# Riesz transform associated with the fractional Fourier transform and applications

Since Zayed <cit.> introduced the fractional Hilbert transform related to the fractional Fourier transform, this transform has been widely concerned and applied in the field of signal processing. Recently, Chen, the first, second and fourth authors <cit.> attribute it to the operator corresponding to fractional multiplier, but it is only limited to 1-dimensional case. This paper naturally considers the high-dimensional situation. We introduce the fractional Riesz transform associated with fractional Fourier transform, in which the chirp function is the key factor and the technical barriers to be overcome. Furthermore, after equipping with chirp functions, we introduce and investigate the boundedness of singular integral operators, the dual properties of Hardy spaces and BMO spaces as well as the applications of theory of fractional multiplier in partial differential equation, which completely matched some classical results. Through numerical simulation, we give the physical and geometric interpretation of the high-dimensional fractional multiplier theorem. Finally, we present the application of the fractional Riesz transform in edge detection, which verifies the prediction proposed in <cit.>. Moreover, the application presented in this paper can also be considered as the high-dimensional case of the application of the continuous fractional Hilbert transform in edge detection in <cit.>.

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## 1 Introduction

One of the fundamental operators in Fourier analysis theory is the Hilbert transform, which can be formally written as

 H(f)(x)=1π p.v.∫Rf(y)x−ydy,    f∈S(R),

and it is related to the study of the conjugate Fourier series of functions. The main method used in the early study of the Hilbert transform is complex analysis method, and the discussion of real analysis method began around 1920s. The Hilbert transform is the prototype of all singular integrals, which provides inspiration for the subsequent development of this subject. In 1952, the work of Calderón and Zygmund in cz extending this operator to . In particular, it has made a great contribution to the application of partial differential equation theory. The singular integral operator is the most wonderful piece of contemporary harmonic analysis. The Hilbert transform plays an key role not only in Fourier analysis, but also in communication systems and digital signal processing systems, such as in filter, edge detection and modulation theory; see g3 ; h . In signal processing, the Hilbert transform of a real-valued function is the convolution of the signal with . Therefore, the Hilbert transform can be understood as the output of linear time invariant system with impulse response is .

###### Definition 1.1.

Given in we define

 ^f(ξ)=1(√2π)n∫Rnf(x)e−ix⋅ξdx.

We call or the Fourier transform of .

The Fourier transform is a standard and powerful tool for analyzing and processing stationary signals. From the point of view of time-frequency analysis, Hilbert transform is also known as phase shifter. The Hilbert transform is a multiplier operator.

 F(Hf)(x)=−isgn(x)F(f)(x). (1.1)

It can be seen from (1.1) that the Hilbert transform is a phase-shift converter that multiplies the positive frequency portion of the original signal by , that is, maintaining the same amplitude, shifts the phase by , while the negative frequency portion is shifted by .

The Riezs transform is a generalization of the Hilbert transform in -dimensional case, and is also a singular integral operator, with properties analogous to those of the Hilbert transform on . It is defined as follows

 Rj(f)(x)=cn p.v.∫Rnxj−yj|x−y|n+1f(y)dy,  1≤j≤n,

where . The Riesz transform is also a multiplier operator:

 F(Rjf)(x)=−ixj|x|F(f)(x).
###### Remark 1.1.

The multiplier of the Hilbert transform is , it is just a phase-shift converter. The multiplier of the Riesz transform is , so the Riesz transform is not only a phase-shift converter, but also an amplitude attenuator.

The Riesz transform has wide applications in image edge detection, image quality assessment and biometric features recognition, which refer to la ; zl ; zzm .

The Fourier transform is limited in processing and analyzing non-stationary signals. The fractional Fourier transform (FRFT for short) was proposed and developed by some scholars mainly because of the need for non-stationary signals. FRFT originated from the work of Wiener in w2 . Namias in n

gave the FRFT through the method primarily on eigenfunction expansions in 1980. McBride-Kerr in

mk and Kerr in k gave the integral expressions of the FRFT on and , respectively. In 2021, Chen, the first, second and fourth authors in cfgw established the behavior of FRFT on for .

A chirp function is a non-stationary signal in which the frequency increases (up-chirp) or decreases (down-chirp) with time. The chirp signal is the most common non-stationary signal. In 1998, Zayed in z2 gave the definition of the fractional Hilbert transform associated with the FRFT.

 Hα(f)(x)=1π p.v. e−α(x)∫Rf(y)x−yeα(y)dy,

where is a chirp function.

