FRACTIONAL FOURIER SERIES ON THE TORUS AND APPLICATIONS CHEN WANG XIANMING HOU QINGYAN WU PEI DANG AND ZUNWEI FU

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FRACTIONAL FOURIER SERIES ON THE TORUS AND

APPLICATIONS

CHEN WANG, XIANMING HOU, QINGYAN WU, PEI DANG AND ZUNWEI FU

Abstract. In this paper, we introduce the fractional Fourier series on the frac-

tional torus and proceed to investigate several fundamental aspects. Our study

includes topics such as fractional convolution, fractional approximation, fractional

Fourier inversion, and the Poisson summation formula. We also explore the rela-

tionship between the decay of fractional Fourier coeﬃcients and the smoothness of

a function. Additionally, we establish the pointwise convergence of the fractional

Fourier series using the properties of the fractional F´ejer kernel. Finally, we demon-

strate the practical applications of the fractional Fourier series, particularly in the

context of solving fractional partial diﬀerential equations with periodic boundary

conditions, and showcase the utility of approximation methods on the fractional

torus for recovering non-stationary signals.

1. Introduction and statement of main results

It is well known that the Fourier series plays a crucial role in studying boundary

value problems, like the vibrating string and the heat equation. Fourier’s approach to

solving the problem of heat distribution in the cube T3by using the triple sine series

of three-variable functions. Dirichlet [4] made signiﬁcant contributions by studying

the pointwise convergence of the Fourier series for piecewise monotonic functions in

terms of the certain kernel on the circle. Inspired by the periodicity observed in

astronomical and geophysical phenomena, many scholars have dedicated their eﬀorts

to studying expansions of periodic functions, as mentioned in [32]. The relevance of

these expansions in both theoretical frameworks and practical applications underscores

the importance of analyzing Fourier series properties on the torus.

The n-torus Tnis deﬁned as the cube [0,1]nwith opposite sides identiﬁed. It is

important to note that functions on Tnexhibit periodicity with a period of 1 in every

coordinate. The m-th Fourier coeﬃcient can be deﬁned using the Fourier transform

as follows:

F(f)(m) = ZTn

f(x)e−2πim·xdx,

where f∈L1(Tn), m∈Zn. The Fourier series of fis the series

m∈ZnF(f)(m)e2πim·x.(1.1)

Kolmogorov [18, 19] demonstrated the existence of a function f∈L1(T1) whose

Fourier series diverges almost everywhere. For 1 <p<∞, Carleson and Hunt [2, 14]

observed that the Fourier series of an Lp(T1) function converges almost everywhere.

2010 Mathematics Subject Classiﬁcation. 42A20, 41A35.

Key words and phrases. fractional Fourier series, fractional approximate identity, fractional Fej´er

kernel, fractional Fourier inversion, convergence.

arXiv:2210.14720v3 [math.FA] 4 Jul 2024

2 CHEN WANG, XIANMING HOU, QINGYAN WU, PEI DANG AND ZUNWEI FU

Feﬀerman [7] presented an alternative proof for this result. Lacey and Thiele [20],

by combining ideas from [2] and [7], introduced a third approach to Carleson’s the-

orem on the pointwise convergence of the Fourier series, providing valuable insights

into the subject. For a comprehensive understanding of the essential properties and

applications of the Fourier series on the torus, one can refer to [5, 13, 15, 24] and the

references therein.

With the development of signal processing, the Fourier transform has revealed lim-

itations in its ability to handle non-stationary signals. To address this challenge, the

fractional Fourier transform (FRFT) was introduced. In this situation, the fractional

Fourier transform was proposed to overcome this problem. Additionally, when a signal

fexhibits t-periodicity with t < 1, it is not well-deﬁned on Tn. In such cases, we can

not obtain its frequency components. Therefore, we need to introduce the matched

fractional torus to study such a kind of signal (or function).

Deﬁnition 1.1. Let α∈R,α̸=πZ. The fractional torus of order α, denoted by Tn

α,

is the cube [0,|sin α|]nwith opposite sides identiﬁed.

Note that Tn

α=Tnfor α=π/2 + 2πZ, see Figure 1 (a) for two-dimensional torus.

The graphs of fractional torus T2

αfor α=π/2, π/3, π/6 are shown in Figure 1 (b).

Functions on Tn

αare functions fon Rnthat satisfy f(x+|sin α|m) = f(x) for any

x∈Rnand m∈Zn. Such functions are called |sin α|-periodic in every coordinate for

ﬁxed α.

