{F}_{\alpha}(f+g)(m)=\mathcal{F}_{\alpha}(f)(m)+\mathcal{F}_{\alpha}(g)(m)$;

  2. 2.

$\mathcal{F}_{\alpha}(\lambda f)(m)=\lambda\mathcal{F}_{\alpha}(f)(m)$;

  3. 3.

$\mathcal{F}_{\alpha}(\overline{f})(m)=\overline{{\mathcal{F}_{-\alpha}}(f)(m)}$;

  4. 4.

$\mathcal{F}_{\alpha}(\widetilde{f})(m)=\mathcal{F}_{\alpha}(f)(-m)$;

  5. 5.

$\mathcal{F}_{\alpha}[e_{-\alpha}\tau_{y}(e_{\alpha}f)](m)=\mathcal{F}_{\alpha}(f)(m)e_{\alpha}(m,y)$;

  6. 6.

$\mathcal{F}_{\alpha}[e_{-\alpha}(k,\cdot)f](m)e^{-2\pi i(m\cdot
k)\cot\alpha}e_{\alpha}({k})=\mathcal{F}_{\alpha}(f)(m-k)$;

  7. 7.

$\mathcal{F}_{\alpha}(f)(0)=A_{\alpha}^{n}\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}e_{-\alpha}f(x)dx$;

  8. 8.

$\sup_{m\in\mathbf{Z}^{n}}|\mathcal{F}_{\alpha}(f)(m)|\leq|\csc\alpha|^{n/2}\|f\|_{L^{1}(\mathbf{T}^{n}_{\alpha})}$;

  9. 9.

$\mathcal{F}_{\alpha}[e_{-\alpha}\partial^{\beta}(e_{\alpha}f)](m)=(2\pi
im\csc\alpha)^{\beta}\mathcal{F}_{\alpha}(f)(m),$ whenever $f\in
e_{-\alpha}C^{\beta}(\mathbf{T}^{n}_{\alpha})$.

###### Proof.

It is obvious that properties (1)-(4), (7) and (8) hold. We now pay attention
to (5). Note that

$\displaystyle\mathcal{F}_{\alpha}[e_{-\alpha}\tau_{y}\big{(}e_{\alpha}f\big{)}](m)=$
$\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}(e_{\alpha}f)(x-y)e_{\alpha}(m,x)dx$
$\displaystyle=$ $\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}-y}(e_{\alpha}f)(x^{\prime})e_{\alpha}(m,x^{\prime}+y)dx^{\prime}$
$\displaystyle=$ $\displaystyle\mathcal{F}_{\alpha}(f)(m)e_{\alpha}(m,y),$

where we make the variable change $x^{\prime}=x-y$ in the second equality, and
the third equality follows from the periodicity of function.

Next, we deal with (6).

$\displaystyle\mathcal{F}_{\alpha}(f)(m-k)=$ $\displaystyle
A_{\alpha}^{n}\int_{\mathbf{T}^{n}_{\alpha}}f(x)e_{\alpha}(x)e_{\alpha}(m-k)e_{\alpha}(m-k,x)dx$
$\displaystyle=$ $\displaystyle e_{\alpha}(k)e^{-2\pi i(m\cdot
k)\cot\alpha}\int_{\mathbf{T}^{n}_{\alpha}}e_{-\alpha}(k,x)f(x)K_{\alpha}(m,x)dx$
$\displaystyle=$
$\displaystyle\mathcal{F}_{\alpha}[e_{-\alpha}(k,\cdot)f](m)e^{-2\pi i(m\cdot
k)\cot\alpha}e_{\alpha}(k).$

Now we turn to (9). By the periodicity and integration by parts, we have

$\displaystyle\mathcal{F}_{\alpha}[e_{-\alpha}\partial^{\beta}(e_{\alpha}f)](m)=$
$\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}\partial^{\beta}(e_{\alpha}f)(x)e^{-2\pi
i(m\cdot x)\csc\alpha}dx$ $\displaystyle=$ $\displaystyle(2\pi
im\csc\alpha)^{\beta}A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}e_{\alpha}f(x)e^{-2\pi
i(m\cdot x)\csc\alpha}dx$ $\displaystyle=$ $\displaystyle(2\pi
im\csc\alpha)^{\beta}\mathcal{F}_{\alpha}(f)(m).$

This completes the proof. ∎

######  Remark 2.1.

Suppose $f_{1}\in e_{-\alpha}L^{1}(\mathbf{T}^{n_{1}}_{\alpha})$ and $f_{2}\in
e_{-\alpha}L^{1}(\mathbf{T}^{n_{2}}_{\alpha})$. We obtain that the tensor
function

$(f_{1}\otimes f_{2})(x_{1},x_{2})=f_{1}(x_{1})f_{2}(x_{2})$

is in $e_{-\alpha}L^{1}(\mathbf{T}^{n_{1}+n_{2}}_{\alpha})$. Moreover, it is
easy to check

$\displaystyle\mathcal{F}_{\alpha}\big{(}f_{1}\otimes
f_{2}\big{)}(m_{1},m_{2})=\mathcal{F}_{\alpha}(f_{1})(m_{1})\mathcal{F}_{\alpha}(f_{2})(m_{2}),$
(2.1)

where $m_{1}\in\mathbf{Z}^{n_{1}}$ and $m_{2}\in\mathbf{Z}^{n_{2}}$.

######  Definition 2.5.

For $\alpha\neq n\pi$, a trigonometric polynomials of order $\alpha$ on
$\mathbf{T}^{n}_{\alpha}$ is a function of the form

$\displaystyle
P_{\alpha}(x)=\sum_{m\in\mathbf{Z}^{n}}c_{m,\alpha}K_{-\alpha}(m,x),$

where $m=(m_{1},\ldots,m_{n})$ and $c_{m,\alpha}$ is a constant depending on
$m$ and $\alpha$. The degree of $P_{\alpha}$ is the largest number
$|m_{1}|+\cdots+|m_{n}|$ such that $c_{m,\alpha}$ is nonzero. Observe that

$\displaystyle\mathcal{F}_{\alpha}(P_{\alpha})(m)=c_{m,\alpha}.$

For $x\in\mathbf{T}^{n}_{\alpha}$, $f\in
e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$, the fractional Fourier series of
order $\alpha$ of $f$ is the series

$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x).$
(2.2)

Now, we wonder (2.2) converges in which sense? The convergence of the
fractional Fourier series is the main topic in this paper. Next, we introduce
a kind of the fractional convolution of order $\alpha$ to study the
convergence of fractional Fourier series.

######  Definition 2.6.

Let $f\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$ and $g\in
L^{1}(\mathbf{T}^{n})$. Define the fractional convolution of order $\alpha$ as
follows:

$\displaystyle(f\overset{\alpha}{\ast}g)(x)=|\csc\alpha|^{n}{e_{-\alpha}(x)\int_{\mathbf{T}^{n}_{\alpha}}e_{\alpha}(y)f(y)(\delta^{\alpha}g)(x-y)dy},$

where $(\delta^{\alpha}g)(x)=g(x\csc\alpha)$.

Let $f\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$. We have

$\displaystyle\sum_{|m|\leq N}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x)$
$\displaystyle=\sum_{|m|\leq
N}\int_{\mathbf{T}^{n}_{\alpha}}f(y)K_{\alpha}(m,y)dyK_{-\alpha}(m,x)$
$\displaystyle=|\csc\alpha|^{n}e_{-\alpha}(x)\sum_{|m|\leq
N}\int_{\mathbf{T}^{n}_{\alpha}}e_{\alpha}(y)f(y)e^{2\pi
im\cdot(x-y)\csc\alpha}dy$
$\displaystyle=|\csc\alpha|^{n}e_{-\alpha}(x)\int_{\mathbf{T}^{n}_{\alpha}}e_{\alpha}(y)f(y)\sum_{|m|\leq
N}e^{2\pi im\cdot(x-y)\csc\alpha}dy$
$\displaystyle=:\big{(}f\overset{\alpha}{\ast}{D}_{N}^{n}\big{)}(x),$

where ${D}^{n}_{N}(x)$ is the classical multidimensional Dirichlet kernel.

Noting that one-dimensional Dirichlet kernel ${{D}}_{N}^{1}$ is not an
approximate identity. This fact led Cesàro and independently Fejér to consider
the arithmetic means of the Dirichlet kernel in dimension $1$ as follows

$\displaystyle{F}^{1}_{N}(x):=\frac{1}{N+1}\big{[}{D}_{0}^{1}(x)+{D}_{1}^{1}(x)+\cdots+{D}_{N}^{1}(x)\big{]}.$

By Proposition 3.1.7 in [7], we have the following equivalent way to write the
kernel ${F}^{1}_{N}$:

$\displaystyle{{F}}^{1}_{N}(x)=\sum_{j=-N}^{N}\Big{(}1-\frac{|j|}{N+1}\Big{)}e^{2\pi
ijx}={\frac{1}{N+1}\Big{(}\frac{\sin\big{(}(N+1)\pi x\big{)}}{\sin(\pi
x)}\Big{)}^{2}}.$ (2.3)

The function ${{F}}^{1}_{N}$ given by (2.3) is called the Fejér kernel. Let

${{F}}^{1,\alpha}_{N}(x):=|\csc\alpha|F^{1}_{N}(x\csc\alpha).$

We have the following identity for the kernel ${{F}}^{1,\alpha}_{N}$:

######  Proposition 2.7.

For every nonnegative integer $N$, the following identity holds

$\displaystyle{{F}}^{1,\alpha}_{N}(x)=|\csc\alpha|\sum_{j=-N}^{N}\Big{(}1-\frac{|j|}{N+1}\Big{)}e^{2\pi
ijx\csc\alpha}={\frac{|\csc\alpha|}{N+1}\Big{(}\frac{\sin\big{(}(N+1)\pi
x\csc\alpha\big{)}}{\sin(\pi x\csc\alpha)}\Big{)}^{2}}$ (2.4)

for all $x\in\mathbf{T}^{1}_{\alpha}$. Thus

$\mathcal{F}_{\alpha}\big{(}e_{-\alpha}{{F}}^{1,\alpha}_{N}\big{)}(m)=A_{\alpha}e_{\alpha}(m)\Big{(}1-\frac{|m|}{N+1}\Big{)}$

if $|m|\leq N$ and zero otherwise.