(a) Torus T2(b) fractional torus of order αT2

Figure 1. The two-dimensional fractional torus of order αT2

The idea of the fractional power of the Fourier operator ﬁrst appeared in the work

of Wiener [33]. Namias [26] introduced the FRFT to address speciﬁc types of ordi-

nary and partial diﬀerential equations encountered in quantum mechanics. McBride

and Kerr [25] provided a rigorous deﬁnition of the FRFT in integral form on the

Schwartz space S(R) based on a modiﬁcation of Namias’ fractional operators. Sub-

sequently, Kerr [17] discussed the L2(R) theory of FRFT. Zayed [40] introduced a

novel convolution structure for the FRFT that preserves the convolution theorem for

the Fourier transform. In [41], Zayed presented a new class of fractional integral

transforms, encompassing the fractional Fourier and Hankel transforms. Addition-

ally, Zayed [42] explored the two-dimensional FRFT and investigated its properties,

including the inversion theorem, convolution theorem, and Poisson summation for-

mula. Kamalakkannan and Roopkumar [16] established the convolution theorem and

product theorem for the multidimensional fractional Fourier transform. The Lp(R)

theory of FRFT for 1 ≤p < 2 was established in [3]. Fu et al [10, 12] introduced

the Riesz transform associated with the FRFT and explored its applications in image

edge detection. The FRFT also has various applications in many ﬁelds, such as optics

[27], signal processing [28, 31], image processing [21, 23, 38, 39], and so on. In this

paper, our focus centers on studying the convergence and applications of the 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 Tn

α.

In Section 3, we establish the fractional Fourier inversion and Poisson summation

formula. Section 4 is devoted to the relationship between the decay of fractional

Fourier coeﬃcients and the smoothness of a function. The pointwise convergences of

fractional Fej´er means are given in Section 5. In Section 6, using fractional Fourier

series on Tn

α, we obtain the solutions of the fractional heat equation and fractional

Dirichlet problem. Finally, we present a non-stationary signal on Tn

α, which can be

recovered by an approximating method.

2. Fractional Fourier series on the torus

In this section, we begin by introducing the deﬁnition of fractional Fourier series

in the setting Tn

α. Subsequently, we establish some basic facts of fractional Fourier

analysis. Additionally, we will give the fractional convolution and obtain fractional

approximation in Lp(Tn

α), 1 ≤p≤ ∞.

Let α∈R,α̸=πZ. Set eα(x) := eπi|x|2cot αand eαf(x) := eα(x)f(x) for a function

fon Tn

α.

Deﬁnition 2.1. Let 1≤p≤ ∞. We say that a function fon Tn

αlies in the space

e−αLp(Tn

α)if

f(x) = e−αg(x), g ∈Lp(Tn

α)

and satisﬁes ∥f∥Lp(Tn

α)<∞.

Deﬁnition 2.2. For a complex-valued function f∈e−αL1(Tn

α),α∈Rand m∈Zn,

we deﬁne

Fα(f)(m) = 









ZTn

f(x)Kα(m, x)dx, α ̸=πZ,

f(m), α = 2πZ,

f(−m), α = 2πZ+π,

where

Kα(m, x) := An

αeα(x)eα(m, x)eα(m),

here Aα=√1−icot αand eα(m, x) = e−2πi(m·x) csc α. We call Fα(f)(m)the m-th

fractional Fourier coeﬃcient of order αof f.

In order to state our results, we recall some notations. The spaces Ck(Tn

α), k∈Z+,

are deﬁned as the sets of functions ϕfor which ∂βϕexist and are continuous for all

|β| ≤ k. When k= 0 we set C0(Tn

α) = C(Tn

α) to be the space of continuous functions

4 CHEN WANG, XIANMING HOU, QINGYAN WU, PEI DANG AND ZUNWEI FU

on Tn

α. Let

C∞(Tn

α) := ∞

k=0

Ck(Tn

α).

Deﬁnition 2.3. For 0≤k≤ ∞. The space e−αCk(Tn

α)is deﬁned to be the space of

functions fon Tn

αsuch that

f(x) = e−αg(x), g ∈Ck(Tn

α).

Notice that the spaces e−αCk(Tn

α) are contained in e−αLp(Tn

α) for all 1 ≤p≤ ∞.

We denote by fthe complex conjugate of the function f, by ˜

fthe function ˜

f(x) =

f(−x), and by τy(f)(x) the function τy(f)(x) = f(x−y) for all y∈Tn

α. Since

α=πZ, we get Tn

α={0}. Hence, throughout this paper, for α̸=πZ, we always

assume Tn

α= [−|sin α|/2,|sin α|/2]n. Next, we give some elementary properties of

fractional Fourier coeﬃcients.

Proposition 2.4. Let f, g be in e−αL1(Tn

α). Then for all m, k ∈Zn,λ∈C,y∈Tn

α,

we have

(1) Fα(f+g)(m) = Fα(f)(m) + Fα(g)(m);

(2) Fα(λf)(m) = λFα(f)(m);

(3) Fα(f)(m) = F−α(f)(m);

(4) Fα(e

f)(m) = Fα(f)(−m);

(5) Fα[e−ατy(eαf)](m) = Fα(f)(m)eα(m, y);

(6) Fα[e−α(k, ·)f](m)e−2πi(m·k) cot αeα(k) = Fα(f)(m−k);

(7) Fα(f)(0) = An

αZTn

e−αf(x)dx;

(8) supm∈Zn|Fα(f)(m)| ≤ |csc α|n/2∥f∥L1(Tn

α);

(9) Fα[e−α∂β(eαf)](m) = (2πim csc α)βFα(f)(m),whenever f∈e−αCβ(Tn

α).

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

Note that

Fα[e−ατyeαf](m) =An

αeα(m)ZTn

(eαf)(x−y)eα(m, x)dx

=An

αeα(m)ZTn

α−y

(eαf)(x′)eα(m, x′+y)dx′

=Fα(f)(m)eα(m, y),

where we make the variable change x′=x−yin the second equality, and the third

equality follows from the periodicity of function.

Next, we deal with (6).

Fα(f)(m−k) =An

αZTn

f(x)eα(x)eα(m−k)eα(m−k, x)dx

=eα(k)e−2πi(m·k) cot αZTn

e−α(k, x)f(x)Kα(m, x)dx

=Fα[e−α(k, ·)f](m)e−2πi(m·k) cot αeα(k).

It follows from the fact |Kα(m, x)|≤|csc α|n/2that (8) holds.

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

Fα[e−α∂β(eαf)](m) =An

αeα(m)ZTn

∂β(eαf)(x)e−2πi(m·x) csc αdx

=(2πim csc α)βAn

αeα(m)ZTn

eαf(x)e−2πi(m·x) csc αdx

=(2πim csc α)βFα(f)(m).

This completes the proof. □

Remark 2.5. Suppose f1∈e−αL1(Tn1

α)and f2∈e−αL1(Tn2

α). We obtain that the

tensor function

(f1⊗f2)(x1, x2) = f1(x1)f2(x2)

is in e−αL1(Tn1+n2

α). Moreover, it is easy to check

Fαf1⊗f2(m1, m2) = Fα(f1)(m1)Fα(f2)(m2),(2.1)

where m1∈Zn1and m2∈Zn2.

Deﬁnition 2.6. For α̸=nπ, a trigonometric polynomials of order αon Tn

αis a

function of the form

Pα(x) = X

m∈Zn

cm,αK−α(m, x),

where m= (m1, . . . , mn)and cm,α is a constant depending on mand α. The degree

of Pαis the largest number |m1|+··· +|mn|such that cm,α is nonzero. Observe that

Fα(Pα)(m) = cm,α.

For x∈Tn

α,f∈e−αL1(Tn

α), the fractional Fourier series of order αof fis the series

m∈ZnFα(f)(m)K−α(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 αto study the convergence of fractional Fourier series.

Deﬁnition 2.7. Let f∈e−αL1(Tn

α)and g∈L1(Tn). Deﬁne the fractional convolu-

tion of order αas follows:

(fα

∗g)(x) = |csc α|ne−α(x)ZTn

eα(y)f(y)(δαg)(x−y)dy,

where (δαg)(x) = g(xcsc α).

Let f∈e−αL1(Tn

α). We have

|m|≤NFα(f)(m)K−α(m, x) = X

|m|≤NZTn

f(y)Kα(m, y)dyK−α(m, x)

=|csc α|ne−α(x)X

|m|≤NZTn

eα(y)f(y)e2πim·(x−y) csc αdy

=|csc α|ne−α(x)ZTn

eα(y)f(y)X

|m|≤N

e2πim·(x−y) csc αdy

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FRACTIONALFOURIERSERIESONTHETORUSANDAPPLICATIONSCHENWANG,XIANMINGHOU,QINGYANWU,PEIDANGANDZUNWEIFUAbstract.Inthispaper,weintroducethefractionalFourierseriesonthefrac-tionaltorusandproceedtoinvestigateseveralfundamentalaspects.Ourstudyincludestopicssuchasfractionalconvolution,fractionalapproximation,f...

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FRACTIONAL FOURIER SERIES ON THE TORUS AND APPLICATIONS CHEN WANG XIANMING HOU QINGYAN WU PEI DANG AND ZUNWEI FU

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