###### Proof.

Note that ${{F}}^{1,\alpha}_{N}\in L^{1}(\mathbf{T}^{1}_{\alpha})$. We have

$\displaystyle\mathcal{F}_{\alpha}\big{(}e_{-\alpha}{{F}}^{1,\alpha}_{N}\big{)}(m)=$
$\displaystyle\int_{\mathbf{T}^{1}_{\alpha}}e_{-\alpha}(x){{F}}^{1,\alpha}_{N}(x)K_{\alpha}(m,x)dx$
$\displaystyle=$ $\displaystyle
A_{\alpha}|\csc\alpha|e_{\alpha}(m)\sum_{j=-N}^{N}\Big{(}1-\frac{|j|}{N+1}\Big{)}\int_{\mathbf{T}^{1}_{\alpha}}e_{\alpha}(m,x)e_{-\alpha}(j,x)dx$
$\displaystyle=$ $\displaystyle
A_{\alpha}e_{\alpha}(m)\Big{(}1-\frac{|m|}{N+1}\Big{)},$ (2.5)

where

$\displaystyle|\csc\alpha|\sum_{j=-N}^{N}\Big{(}1-\frac{|j|}{N+1}\Big{)}\int_{\mathbf{T}^{1}_{\alpha}}e_{\alpha}(m,x)e_{-\alpha}(j,x)dx=0,\qquad
m\neq j,$
$\displaystyle|\csc\alpha|\sum_{j=-N}^{N}\Big{(}1-\frac{|j|}{N+1}\Big{)}\int_{\mathbf{T}^{1}_{\alpha}}e_{\alpha}(m,x)e_{-\alpha}(j,x)dx=1-\frac{|m|}{N+1},\qquad
m=j.$

∎

######  Definition 2.8.

Let $N$ be a nonnegative integer. The function ${{F}}^{1,\alpha}_{N}$ on
$\mathbf{T}^{1}_{\alpha}$ given by $(\ref{eq2.3})$ is called the fractional
Fejér kernel of order $\alpha$.

The Fejér kernel ${{F}}^{n}_{N}$ on the torus is defined as the product of the
$1$-dimensional Fejér kernels. Similarly, we define fractional Fejér kernel
${{F}}^{n,\alpha}_{N}$ on $\mathbf{T}^{n}_{\alpha}$ as the product of the
$1$-dimensional fractional Fejér kernels, namely

$\displaystyle{{F}}^{n,\alpha}_{N}(x_{1},\ldots,x_{n}):=$
$\displaystyle\prod_{j=1}^{n}{{F}}^{1,\alpha}_{N}(x_{j}).$

######  Remark 2.2.

For all $N\geq 0$, by $(\ref{eq2.3})$, we get

$\displaystyle{{F}}^{n,\alpha}_{N}(x_{1},\ldots,x_{n})=$
$\displaystyle|\csc\alpha|^{n}\sum_{m\in\mathbf{Z}^{n},|m_{j}|\leq
N}\prod_{j=1}^{n}\Big{(}1-\frac{|m_{j}|}{N+1}\Big{)}e^{2\pi i(m\cdot
x)\csc\alpha}$ $\displaystyle=$
$\displaystyle\frac{|\csc\alpha|^{n}}{(N+1)^{n}}\prod_{j=1}^{n}\Big{(}\frac{\sin\big{(}(N+1)\pi
x_{j}\csc\alpha\big{)}}{\sin(\pi x_{j}\csc\alpha)}\Big{)}^{2}.$

Thus ${{F}}^{n,\alpha}_{N}\geq 0$. Note that
${{F}}^{n,\alpha}_{0}(x)=|\csc\alpha|^{n}$ and
${{F}}^{n,\alpha}_{N}(0)=|\csc\alpha|^{n}(N+1)^{n}$.

######  Proposition 2.9.

The family of fractional Fejér kernels
$\\{{{F}}^{n,\alpha}_{N}\\}_{N=0}^{\infty}$ is an approximate identity on
$\mathbf{T}^{n}_{\alpha}$.

###### Proof.

It is easy to see that

$\displaystyle\|{F}^{n,\alpha}_{N}\|_{L^{1}(\mathbf{T}^{n}_{\alpha})}=\int_{\mathbf{T}^{n}_{\alpha}}{F}^{n,\alpha}_{N}(x)dx=\prod_{j=1}^{N}\int_{\mathbf{T}^{1}_{\alpha}}{F}^{1,\alpha}_{N}(x_{j})dx_{j}=1.$

By noting that $1\leq{|t|}{/|\sin t|}\leq{\pi}{/2}$ when $|t|\leq{\pi}{/2}$,
we obtain

$\displaystyle{F}^{1,\alpha}_{N}(x)\leq$
$\displaystyle\frac{|\csc\alpha|}{N+1}\min\Big{(}\frac{(N+1)|\pi
x\csc\alpha|}{|\sin(\pi x\csc\alpha)|},\frac{1}{|\sin(\pi
x\csc\alpha)|}\Big{)}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{|\csc\alpha|}{N+1}\frac{\pi^{2}}{4}\min\Big{(}N+1,\frac{1}{|\pi
x\csc\alpha|}\Big{)}^{2},$

where $|x|\leq\frac{1}{2|\csc\alpha|}$. For $\delta>0$, we get

$\displaystyle\int_{\delta\leq|x|\leq\frac{1}{2|\csc\alpha|}}{F}^{1,\alpha}_{N}(x))dx\leq\frac{|\csc\alpha|}{N+1}\frac{\pi^{2}}{4}\int_{\delta\leq|x|\leq\frac{1}{2|\csc\alpha|}}\frac{1}{|\pi\delta\csc\alpha|^{2}}dx\leq\frac{1}{4\delta^{2}}\frac{1}{N+1}\rightarrow
0$

as $N\rightarrow\infty$.

In higher dimensions, given
$x=(x_{1},\ldots,x_{n})\in[-{|\sin\alpha|}/{2},{|\sin\alpha|}/{2}]^{n}$ with
$|x|\geq\delta,$ there exists a $j\in\\{1,\ldots,n\\}$ such that
$|x_{j}|\geq\delta/\sqrt{n}$ and thus

$\displaystyle\int_{|x|\geq\delta}{F}^{n,\alpha}_{N}(x)dx\leq$
$\displaystyle\int_{|x_{j}|\geq\frac{\delta}{\sqrt{n}}}{F}^{1,\alpha}_{N}(x_{j})dx_{j}\prod_{k\neq
j}\int_{\mathbf{T}^{1}_{\alpha}}{F}^{1,\alpha}_{N}(x_{k})dx_{k}\leq\frac{n}{4\delta^{2}}\frac{1}{N+1}\rightarrow
0,$ (2.6)

which completes the proof. ∎

######  Theorem 2.10.

Let $\alpha\in\mathbb{R}$ and $\alpha\neq\pi\mathbf{Z}$.
(1) If $f\in e_{-\alpha}L^{p}(\mathbf{T}^{n}_{\alpha})$, $1\leq p<\infty$,
then

$\displaystyle\lim_{N\rightarrow\infty}\Big{\|}f\overset{\alpha}{\ast}{{F}}^{n}_{N}-f\Big{\|}_{L^{p}(\mathbf{T}^{n}_{\alpha})}=0.$

(2) If $f\in e_{-\alpha}L^{\infty}(\mathbf{T}^{n}_{\alpha})$ is uniformly
continuous on a subset $K$ of $\mathbf{T}^{n}_{\alpha}$, then

$\displaystyle\lim_{N\rightarrow\infty}\Big{\|}f\overset{\alpha}{\ast}{{F}}^{n}_{N}-f\Big{\|}_{L^{\infty}(K)}=0.$

.

###### Proof.

We first deal with the case $1\leq p<\infty$. It is easy to check that

$\displaystyle(f\overset{\alpha}{\ast}{{F}}^{n}_{N})(x)-f(x)=$ $\displaystyle
e_{-\alpha}(x)\int_{\mathbf{T}^{n}_{\alpha}}e_{\alpha}f(t){{F}}^{n,\alpha}_{N}(x-t)\mathrm{d}t-\int_{\mathbf{T}^{n}_{\alpha}}f(x){{F}}^{n,\alpha}_{N}(t)\mathrm{d}t$
$\displaystyle=$ $\displaystyle
e_{-\alpha}(x)\int_{\mathbf{T}^{n}_{\alpha}}\big{(}(e_{\alpha}f)(x-t)-(e_{\alpha}f)(x)\big{)}{{F}}^{n,\alpha}_{N}(t)\mathrm{d}t.$

Now approximate a given $e_{\alpha}f\in L^{p}(\mathbf{T}^{n}_{\alpha})$ by a
continuous function with compact support to deduce that

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\big{|}(e_{\alpha}f)(x-h)-(e_{\alpha}f)(x)\big{|}^{p}dx\rightarrow
0\quad as\quad h\rightarrow 0.$

For given $\varepsilon>0$, there exists $\delta>0$ such that

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\big{|}(e_{\alpha}f)(x-h)-(e_{\alpha}f)(x)\big{|}^{p}dx<\frac{\varepsilon^{p}}{2^{p}},\quad
h\in[-\delta,\delta].$ (2.7)

Then

$\displaystyle(f\overset{\alpha}{\ast}{{F}}^{n}_{N})(x)-f(x)$
$\displaystyle=e_{\alpha}(x)\int_{|t|\leq\delta}\big{(}(e_{\alpha}f)(x-t)-(e_{\alpha}f)(x)\big{)}{{F}}^{n,\alpha}_{N}(t)\mathrm{d}t$
$\displaystyle\qquad+e_{\alpha}(x)\int_{|t|>\delta}\big{(}(e_{\alpha}f)(x-t)-(e_{\alpha}f)(x)\big{)}{{F}}^{n,\alpha}_{N}(t)dt$
$\displaystyle=:I_{1}+I_{2}.$

By Minkowski’s integral inequality and (2.6), we obtain

$\displaystyle\|I_{2}\|_{p}$
$\displaystyle\leq\int_{|t|>\delta}\Big{(}\int_{\mathbf{T}^{n}_{\alpha}}\big{|}\big{(}e_{\alpha}f\big{)}(x-t)-\big{(}e_{\alpha}f\big{)}(x)\big{|}^{p}dx\Big{)}^{1/p}{{F}}^{n,\alpha}_{N}(t)dt$
(2.8) $\displaystyle\leq
2\|f\|_{p}\int_{|t|>\delta}{{F}}^{n,\alpha}_{N}(t)dt\rightarrow 0\qquad
as\quad N\rightarrow\infty.$

In addition, by $(\ref{eq2.2})$, we get

$\displaystyle\|I_{1}\|_{p}$
$\displaystyle\leq\int_{|t|\leq\delta}\Big{(}\int_{\mathbf{T}^{n}_{\alpha}}\big{|}\big{(}e_{\alpha}f\big{)}(x-t)-\big{(}e_{\alpha}f\big{)}(x)\big{|}^{p}dx\Big{)}^{1/p}{{F}}^{n,\alpha}_{N}(t)dt<\frac{\varepsilon}{2}.$

This together with (2.8) implies the required conclusion.

Now, we turn to conclusion $(2)$. Let $e_{\alpha}f$ be a bounded function on
$\mathbf{T}^{n}_{\alpha}$ that is uniformly continuous on K. Given
$\varepsilon>0$, there exists a neighborhood $V$ of $0$ such that

$\displaystyle\big{|}(e_{\alpha}f)(x-h)-(e_{\alpha}f)(x)\big{|}<\frac{\varepsilon}{2}{\rm{\;for\;all\;}}h\in
V\;{\rm{and}}\;x\in K.$

Applying this along with $(\ref{eq1.5})$, we get that as $N\to\infty$

$\displaystyle\sup_{x\in
K}\big{|}(f\overset{\alpha}{\ast}{{F}}^{n,\alpha}_{N})(x)-f(x)\big{|}$
$\displaystyle\leq\int_{|t|\leq\delta}\sup_{x\in
K}\big{|}(e_{\alpha}f)(x-t)-(e_{\alpha}f)(x)\big{|}{{F}}^{n,\alpha}_{N}(t)\mathrm{d}t$
$\displaystyle\qquad+\int_{|t|>\delta}\sup_{x\in
K}\big{|}(e_{\alpha}f)(x-t)-(e_{\alpha}f)(x)\big{|}{{F}}^{n,\alpha}_{N}(t)dt$
$\displaystyle\leq\frac{\varepsilon}{2},$

which completes the proof. ∎

######  Proposition 2.11.

The set of trigonometric polynomials of order $\alpha$ is dense in
$e_{-\alpha}L^{p}(\mathbf{T}^{n}_{\alpha})$ for $1\leq p<\infty$.

###### Proof.

For $1\leq p<\infty$, we claim that $f\overset{\alpha}{\ast}{{F}}^{n}_{N}$ is
also a trigonometric polynomial of order $\alpha$. In fact,

$\displaystyle(f\overset{\alpha}{\ast}{{F}}^{n}_{N})(x)=$
$\displaystyle\underset{|m_{j}|\leq
N}{\sum_{m\in\mathbf{Z}^{n}}}\prod_{j=1}^{n}\Big{(}1-\frac{|m_{j}|}{N+1}\Big{)}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x).$

It follows from Theorem 2.10 that
$f\overset{\alpha}{\ast}{\mathcal{F}}^{n}_{N}$ converges to $f$ as
$N\to\infty$. This finishes the proof. ∎

######  Corollary 2.12.

(Weierstrass approximation theorem for trigonometric polynomials of order
$\alpha$) Every continuous function on $\mathbf{T}^{n}_{\alpha}$ is a uniform
limit of trigonometric polynomials with order $\alpha$.

###### Proof.

Note that $\mathbf{T}^{n}_{\alpha}$ is a compact set. By Theorem 2.10 (2), we
know that $f\overset{\alpha}{\ast}{\mathcal{F}}^{n}_{N}$ converges uniformly
to $f$ as $N\to\infty$. In Proposition 2.11, we obtain that
$f\overset{\alpha}{\ast}{\mathcal{F}}^{n}_{N}$ is a trigonometric polynomial
of order $\alpha$. Therefore, we conclude that every continuous function on
$\mathbf{T}^{n}_{\alpha}$ can be uniformly approximated by trigonometric
polynomials of order $\alpha$. ∎

##  3 Reproduction of Functions from Their Fractional Fourier Coefficients

In this section, we establish the fractional Fourier inversion on
$e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$ and study basic properties of
$e_{-\alpha}L^{2}(\mathbf{T}^{n}_{\alpha})$. Moreover, to explore the
connections between fractional Fourier analysis on $\mathbf{T}^{n}_{\alpha}$
and fractional Fourier analysis on ${\mathbb{R}^{n}}$, we give the fractional
Poisson summation formula.

###  3.1 Fractional Fourier inversion

We now define the partial sums of fractional Fourier series.

######  Definition 3.1.

Let $N\in\mathbb{N}$, $\alpha\in\mathbb{R}$ and $\alpha\neq\pi\mathbf{Z}$. The
expressions

$\displaystyle(f\overset{\alpha}{\ast}{F}_{N}^{n})(x)=$
$\displaystyle\underset{|m_{j}|\leq
N}{\sum_{m\in\mathbf{Z}^{n}}}\prod_{j=1}^{n}\Big{(}1-\frac{|m_{j}|}{N+1}\Big{)}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x)$
(3.1)

are called the fractional Fejér means (or fractional Cesàro means) of $f$.

In the following propositions, we obtain that the fractional Fourier series
uniquely determine the function.

######  Proposition 3.2.

If $f,g\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$ satisfy
$\mathcal{F}_{\alpha}(f)(m)=\mathcal{F}_{\alpha}(g)(m)$ for all
$m\in\mathbf{Z}^{n}$, then $f=g$ a.e.

###### Proof.

By linearity of $\mathcal{F}_{\alpha}$, we can set $g=0$. If
$\mathcal{F}_{\alpha}(f)(m)=0$ for all $m\in\mathbf{Z}^{n}$, we know from
(3.1) that $f\overset{\alpha}{\ast}{F}_{N}^{n}=0$ for all
$N\in\mathbf{Z}^{+}$. By Theorem 2.10, we have

$\displaystyle\big{\|}f\overset{\alpha}{\ast}{{F}}^{n}_{N}-f\big{\|}_{L^{1}(\mathbf{T}^{n}_{\alpha})}\rightarrow
0$

as $N\rightarrow\infty$. Then

$\displaystyle\|f\|_{L^{1}(\mathbf{T}^{n}_{\alpha})}\leq\big{\|}f\overset{\alpha}{\ast}{{F}}^{n}_{N}-f\big{\|}_{L^{1}(\mathbf{T}^{n}_{\alpha})}+\big{\|}f\overset{\alpha}{\ast}{{F}}^{n}_{N}\big{\|}_{L^{1}(\mathbf{T}^{n}_{\alpha})}\rightarrow
0,$

from which we conclude that $f=0$ a.e. ∎

Whether the partial sums of the fractional Fourier series converge to the
function as $N\rightarrow\infty$? We give a firm answer to this question.

######  Corollary 3.3.

(Fractional Fourier inversion) Suppose $f\in
e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$ and

$\sum_{m\in\mathbf{Z}^{n}}|\mathcal{F}_{\alpha}(f)(m)|<\infty.$

Then

$\displaystyle
f(x)=\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x)\quad
a.e.,$ (3.2)

and therefore $f$ is almost everywhere equal to a continuous function.

###### Proof.

For $m\neq n$, it is easy to see that

$\displaystyle
K_{-\alpha}(m,x)K_{\alpha}(n,x)=|\csc\alpha|^{n}e_{-\alpha}(m)e_{\alpha}(n)e_{\alpha}(n-m,x).$

This implies

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}K_{-\alpha}(m,x)K_{\alpha}(n,x)dx=0,\quad
m\neq n.$

Meanwhile

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}K_{-\alpha}(m,x)K_{\alpha}(m,x)dx=$
$\displaystyle|\csc\alpha|^{n}\int_{\mathbf{T}^{n}_{\alpha}}dx=1.$

Set

$G(x):=\sum_{j\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(j)K_{-\alpha}(j,x).$

It is obvious that $G\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$.
Consequently,

$\displaystyle\mathcal{F}_{\alpha}(G)(m)=$
$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\Big{(}\sum_{j\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(j)K_{-\alpha}(j,x)\Big{)}K_{\alpha}(m,x)dx=\mathcal{F}_{\alpha}(f)(m).$

By Proposition 3.2, we have

$\displaystyle
f(x)=\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x)\quad
a.e.$

This completes the proof. ∎

###  3.2 Fractional Fourier series of square summable functions

Now consider the space $e_{-\alpha}L^{2}(\mathbf{T}^{n}_{\alpha})$ with inner
product

$\displaystyle\langle
f|g\rangle=\int_{\mathbf{T}^{n}_{\alpha}}f(t)\overline{g(t)}dt.$

It is easy to check that

$\int_{\mathbf{T}^{n}_{\alpha}}K_{-\alpha}(m,x)\overline{K_{-\alpha}(n,x)}dx=\left\\{\begin{array}[]{ll}1,&when\;m=n,\\\
0,&when\;m\neq n.\end{array}\right.$

This implies that the sequence $\\{K_{-\alpha}(m,\cdot)\\}$ is orthonormal.
Meanwhile, for all $f\in e_{-\alpha}L^{2}(\mathbf{T}^{n}_{\alpha})$, we have

$\displaystyle\langle f|K_{-\alpha}(m,\cdot)\rangle=$
$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}f(y)\overline{K_{-\alpha}(m,y)}dy=\mathcal{F}_{\alpha}(f)(m).$

If $\langle f|K_{-\alpha}(m,\cdot)\rangle=0$ for all $m\in\mathbf{Z}^{n}$, we
know from Proposition 3.2 that $f=0$ a.e. Therefore, the completeness of the
sequence $\\{K_{-\alpha}(m,\cdot)\\}$ holds.

######  Proposition 3.4.

The following are valid for $f,g\in e_{-\alpha}L^{2}(\mathbf{T}^{n}_{\alpha})$
and
(1) (Plancherel’s identity)

$\displaystyle\|f\|_{2}^{2}=\sum_{m\in\mathbf{Z}^{n}}|\mathcal{F}_{\alpha}(f)(m)|^{2}.$

(2) The function $f(x)$ is a.e. equal to the
$e_{-\alpha}L^{2}(\mathbf{T}^{n}_{\alpha})$ limit of the sequence

$\displaystyle\sum_{|m|\leq M}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x).$

(3) (Parseval’s relation)

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}f(t)\overline{g(t)}dt=\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)\overline{\mathcal{F}_{\alpha}(g)(m)}.$

(4) The map $f\mapsto\big{\\{}\mathcal{F}_{\alpha}(f)(m)\big{\\}}$ is an
isometry from $e_{-\alpha}L^{2}(\mathbf{T}^{n}_{\alpha})$ onto $l^{2}$.
(5) For all $k\in\mathbf{Z}^{n}$, we have

$\displaystyle\mathcal{F}_{\alpha}\big{[}e_{\alpha}(\cdot)fg\big{]}(m)={A_{\alpha}^{n}}e_{\alpha}^{2}(m)\sum_{j\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(j)e^{-2\pi
i(j\cdot m)\cot\alpha}{\mathcal{F}_{-\alpha}(e_{\alpha}^{2}(\cdot)g)(j-m)}.$

###### Proof.

The proofs of (1), (2), (3) and (4) are quite similar to those in the case of
Fourier series, so we omit them. Now we turn to (5). Let
$\overline{G(x)}=e_{\alpha}(x)g(x)K_{\alpha}(m,x)$. From (3), we have

$\displaystyle\mathcal{F}_{\alpha}\big{[}e_{\alpha}(fg)\big{]}(m)=$
$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}e_{\alpha}(x)f(x)g(x)K_{\alpha}(m,x)dx$
$\displaystyle=$
$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}f(x)\overline{G(x)}dx$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(j)\overline{\mathcal{F}_{\alpha}(G)(j)}.$

In view of Proposition 2.4 (6), we get

$\displaystyle{\mathcal{F}_{\alpha}(G)(j)}=$
$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\overline{e_{\alpha}(x)g(x)K_{\alpha}(m,x)}K_{\alpha}(j,x)dx$
$\displaystyle=$
$\displaystyle\overline{A_{\alpha}^{n}}e_{-\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}\big{[}\overline{e^{2}_{\alpha}(x)g(x)}e^{2\pi
imx\csc\alpha}\big{]}K_{\alpha}(j,x)dx$ $\displaystyle=$
$\displaystyle\overline{A_{\alpha}^{n}}e_{-\alpha}(m)\mathcal{F}_{\alpha}\big{(}\overline{e_{\alpha}^{2}g}e_{-\alpha}(m,\cdot)\big{)}(j)$
$\displaystyle=$
$\displaystyle\overline{A_{\alpha}^{n}}e_{-\alpha}^{2}(m)e^{2\pi i(j\cdot
m)\cot\alpha}\mathcal{F}_{\alpha}\big{(}\overline{e^{2}_{\alpha}g}\big{)}(j-m).$

By Proposition 2.4(3), we have

$\displaystyle\overline{\mathcal{F}_{\alpha}(G)(j)}=$
$\displaystyle{A_{\alpha}^{n}}e^{2}_{\alpha}(m)e^{-2\pi i(j\cdot
m)\cot\alpha}\overline{\mathcal{F}_{\alpha}\big{(}\overline{e^{2}_{\alpha}g}\big{)}(j-m)}$
$\displaystyle=$ $\displaystyle{A_{\alpha}^{n}}e^{2}_{\alpha}(m)e^{-2\pi
i(j\cdot
m)\cot\alpha}{\mathcal{F}_{-\alpha}\big{(}e^{2}_{\alpha}g\big{)}(j-m)}.$

∎

###  3.3 The fractional Poisson summation formula

In this subsection, we establish an important connection between fractional
Fourier analysis on $\mathbf{T}^{n}_{\alpha}$ and Fourier analysis on
${\mathbb{R}^{n}}$.

######  Theorem 3.5.

(Fractional Poisson summation formula) Suppose that $f$ is a continuous
function on ${\mathbb{R}^{n}}$. If there exist $C,\delta>0$ such that

$\displaystyle|f(x)|\leq\frac{C}{(1+|x|)^{n+\delta}},\quad\forall
x\in{\mathbb{R}^{n}}$ (3.3)

and the fractional Fourier transform $\mathcal{F}_{\alpha}(e_{-\alpha}f)$
restricted on $\mathbf{Z}^{n}$ satisfies

$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\big{|}\mathcal{F}_{\alpha}\big{(}e_{-\alpha}f\big{)}(m)\big{|}<\infty.$
(3.4)

Then for all $x\in{\mathbb{R}^{n}}$ we have

$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}\big{(}e_{-\alpha}f\big{)}(m)K_{-\alpha}(m,x)=e_{-\alpha}(x)\sum_{k\in\mathbf{Z}^{n}}f(x+k|\sin\alpha|),$
(3.5)

and in particular

$\displaystyle
A_{-\alpha}^{n}\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}\big{(}e_{-\alpha}f\big{)}(m)e_{-\alpha}(m)=\sum_{k\in\mathbf{Z}^{n}}f(k|\sin\alpha|).$

###### Proof.

Suppose

$F(x)=e_{-\alpha}(x)\sum_{k\in\mathbf{Z}^{n}}f(x+k|\sin\alpha|).$

It is easy to check that $F\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$. We
show that the sequence of the fractional Fourier coefficients of $F$ coincides
with the restriction of the fractional Fourier transform of $f$ on
$\mathbf{Z}^{n}$. In fact, for any $k\in\mathbf{Z}^{n}$, we get

$\displaystyle|k|\sin\alpha|+x|\geq|k\sin\alpha|-|x|\geq|k\sin\alpha|-\frac{|\sin\alpha|}{2}\sqrt{n}$

for all $x\in[-|\sin\alpha|/2,|\sin\alpha|/2]^{n}$. This together with (3.3)
implies

$\displaystyle\sum_{k\in\mathbf{Z}^{n}}\big{|}f(x+k|\sin\alpha|)\big{|}\leq$
$\displaystyle\sum_{k\in\mathbf{Z}^{n}}\frac{C_{n,\delta}}{(1+|k\sin\alpha|)^{n+\delta}}<\infty.$
(3.6)

Therefore,

$\displaystyle\mathcal{F}_{\alpha}(F)(m)=$ $\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}e^{-2\pi i(m\cdot
x)\csc\alpha}\sum_{k\in\mathbf{Z}^{n}}f(x+k|\sin\alpha|)dx$ $\displaystyle=$
$\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\sum_{k\in\mathbf{Z}^{n}}\int_{\mathbf{T}^{n}_{\alpha}}e^{-2\pi
i(m\cdot x)\csc\alpha}f(x+k|\sin\alpha|)dx$ $\displaystyle=$ $\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\sum_{k\in\mathbf{Z}^{n}}\int_{\mathbf{T}^{n}_{\alpha}+k|\sin\alpha|}e^{-2\pi
i(m\cdot x)\csc\alpha}f(x)dx$ $\displaystyle=$ $\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbb{R}^{n}}e^{-2\pi i(m\cdot
x)\csc\alpha}f(x)dx$ $\displaystyle=$
$\displaystyle\mathcal{F}_{\alpha}(e_{-\alpha}f)(m),$

where the second equality follows from (3.6). Meanwhile, we also obtain that
$F$ is continuous. It follows that (3.4) holds with
$|\mathcal{F}_{\alpha}(F)|$ in place of
$|\mathcal{F}_{\alpha}(e_{-\alpha}f)|$. By Corollary 3.3, we obtain that (3.5)
holds for $x\in\mathbf{T}^{n}_{\alpha}$ and then by periodicity, for all
$x\in{\mathbb{R}^{n}}$. ∎

##  4 Decay of Fractional Fourier coefficient

In this section, we study the decay of the fractional Fourier coefficients
associated with the smoothness of a function.

###  4.1 Decay of fractional Fourier coefficients of integrable function

######  Proposition 4.1.

(Riemann-Lebesgue lemma) Suppose $f\in
e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$. we have

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|\to 0\quad as\quad|m|\to\infty.$

###### Proof.

For all $\varepsilon>0$, there exists a trigonometric polynomial $P_{\alpha}$
such that $\|f-P_{\alpha}\|_{L^{1}}<\varepsilon$. If
$|m|>\text{degree}(P_{\alpha})$, then we have
$\mathcal{F}_{\alpha}(P_{\alpha})(m)=0$. Hence

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|=|\mathcal{F}_{\alpha}(f)(m)-\mathcal{F}_{\alpha}(P_{\alpha})(m)|\leq\|f-P_{\alpha}\|_{L^{1}}<\varepsilon.$

This implies that $|\mathcal{F}_{\alpha}(f)(m)|\to 0$ as $|m|\to\infty$. ∎

For $f\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$, we claim that
$|\mathcal{F}_{\alpha}(f)(m)|$ may tend to zero arbitrarily slowly. More
precise, we have the following theorem.

######  Theorem 4.2.

Let $(d_{m})_{m\in\mathbf{Z}^{n}}$ be a sequence of positive real numbers with
$d_{m}\to 0$ as $|m|\to\infty$. Then there exists a function $f\in
e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$ such that
$|\mathcal{F}_{\alpha}(f)(m)|\geq d_{m}$ for all $m\in\mathbf{Z}^{n}$. In
other words, given any rate of decay, there exists an integrable function on
$\mathbf{T}^{n}_{\alpha}$ whose absolute value of fractional Fourier
coefficients have slower rate of decay.

###### Proof.

We first deal with $n=1$. Let $\\{a_{m}\\}_{m\in\mathbf{Z}}$ be a sequence of
positive numbers that converges to zero as $|m|\to\infty$. Apply Lemma 3.3.2
in [7] to the sequence $\\{a_{m}+a_{-m}\\}_{m\geq 0}$ to find a convex
sequence $\\{c_{m}\\}_{m\geq 0}$ that dominates $\\{a_{m}+a_{-m}\\}_{m\geq 0}$
and decreases to zero as $|m|\to\infty$. Extend $c_{m}$ for $m<0$ by taking
$c_{m}=c_{|m|}$. Set

$\displaystyle
f(x)=e_{-\alpha}(x)\sum_{j=0}^{\infty}(j+1)(c_{j}+c_{j+2}-2c_{j+1})F_{j}^{1,\alpha}(x).$
(4.1)

Using Lemma 3.3.3 in [7] with $s=0$, we have

$\displaystyle\sum_{j=0}^{\infty}(j+1)(c_{j}+c_{j+2}-2c_{j+1})\|F_{j}^{1,\alpha}\|_{L^{1}(\mathbf{T}^{1}_{\alpha})}=c_{0}<\infty,$
(4.2)

since $\|F_{j}^{1,\alpha}\|_{L^{1}(\mathbf{T}^{1}_{\alpha})}=1$ for all $j$.
Therefore, we obtain that $f\in{e_{-\alpha}L^{1}(\mathbf{T}^{1}_{\alpha})}$.
By noting that the series in (4.1) converges in
${L^{1}(\mathbf{T}^{1}_{\alpha})}$. For $m\in\mathbf{Z}$, we have

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|=$
$\displaystyle\sum_{j=0}^{\infty}(j+1)(c_{j}+c_{j+2}-2c_{j+1})\big{|}\mathcal{F}_{\alpha}(e_{-\alpha}F_{j}^{1,\alpha})(m)\big{|}$
$\displaystyle=$
$\displaystyle\sum_{j=|m|}^{\infty}(j+1)(c_{j}+c_{j+2}-2c_{j+1})\Big{|}A_{\alpha}e_{\alpha}(m)\Big{(}1-\frac{|m|}{j+1}\Big{)}\Big{|}$
$\displaystyle=$
$\displaystyle|\csc\alpha|^{1/2}\sum_{r=0}^{\infty}(r+1)(c_{r+|m|}+c_{r+|m|+2}-2c_{r+|m|+1})$
$\displaystyle=$
$\displaystyle|\csc\alpha|^{1/2}c_{|m|}=|\csc\alpha|^{1/2}c_{m}.$ (4.3)

where the second equality follows from Proposition 2.7 and the third equality
is due to Lemma 3.3.3 in [7] with $s=|m|$.

Next, we turn to $n\geq 2$. Let $(d_{m})_{m\in\mathbf{Z}^{n}}$ be a sequence
of positive real numbers with $d_{m}\to 0$ as $|m|\to\infty$. There exists a
positive sequence $\\{a_{j}\\}_{j\in\mathbf{Z}}$ such that $a_{m_{1}}\cdots
a_{m_{n}}\geq d_{({m_{1}},\ldots,{m_{n}})}$ and $a_{j}\to 0$ as
$|j|\to\infty$. Set

$\mathbf{f}(x_{1},\ldots,x_{n})=f(x_{1})\cdots f(x_{n}),$

where $f$ is defined as in (4.1) such that $|\mathcal{F}_{\alpha}(f)(m)|\geq
a_{m}$. This together with (2.1) implies
$|\mathcal{F}_{\alpha}(\mathbf{f})(m)|\geq d_{m}$. ∎

###  4.2 Decay of Fractional Fourier Coefficients of Smooth Functions

In this subsection, we are devoted to studying the relationship between the
decay of fractional Fourier coefficients and the smoothness of a function.

######  Definition 4.3.

For $0<\gamma<1$, the homogeneous Lipschitz space of order $\gamma$ on
$\mathbf{T}^{n}_{\alpha}$ is defined by

$\dot{\Lambda}_{\gamma}(\mathbf{T}^{n}_{\alpha})=\\{f:\mathbf{T}^{n}_{\alpha}\to
C\;with\;\|f\|_{\dot{\Lambda}_{\gamma}}<\infty\\},$

where

$\|f\|_{\dot{\Lambda}_{\gamma}}:=\underset{h\neq
0}{\sup_{x,h\in\mathbf{T}^{n}_{\alpha}}}\frac{|f(x+h)-f(x)|}{|h|^{\gamma}}.$

Next, we discuss the decay of fractional Fourier coefficients of Lipschitz
functions.

######  Theorem 4.4.

Let $s\in\mathbf{Z}$ and $s\geq 0$.
(a) Suppose that $f\in e_{-\alpha}C^{s}(\mathbf{T}^{n}_{\alpha})$. Then

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|\leq\Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha|}\Big{)}^{s}\frac{\max\limits_{|\beta|=s}{|\mathcal{F}_{\alpha}[e_{-\alpha}\partial^{\beta}(e_{\alpha}f)](m)|}}{|m|^{s}}$
(4.4)

and thus

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|(1+|m|^{s})\to 0$

as $|m|\to\infty$.
(b) Suppose that $f\in e_{-\alpha}C^{s}(\mathbf{T}^{n}_{\alpha})$ and whenever
$|\beta|=s$, $\partial^{\beta}(e_{\alpha}f)$ are in
$\dot{\Lambda}_{\gamma}(\mathbf{T}^{n}_{\alpha})$ for some $0<\gamma<1$. Then

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|\leq\frac{(\sqrt{n})^{s+\gamma}}{(2\pi)^{s}|\csc\alpha|^{s+\gamma+n-1}2^{\gamma+1}}\frac{\max\limits_{|\beta|=s}{\|\partial^{\beta}(e_{\alpha}f)\|_{\dot{\Lambda}_{\gamma}}}}{|m|^{s+\gamma}},\qquad
m\neq 0.$ (4.5)

###### Proof.

For fixed $m\in\mathbf{Z}^{n}\backslash\\{0\\}$, there exists a $j$ such that
$|m_{j}|=\sup_{1\leq k\leq n}|m_{k}|$. Therefore, $|m|\leq\sqrt{n}|m_{j}|$.
For $f\in e_{-\alpha}C^{s}(\mathbf{T}^{n}_{\alpha})$, integrating by parts $s$
times with respect to the variable $x_{j}$, we get

$\displaystyle\mathcal{F}_{\alpha}(f)(m)=$ $\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}(e_{\alpha}f)(x)e^{-2\pi
i(m\cdot x)\csc\alpha}dx$ $\displaystyle=$ $\displaystyle
A_{\alpha}^{n}e_{\alpha}(m)\int_{\mathbf{T}^{n}_{\alpha}}\partial_{j}^{s}(e_{\alpha}f)(x)\frac{e^{-2\pi
i(m\cdot x)\csc\alpha}}{(2\pi im_{j}\csc\alpha)^{s}}dx$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi
im_{j}\csc\alpha)^{s}}\int_{\mathbf{T}^{n}_{\alpha}}e_{-\alpha}(x)\partial_{j}^{s}(e_{\alpha}f)(x)K_{\alpha}(m,x)dx$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi
im_{j}\csc\alpha)^{s}}\mathcal{F}_{\alpha}[e_{-\alpha}\partial_{j}^{s}(e_{\alpha}f)](m).$
(4.6)

Then

$|\mathcal{F}_{\alpha}(f)(m)|\leq\Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha|}\Big{)}^{s}\frac{\max\limits_{|\beta|=s}{|\mathcal{F}_{\alpha}[e_{-\alpha}\partial^{\beta}(e_{\alpha}f)](m)|}}{|m|^{s}}.$

We now turn to conclusion (b). Let $e_{j}=(0,...,|\sin\alpha|,...,0)$ be the
element of the torus $\mathbf{T}^{n}_{\alpha}$ whose $j$th coordinate is
$|\sin\alpha|$ and all the others are zero. By noting that $e^{\pi i}=e^{-\pi
i}=-1$, we have

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\partial_{j}^{s}(e_{\alpha}f)(x)e^{-2\pi
i(m\cdot
x)\csc\alpha}dx=-\int_{\mathbf{T}^{n}_{\alpha}}\partial_{j}^{s}(e_{\alpha}f)\Big{(}x-\frac{e_{j}}{2m_{j}}\Big{)}e^{-2\pi
i(m\cdot x)\csc\alpha}dx.$

Therefore,

$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\partial_{j}^{s}\big{(}e_{\alpha}f\big{)}(x)e^{-2\pi
i(m\cdot x)\csc\alpha}dx$
$\displaystyle\quad=\frac{1}{2}\int_{\mathbf{T}^{n}_{\alpha}}\Big{[}\partial_{j}^{s}\big{(}e_{\alpha}f\big{)}(x)-\partial_{j}^{s}\big{(}e_{\alpha}f\big{)}\Big{(}x-\frac{e_{j}}{2m_{j}}\Big{)}\Big{]}e^{-2\pi
i(m\cdot x)\csc\alpha}dx.$

Notice that

$\displaystyle\Big{|}\partial_{j}^{s}(e_{\alpha}f)(t)-\partial_{j}^{s}(e_{\alpha}f)\Big{(}t-\frac{e_{j}}{2m_{j}}\Big{)}\Big{|}\leq\frac{\|\partial_{j}^{s}(e_{\alpha}f)\|_{\dot{\Lambda}_{\gamma}}}{(2|m_{j}\csc\alpha|)^{\gamma}}.$

This combined with (4.2) leads to

$\displaystyle\big{|}\mathcal{F}_{\alpha}(f)(m)\big{|}=$
$\displaystyle\Big{|}\frac{A_{\alpha}^{n}e_{\alpha}(m)}{(2\pi
im_{j}\csc\alpha)^{s}}\int_{\mathbf{T}^{n}_{\alpha}}\partial_{j}^{s}\big{(}e_{\alpha}f\big{)}(x)e^{-2\pi
im\cdot x\csc\alpha}dx\Big{|}$ $\displaystyle\leq$
$\displaystyle\Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha||m|}\Big{)}^{s}\frac{|A_{\alpha}^{n}|}{2|\csc\alpha|^{n}}\frac{\|\partial_{j}^{s}(e_{\alpha}f)\|_{\dot{\Lambda}_{\gamma}}}{(2|m_{j}\csc\alpha|)^{\gamma}}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha||m|}\Big{)}^{s}\frac{1}{2|\csc\alpha|^{n/2}}\Big{(}\frac{\sqrt{n}}{2|\csc\alpha||m|}\Big{)}^{\gamma}{\|\partial_{j}^{s}\big{(}e_{\alpha}f\big{)}\|_{\dot{\Lambda}_{\gamma}}}$
$\displaystyle\leq$
$\displaystyle\frac{(\sqrt{n})^{s+\gamma}}{(2\pi)^{s}|\csc\alpha|^{s+\gamma+n/2}2^{\gamma+1}}\frac{\max\limits_{|\beta|=s}{\|\partial_{j}^{s}(e_{\alpha}f)\|_{\dot{\Lambda}_{\gamma}}}}{|m|^{s+\gamma}}.$

This completes the proof. ∎

The following corollary is a consequence of Theorem 4.4.

######  Corollary 4.5.

Let $s\in\mathbf{Z}$ with $s\geq 0$.
(a) Suppose that $f\in e_{-\alpha}C^{s}(\mathbf{T}^{n}_{\alpha})$. Then

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|\leq
c_{n,s,\alpha}\frac{\max(\|f\|_{1},\max_{|\beta|=s}{\|\partial^{\beta}(e_{\alpha}f)\|_{1}})}{(1+|m|)^{s}}$

(b) Suppose that $f\in e_{-\alpha}C^{s}(\mathbf{T}^{n}_{\alpha})$ and whenever
$|\beta|=s$, $\partial^{\beta}(e_{\alpha}f)$ are in
$\dot{\Lambda}_{\gamma}(\mathbf{T}^{n}_{\alpha})$ for some $0<\gamma<1$. Then

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|\leq$ $\displaystyle
c_{n,s,\alpha}\frac{\max(\|f\|_{1},\max_{|\beta|=s}{\|\partial_{j}^{s}(e_{\alpha}f)\|_{\dot{\Lambda}_{\gamma}}})}{(1+|m|)^{s+\gamma}}.$

The following proposition provides a partial converse to Theorem 4.4. We
denote below by $[[s]]$ the largest integer strictly less than a given real
number s. Then $[[s]]$ is equal to the integer part $[s]$ of $s$, unless $s$
is an integer, in which case $[[s]]=[s]-1$.

######  Proposition 4.6.

Suppose $s>0$. If $f\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$ and
satisfies

$\displaystyle|\mathcal{F}_{\alpha}(f)(m)|\leq C(1+|m|)^{-s-n}$ (4.7)

for all $m\in\mathbf{Z}^{n}$. Then $e_{\alpha}f$ has partial derivatives of
all orders $|\beta|\leq[[s]]$. Moreover,
$\partial^{\beta}(e_{\alpha}f)\in{\dot{\Lambda}_{\gamma}}$ for all multi-
indices $\beta$ satisfying $|\beta|=[[s]]$ with $0<\gamma<s-[[s]]$.

###### Proof.

Using (4.7) and Corollary 3.3, we have

$\displaystyle
f(x)=\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x)$
(4.8)

for almost all $x\in\mathbf{T}^{n}_{\alpha}$. Let
$f_{m}(x)=\mathcal{F}_{\alpha}(f)(m)K_{-\alpha}(m,x)$. If a series
$g=\sum_{m}g_{m}$ satisfies
$\sum_{m}\|\partial^{\beta}g_{m}\|_{\infty}<\infty$ for all $|\beta|\leq M$,
we obtain $g\in C^{M}$ and $\partial^{\beta}g=\sum_{m}\partial^{\beta}g_{m}$.
Then, we need to check

$\sum_{m\in\mathbf{Z}^{n}}|\partial^{\beta}(e_{\alpha}f_{m})(x)|<\infty.$

Notice that

$\displaystyle\partial^{\beta}(e_{\alpha}f_{m})(x)=$ $\displaystyle
A_{-\alpha}^{n}e_{-\alpha}(m)\mathcal{F}_{\alpha}(f)(m)(2\pi
im\csc\alpha)^{\beta}e^{2\pi i(m\cdot x)\csc\alpha}.$

For $|\beta|\leq[[s]]$, by $(\ref{eq3.11})$, we get

$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\big{|}\partial^{\beta}(e_{\alpha}f_{m})(x)\big{|}\leq$
$\displaystyle|A_{-\alpha}^{n}|\sum_{m\in\mathbf{Z}^{n}}|\mathcal{F}_{\alpha}(f)(m)|\sup_{x\in\mathbf{T}^{n}_{\alpha}}|(2\pi
im\csc\alpha)^{\beta}e_{-\alpha}(m,x)|<\infty.$

Hence $e_{\alpha}f\in C^{[[s]]}(\mathbf{T}^{n}_{\alpha})$, and

$\displaystyle\partial^{\beta}(e_{\alpha}f)(x)=$ $\displaystyle
e_{\alpha}(x)\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)(2\pi
im\csc\alpha)^{\beta}K_{-\alpha}(m,x).$

For $|\beta|=[[s]]$ and $0<\gamma<s-[[s]]$, we get

$\displaystyle|e^{2\pi i(m\cdot
h)\csc\alpha}-1|\leq\min(2,2\pi|m||h||\csc\alpha|)\leq
2^{1-\gamma}(2\pi)^{\gamma}|m|^{\gamma}|h|^{\gamma}|\csc\alpha|^{\gamma}.$

This together with (4.7) implies

$\displaystyle\big{|}\partial^{\beta}(e_{\alpha}f)(x+h)-\partial^{\beta}(e_{\alpha}f)(x)\big{|}$
$\displaystyle=$
$\displaystyle\Big{|}A_{-\alpha}^{n}\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e_{-\alpha}(m)e_{-\alpha}(m,x)\big{(}e^{2\pi
i(m\cdot h)\csc\alpha}-1\big{)}(2\pi im\csc\alpha)^{\beta}\Big{|}$
$\displaystyle\leq$
$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\frac{C|\csc\alpha|^{n/2}}{(1+|m|)^{s+n}}(2\pi)^{[[s]]}|m|^{[[s]]}|\csc\alpha|^{[[s]]}\big{|}e^{2\pi
i(m\cdot h)\csc\alpha}-1\big{|}$ $\displaystyle\leq$
$\displaystyle\sum_{m\in\mathbf{Z}^{n}}\frac{C|\csc\alpha|^{n/2}}{(1+|m|)^{s+n}}(2\pi)^{[[s]]}|m|^{[[s]]}|\csc\alpha|^{[[s]]}2^{1-\gamma}(2\pi)^{\gamma}|m|^{\gamma}|h|^{\gamma}|\csc\alpha|^{\gamma}$
$\displaystyle\leq$ $\displaystyle
2^{1-\gamma}|h|^{\gamma}(2\pi)^{[[s]]+\gamma}|\csc\alpha|^{[[s]]+n/2+\gamma}\sum_{m\in\mathbf{Z}^{n}}\frac{C|m|^{[[s]]+\gamma}}{(1+|m|)^{s+n}}$
$\displaystyle\leq$ $\displaystyle
2^{1-\gamma}|h|^{\gamma}(2\pi)^{s}|\csc\alpha|^{[[s]]+n/2+\gamma}\sum_{m\in\mathbf{Z}^{n}}\frac{C|m|^{[[s]]+\gamma}}{(1+|m|)^{s+n}}$
$\displaystyle\leq$ $\displaystyle
C2^{1-\gamma}(2\pi)^{s}|\csc\alpha|^{[[s]]+n/2+\gamma}|h|^{\gamma}.$

Therefore, $\partial^{\beta}(e_{\alpha}f)\in{\dot{\Lambda}_{\gamma}}$. This
finishes the proof. ∎

##  5 Pointwise Convergence of fractional Fourier series

In this section, we discuss the pointwise convergence of fractional Fejér
means. For the convergence of other means, such as Bochner-Riesz means, one
can refer to [1, 6, 25] and the references therein.

###  5.1 Pointwise Convergence of the fractional Fejér Means

Using properties of the fractional Fejér kernel, we obtain the following
1-dimensional result regarding the convergence of the fractional Fejér means:

######  Theorem 5.1.

If $f\in e_{-\alpha}L^{1}(\mathbf{T}^{1}_{\alpha})$ has both left and right
limits at a point $x_{0}$, denoted by $f(x_{0-})$ and $f(x_{0+})$,
respectively, then

$\displaystyle(f\overset{\alpha}{\ast}{{F}}^{1}_{N})(x_{0})\to\frac{1}{2}\big{(}f(x_{0+})+f(x_{0-})\big{)}\quad
as\quad N\to\infty.$ (5.1)

###### Proof.

Set $\mathbf{T}^{1}_{\alpha}=[-{|\sin\alpha|}/{2},{|\sin\alpha|}/{2}]$. For
fixed $\varepsilon>0$, take $\delta\in(0,{|\sin\alpha|}/{2})$ such that

$\displaystyle\Big{|}\frac{(e_{\alpha}f)(x_{0}+t)+(e_{\alpha}f)(x_{0}-t)}{2}-\frac{e_{\alpha}(x_{0})f(x_{0+})+e_{\alpha}(x_{0})f(x_{0-})}{2}\Big{|}<\frac{\varepsilon}{2},$
(5.2)

where $0<t<\delta$. Follows form Proposition 2.9, there exists $N>N_{0}$ such
that

$\displaystyle\sup_{\delta\leq
t\leq{|\sin\alpha|/2}}{{F}}_{N}^{1,\alpha}(t)<\frac{\varepsilon}{2},$ (5.3)

Meanwhile,

$\displaystyle(f\overset{\alpha}{\ast}{{F}}_{N}^{1})(x_{0})-f(x_{0+})=e_{-\alpha}(x_{0})\int_{\mathbf{T}^{1}_{\alpha}}\big{[}(e_{\alpha}f)(x_{0}+t)-e_{\alpha}(x_{0})f(x_{0+})\big{]}{{F}}_{N}^{1,\alpha}(t)dt,$
$\displaystyle(f\overset{\alpha}{\ast}{{F}}_{N}^{1})(x_{0})-f(x_{0-})=e_{-\alpha}(x_{0})\int_{\mathbf{T}^{1}_{\alpha}}\big{[}(e_{\alpha}f)(x_{0}-t)-e_{\alpha}(x_{0})f(x_{0-})\big{]}{{F}}_{N}^{1,\alpha}(t)dt.$

Using the fact that the integrand is even, we have

$\displaystyle(f\overset{\alpha}{\ast}{{F}}_{N}^{1})(x_{0})-\frac{f(x_{0+})+f(x_{0-})}{2}$
$\displaystyle=2\int_{0}^{\frac{|\sin\alpha|}{2}}\Big{[}\frac{(e_{\alpha}f)(x_{0}+t)+(e_{\alpha}f)(x_{0}-t)}{2}$
$\displaystyle\quad\quad-\frac{e_{\alpha}(x_{0})f(x_{0+})+e_{\alpha}(x_{0})f(x_{0-})}{2}\Big{]}{{F}}_{N}^{1,\alpha}(t)dt.$
(5.4)

We divided the integral in (5.1) into two parts, the integral over
$[0,\delta)$ and the integral over $(\delta,{|\sin\alpha|}/{2})$, which are
denoted by $I_{3}$ and $I_{4}$, respectively. Using (5.2), we get
$I_{3}\leq\varepsilon$. For $N\geq N_{0}$, by (5.3), we obtain

$\displaystyle
I_{4}\leq\varepsilon\big{(}\|e_{\alpha}f-e_{\alpha}(x_{0})f(x_{0_{+}})\|_{L^{1}}+\|e_{\alpha}f-e_{\alpha}(x_{0})f(x_{0_{-}})\|_{L^{1}}\big{)}$
$\displaystyle=:\varepsilon c(f,x_{0},\alpha),$

where $c(f,x_{0},\alpha)$ is a constant depending on $f$, $x_{0}$ and
$\alpha$. This completes the proof. ∎

###  5.2 Almost Everywhere Convergence of the fractional Fejér Means

######  Theorem 5.2.

Suppose $f\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$.
(a) Let

$\mathcal{{H}}_{\alpha}f:=\sup_{N\in\mathbf{Z}^{+}}|f\overset{\alpha}{\ast}{{F}}^{n}_{N}|.$

Then $\mathcal{{H}}_{\alpha}$ maps $e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$
to $L^{1,\infty}(\mathbf{T}^{n}_{\alpha})$ and
$e_{-\alpha}L^{p}(\mathbf{T}^{n}_{\alpha})$ to
$L^{p}(\mathbf{T}^{n}_{\alpha})$ for $1<p\leq\infty$.
(b) For any function $f\in e_{-\alpha}L^{1}(\mathbf{T}^{n}_{\alpha})$, we get
that as $N\to\infty$

$\displaystyle f\overset{\alpha}{\ast}{{F}}^{n}_{N}\to f\quad a.e.$

###### Proof.

Using the fact that $1\leq\frac{|t|}{|\sin t|}\leq\frac{\pi}{2}$ when
$|t|\leq\frac{\pi}{2}$, we obtain

$\displaystyle|{{F}}_{N}^{1,\alpha}(x)|=$
$\displaystyle\frac{|\csc\alpha|}{N+1}\Big{|}\frac{\sin\big{(}(N+1)\pi
x\csc\alpha\big{)}}{\sin(\pi x\csc\alpha)}\Big{|}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{(N+1)|\csc\alpha|}{4}\Big{|}\frac{\sin\big{(}(N+1)\pi
x\csc\alpha\big{)}}{(N+1)x\csc\alpha}\Big{|}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{(N+1)|\csc\alpha|}{4}\min\Big{\\{}\pi^{2},\frac{1}{(N+1)^{2}|x\csc\alpha|^{2}}\Big{\\}}$
$\displaystyle\leq$
$\displaystyle\frac{\pi^{2}}{2}\frac{(N+1)|\csc\alpha|}{1+(N+1)^{2}|x|^{2}}.$

where $|x|\leq\frac{1}{2|\csc\alpha|}$. For $t\in\mathbb{R}$, set
$\varphi(t)=(1+t^{2})^{-1}$ and
$\varphi_{\varepsilon}(t)=\varepsilon^{-1}\varphi(t/\varepsilon)$ for
$\varepsilon>0$. For $x=(x_{1},\ldots,x_{n})$ and $\varepsilon>0$, let

$\Phi(x)=\varphi(x_{1})\cdots\varphi(x_{n})$

and $\Phi_{\varepsilon}(t)={\varepsilon^{-n}}\Phi(t/\varepsilon)$. Therefore,
for $|x|\leq{|\sin\alpha|/2}$ we get
$|{{F}}_{N}^{1,\alpha}(x)|\leq{\pi^{2}}|\csc\alpha|\varphi_{\varepsilon}(x)$
with $\varepsilon=(N+1)^{-1}$. Meanwhile, for
$y\in[-|\sin\alpha|/2,{|\sin\alpha|/2}]^{n}$, we have

$|{{F}}_{N}^{n,\alpha}(y)|\leq\pi^{2n}|\csc\alpha|^{n}\Phi_{\varepsilon}(y),$

where $\varepsilon=(N+1)^{-1}$.

Now let $e_{\alpha}f$ be an integrable function on ${\mathbf{T}^{n}_{\alpha}}$
and let $f_{0}$ denote its periodic extension on ${\mathbb{R}^{n}}$. For
$x\in[-|\sin\alpha|/2,{|\sin\alpha|/2}]^{n}$, we obtain

$\displaystyle\mathcal{{H}}_{\alpha}f(x)=$
$\displaystyle\sup_{N\in\mathbf{Z}^{+}}\Big{|}e_{-\alpha}(x)\int_{\mathbf{T}^{n}_{\alpha}}(e_{\alpha}f)(x-t){{F}}^{n,\alpha}_{N}(t)\mathrm{d}t\Big{|}$
$\displaystyle\leq$
$\displaystyle\pi^{2n}|\csc\alpha|^{n}\sup_{\varepsilon>0}\int_{\mathbf{T}^{n}_{\alpha}}|(e_{\alpha}f)(x-t)|\Phi_{\varepsilon}(t)\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
10^{n}|\csc\alpha|^{n}\sup_{\varepsilon>0}\int_{\mathbf{T}^{n}_{\alpha}}{|f_{0}(x-t)|}\Phi_{\varepsilon}(t)\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
10^{n}|\csc\alpha|^{n}\sup_{\varepsilon>0}\int_{{\mathbb{R}^{n}}}{|(f_{0}\chi_{Q})(x-t)|}\Phi_{\varepsilon}(t)\mathrm{d}t$
$\displaystyle=$
$\displaystyle:10^{n}|\csc\alpha|^{n}\mathfrak{F}(f_{0}\chi_{Q})(x),$

where $Q$ is the cube $[-|\sin\alpha|,|\sin\alpha|]^{n}$ and $\mathfrak{F}$ is
the operator on the set of integrable functions on ${\mathbb{R}^{n}}$, which
is defined by

$\mathfrak{F}(h)=\sup_{\varepsilon>0}|h|\ast\Phi_{\varepsilon}.$

By Lemma 3.4.5 in [7], we have

$\displaystyle\|\mathcal{{H}}_{\alpha}f\|_{L^{1,\infty}(\mathbf{T}^{n}_{\alpha})}\leq
10^{n}|\csc\alpha|^{n}\|\mathfrak{F}(f_{0}\chi_{Q})\|_{L^{1,\infty}(\mathbb{R}^{n})}\leq
C_{n,\alpha}\|f_{0}\chi_{Q}\|_{L^{1}({\mathbb{R}^{n}})}\leq
C_{n,\alpha}\|f\|_{L^{1}(\mathbf{T}^{n}_{\alpha})}.$

Meanwhile, it is easy to see

$\displaystyle\|\mathcal{{H}}_{\alpha}f\|_{L^{\infty}(\mathbf{T}^{n}_{\alpha})}\leq
C_{n,\alpha}\|f\|_{L^{\infty}(\mathbf{T}^{n}_{\alpha})}.$

By the Marcinkiewicz interpolation theorem, we obtain the $L^{p}$ conclusion.

Now we turn to conclusion (b). Since the sequence
$\\{{{F}}_{N}^{n,\alpha}\\}_{N=0}^{\infty}$ is an approximate identity and
$e_{-\alpha}C^{\infty}({\mathbf{T}^{n}_{\alpha}})$ is dense in
$e_{-\alpha}L^{1}({\mathbf{T}^{n}_{\alpha}})$, we have

$f\overset{\alpha}{\ast}{{F}}^{n}_{N}\to f,\quad f\in
e_{-\alpha}C^{\infty}({\mathbf{T}^{n}_{\alpha}}),$

uniformly on ${\mathbf{T}^{n}_{\alpha}}$ as $N\to\infty$. Since
$\mathcal{{H}}_{\alpha}$ maps $e_{-\alpha}L^{1}({\mathbf{T}^{n}_{\alpha}})$ to
$L^{1,\infty}({\mathbf{T}^{n}_{\alpha}})$ and using Theorem 2.1.14 in [7], we
obtain that for $f\in e_{-\alpha}L^{1}({\mathbf{T}^{n}_{\alpha}})$,
$f\overset{\alpha}{\ast}{{F}}^{n}_{N}\to f$ a.e. ∎

##  6 Applications to Partial Differential Equations

In this section, we apply fractional Fourier series to get solutions of
fractional heat equation and fractional Dirichlet problem.

######  Definition 6.1.

For fixed $k>0$, define the fractional heat kernel of order $\alpha$:

$H_{t}^{\alpha}(x)=\sum_{m\in\mathbf{Z}^{n}}e^{-4\pi^{2}|m|^{2}|\csc\alpha|^{2}kt}e^{2\pi
im\cdot x},$

where $t>0$. Note that $H_{t}^{\alpha}$ is absolutely convergent for any
$t>0$.

######  Proposition 6.2.

Let $k>0$ be fixed and $f\in e_{-\alpha}C^{\infty}(\mathbf{T}^{n}_{\alpha})$.
Then the fractional heat equation of order $\alpha$

$\displaystyle\displaystyle\frac{\partial}{\partial
t}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)}=k\triangle_{x}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)},\quad
t\in(0,\infty),x\in\mathbf{T}^{n}_{\alpha},$ (6.1)

under the initial condition

$\displaystyle F_{\alpha}(x,0)=f(x),\quad x\in\mathbf{T}^{n}_{\alpha},$ (6.2)

has a unique solution which is continuous on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$ and $C^{\infty}$ on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$ given by

$\displaystyle
F_{\alpha}(x,t)=(f\overset{\alpha}{\ast}{H_{t}^{\alpha}})(x)=\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-4\pi^{2}|m|^{2}|\csc\alpha|^{2}kt}K_{-\alpha}(m,x).$
(6.3)

###### Proof.

For $f\in e_{-\alpha}C^{\infty}(\mathbf{T}^{n}_{\alpha})$, it is easy to see
that the series in (6.3) is rapidly convergent in $m$ and we obtain that
$F_{\alpha}(x,t)$ is a continuous function on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$. In addition, the series can be
differentiated term by term with $t>0$, and thus it produces a $C^{\infty}$
function on $[0,\infty)\times\mathbf{T}^{n}_{\alpha}$. By Corollary 3.3,
$F_{\alpha}$ satisfies the initial condition. Next, it remains to check (6.1).

$\displaystyle\frac{\partial}{\partial
t}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)}=$
$\displaystyle\frac{\partial}{\partial
t}\Big{[}\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-4\pi^{2}|m|^{2}|\csc\alpha|^{2}kt}A_{-\alpha}^{n}e_{-\alpha}(m)e_{-\alpha}(m,x)\Big{]}$
$\displaystyle=$ $\displaystyle
k\Big{[}\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-4\pi^{2}m^{2}|\csc\alpha|^{2}kt}A_{-\alpha}^{n}e_{-\alpha}(m)\sum_{j=1}^{n}\frac{\partial^{2}}{\partial{x_{j}^{2}}}e_{-\alpha}(m,x)\Big{]}$
$\displaystyle=$ $\displaystyle
k\sum_{j=1}^{n}\frac{\partial^{2}}{\partial{x_{j}^{2}}}\Big{[}\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-4\pi^{2}m^{2}|\csc\alpha|^{2}kt}A_{-\alpha}^{n}e_{-\alpha}(m)e_{-\alpha}(m,x)\Big{]}$
$\displaystyle=$ $\displaystyle
k\triangle_{x}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)},$

where the last equality follows from the convergence of the series.

Finally, assume that there is another solution $G_{\alpha}(x,t)$, which is
continuous on $[0,\infty)\times{\mathbf{T}^{n}_{\alpha}}$ and $C^{\infty}$ on
$[0,\infty)\times{\mathbf{T}^{n}_{\alpha}}$. Then $G_{\alpha}(x,t)$ can be
expanded in fractional Fourier series as follows:

$G_{\alpha}(x,t):=\sum_{m\in\mathbf{Z}^{n}}c_{m}^{\alpha}(t)K_{-\alpha}(m,x),$

where

$c_{m}^{\alpha}(t)=\int_{\mathbf{T}^{n}_{\alpha}}G_{\alpha}(y,t)K_{\alpha}(m,y)dy.$

Then $c_{m}^{\alpha}(t)$ is a smooth function on $(0,\infty)$ since
$G_{\alpha}$ is $C^{\infty}$ on $[0,\infty)\times{\mathbf{T}^{n}_{\alpha}}$.
By equation (6.1), we have

$\displaystyle\frac{d}{dt}{c}_{m}^{\alpha}(t)=$
$\displaystyle\int_{\mathbf{T}^{n}_{\alpha}}\frac{\partial}{\partial
t}\Big{(}e_{\alpha}(x)G_{\alpha}(x,t)\Big{)}e_{-\alpha}(x)K_{\alpha}(m,x)dx$
$\displaystyle=$ $\displaystyle
kA_{\alpha}^{n}e_{\alpha}(m)\sum_{j=1}^{n}\int_{\mathbf{T}^{n}_{\alpha}}\frac{\partial^{2}}{\partial{x_{j}^{2}}}\Big{(}e_{\alpha}(x)G_{\alpha}(x,t)\Big{)}e^{-2\pi
i(m\cdot x)\csc\alpha}dx$ $\displaystyle=$
$\displaystyle-4\pi^{2}|m|^{2}|\csc\alpha|^{2}k{c}_{m}^{\alpha}(t),$

where the last equality follows from an integration by parts in which the
boundary terms cancel each other in view of the periodicity of the integrand
in $x$. In addition, ${c}_{m}^{\alpha}(0)=\mathcal{F}_{\alpha}(f)(m)$.
Therefore, we get

${c}_{m}^{\alpha}(t)=\mathcal{F}_{\alpha}(f)(m)e^{-4\pi^{2}|m|^{2}|\csc\alpha|^{2}kt}.$

Thus $G_{\alpha}=F_{\alpha}$ on $[0,\infty)\times\mathbf{T}^{n}_{\alpha}$. ∎

######  Remark 6.1.

Let $\mathbb{H}_{t}^{\alpha}(x)=|\csc\alpha|^{n}H_{t}^{\alpha}(x\csc\alpha)$.
The family $\big{\\{}\mathbb{H}_{t}^{\alpha}\big{\\}}_{t>0}$ is an approximate
identity on $\mathbf{T}^{n}_{\alpha}$.

######  Definition 6.3.

Define the fractional Poisson kernel of order $\alpha$

$P_{t}^{\alpha}(x)=\sum_{m\in\mathbf{Z}^{n}}e^{-2\pi|m|{|\csc\alpha|}t}e^{2\pi
im\cdot x},$

where $t>0$. Note that $P_{t}^{\alpha}$ is absolutely convergent for any
$t>0$.

######  Proposition 6.4.

Let $f\in e_{-\alpha}C^{\infty}(\mathbf{T}^{n}_{\alpha})$. Then the fractional
Dirichlet problem of order $\alpha$

$\displaystyle\begin{cases}\displaystyle\frac{\partial^{2}}{\partial
t^{2}}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)}+\sum_{j=1}^{n}\frac{\partial^{2}}{\partial{x_{j}}^{2}}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)}=0;\;t\in(0,\infty),x\in\mathbf{T}^{n}_{\alpha},\\\
\quad\quad\quad\quad\quad\quad\quad\quad\;\;F_{\alpha}(x,0)=f(x);\quad
x\in\mathbf{T}^{n}_{\alpha},\end{cases}$ (6.4)

has a solution which is continuous on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$ and $C^{\infty}$ on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$ given by

$\displaystyle
F_{\alpha}(x,t)=(f\overset{\alpha}{\ast}{P_{t}^{\alpha}})(x)=\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-2\pi|m|{|\csc\alpha|}t}K_{-\alpha}(m,x).$

###### Proof.

It is easy to see that $F_{\alpha}(x,t)$ is a continuous function on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$ and $C^{\infty}$ on
$[0,\infty)\times\mathbf{T}^{n}_{\alpha}$. By Corollary 3.3, $F_{\alpha}$
satisfies the initial condition. It suffices to verify (6.4). Notice that

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)}=$
$\displaystyle-\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-2\pi|m|{|\csc\alpha|}t}A_{-\alpha}^{n}e_{-\alpha}(m^{2})\sum_{j=1}^{n}\frac{\partial^{2}}{\partial{x_{j}^{2}}}e_{\alpha}(m,x)$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{n}\frac{\partial^{2}}{\partial{x_{j}^{2}}}\Big{[}e_{\alpha}(x^{2})\sum_{m\in\mathbf{Z}^{n}}\mathcal{F}_{\alpha}(f)(m)e^{-2\pi|m|{|\csc\alpha|}t}K_{-\alpha}(m,x)\Big{]}$
$\displaystyle=$
$\displaystyle-\triangle_{x}\Big{(}e_{\alpha}(x)F_{\alpha}(x,t)\Big{)}.$

This completes the proof. ∎

######  Remark 6.2.

Let $\mathbb{P}_{t}^{\alpha}(x)=|\csc\alpha|^{n}P_{t}^{\alpha}(x\csc\alpha)$.
The family $\\{\mathbb{P}_{t}^{\alpha}\\}_{t>0}$ is an approximate identity on
$\mathbf{T}^{n}_{\alpha}$.

##  7 Applications to non-stationary signals

In this section, we will demonstrate the use of Corollary 3.3 and Theorem 5.2
in the recovery of non-stationary signals. For example, suppose

$g(x)=\displaystyle\left\\{\begin{array}[]{ll}x,&-\frac{1}{4}<x\leq\frac{1}{4},\\\
\frac{1}{4},&x=-\frac{1}{4},\end{array}\right.$

and $g(x+1/4)=g(x)$ for all $x\in\mathbb{R}$. By taking $\alpha=\pi/6$, we
have

$f(x)=e^{-\pi ix^{2}\cot\pi/6}g(x)=\begin{cases}xe^{-\sqrt{3}\pi
ix^{2}},&-\frac{1}{4}<x\leq\frac{1}{4},\\\
\frac{1}{4}e^{-\frac{\sqrt{3}}{16}\pi i},&x=-\frac{1}{4}.\end{cases}$

It is easy to see $f\in e_{-\pi/6}L^{1}(\mathbf{T}^{1}_{-\pi/6})$. Note that

$\mathcal{F}_{\alpha}(f)(m)=\left\\{\begin{array}[c]{cc}\displaystyle
iA_{\alpha}^{1}e^{\sqrt{3}\pi im^{2}}\frac{(-1)^{m}}{8\pi m},&m\neq 0,\\\
\displaystyle 0,&m=0.\end{array}\right.$

Therefore, the series $\sum_{m\in\mathbf{Z}}|\mathcal{F}_{\alpha}(f)(m)|$ is
not convergent and we can not recover the signal $f(x)$ by fractional Fourier
inversion. Meanwhile, the approximating method (Theorem 5.2) can give another
way to recover the non-stationary signal $f(x)$. In fact, for $N\geq 0$, we
obtain

$\displaystyle(f\overset{\pi/6}{\ast}{F}_{N}^{1})(x)=$ $\displaystyle
ie^{-\sqrt{3}\pi
ix^{2}}\sum_{m=-N}^{N}\Big{(}1-\frac{|m|}{N+1}\Big{)}\frac{(-1)^{m}}{4\pi
m}e^{4\pi imx}.$

Figure 7.1 shows the real and imaginary part graph of the fractional
convolution $f\overset{\pi/6}{\ast}{F}_{N}^{1}$ with $N=10,50,100,500$.

(a) real part graph of $(f\overset{\pi/6}{\ast}{F}_{N}^{1})(x)$

(b) imaginary part graph of $(f\overset{\pi/6}{\ast}{F}_{N}^{1})(x)$

Figure 7.1: real and imaginary part graphs of
$(f\overset{\pi/6}{\ast}{F}_{N}^{1})(x)$

By Theorem 5.2, we obtain that $(f\overset{\pi/6}{\ast}{F}_{N}^{1})(x)\to f$
for almost every $x\in\mathbf{T}^{1}_{-\pi/6}$ as $N\to\infty$.

##  8 References

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