In py , Pei and Yeh decomposed the discrete fractional Hilbert transform into the composite of discrete fractional Fourier transform (DFRFT for short), masking DFRFT and inverse DFRFT, and carry out simulation verification on the edge detection of digital images. In cfgw , Chen, the first, second and fourth authors attributed the fractional Hilbert transform to a fractional Fourier multiplier

 Fα(Hαf)(x)=−isgn((π−α)x)Fα(f)(x),

where is FRFT, see Definition 1.2. Similar to the Hilbert transform, the fractional Hilbert transform is also a phase-shift converter. It is also pointed out that the continuous fractional Hilbert transform can be decomposed into a combination of FRFT, multiplier and inverse FRFT, which confirms the prediction in py . The fractional Hilbert transform can also be used in single sideband communication system and image encryption system. The rotation angle can be used as the encryption key to improve the communication security and image encryption effect in tlw .

In recent years, the topic of multidimensional FRFT has attracted much attention. Zayed in z3 ; z4 introduced a new two dimensional FRFT. In kr , Kamalakkannan and Roopkumar introduced the definition of multidimensional FRFT.

###### Definition 1.2.

(see kr ) The multidimensional FRFT with order on is defined by

 Fα(f)(u)=∫Rnf(x)Kα(x,u)dx,

where and defined by

 Kαk(xk,uk)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩c(αk)√2πei(a(αk)(x2k+u2k−2b(αk)xkuk)),αk∉πZ,δ(xk−uk),αk∈2πZ,δ(xk+uk),αk∈2πZ+π,

, , , .

###### Remark 1.2.

Suppose with . Denote chirp function as follows

 eα(x)=ei∑nk=1a(αk)x2k,

for any It is easy to observe that FRFT of can be written with

 Fα(f)(u)=c(α)eα(u)F(eαf)(~u),

where , . From the above identity we can see that from to . We rewrite

 Kα(x,u)=c(α)(√2π)neα(x)eα(u)e−i∑nk=1xkukcscαk.

Motivated by these works, we define the fractional Riesz transform associated with the multidimensional FRFT as follows.

###### Definition 1.3.

For , the th fractional Riesz transform of is given by

 Rαj(f)(x)=cn p.v. e−α(x)∫Rnxj−yj|x−y|n+1f(y)eα(y)dy,

where and with .

###### Remark 1.3.

The fractional Riesz transform will be reduced to the fractional Hilbert transform for , while the fractional Riesz transform will be reduced to the classical Riesz transform for , .

This paper will be organized as follows. In Section 2, we give a characterization of the fractional Riesz transform by FRFT, and point out that the fractional Riesz transform is not only a phase shift converter, but also an amplitude attenuator. We obtain an identity property of the fractional Riesz transform and the boundedness of singular integral operators with chirp function in rotation invariant spaces. In Section 3, we introduce the definition of chirp Hardy space by taking possion maximum for the function with chirp factor, and study the dual spaces of chirp Hardy spaces. We also give a characterization of the boundedness of singular integral operators with chirp function in chirp Hardy spaces. In Section 4, the derivation formula of the high-dimensional FRFT is established and the fractional Riesz transform is applied to the partial differential equation. In Section 5, we carry out a simulation experiment of the fractional Riesz transform on image, and give the physical and geometric interpretation of the high-dimensional fractional multiplier theorem. In Section 6, it is difficult to directly use fractional Riesz transform for edge detection, the fractional multiplier theorem provides the possibility. The use of the fractional Riesz transform is completely equivalent to the compound operation of FRFT, inverse FRFT and multiplier, FRFT and inverse FRFT can realize fast operation.

## 2 Fractional Riesz transform

### 2.1 The properties of the fractional Riesz transform

We now give a fractional Fourier multiplier theorem in the FRFT context.

###### Theorem 2.1.

The th fractional Riesz transform is given on the FRFT side by multiplication by the function . That is, for any we have

 Fα(Rαjf)(u)=−i~uj|~u|Fα(f)(u),

where with ; and .

###### Proof.

Fix a . For , we have

 Fα(Rαjf)(u)= ∫RnRαjf(x)c(α)(√2π)neα(x)eα(u)e−i∑nj=1xjujcscαjdx = ∫RnΓ(n+12)πn+12e−α(x)limε→0∫|y|≥εyj|y|n+1f(x−y)eα(x−y)dy ×c(α)(√2π)neα(x)eα(u)e−i∑nj=1xjujcscαjdx = Γ(n+12)πn+12limε→0∫|y|≥εyj|y|n+1∫Rnf(x−y)eα(x−y)c(α)(√2π)neα(u) ×e−i∑nj=1xjujcscαjdxdy.

From the substitution of variables, we have

 Fα(Rαjf)(u)= Γ(n+12)πn+12limε→0∫|y|≥εyj|y|n+1∫Rnf(w)eα(w)c(α)(√2π)neα(u) ×e−i∑nj=1(yj+wj)ujcscαjdwdy = Γ(n+12)πn+12limε→0∫|y|≥εyj|y|n+1e−i∑nj=1yjujcscαjdy∫Rnc(α)(√2π)n ×f(w)eα(u)eα(w)e−i∑nj=1wjujcscαjdw = Γ(n+12)πn+12Fα(f)(u)limε→0∫|y|≥εyj|y|n+1e−i∑nj=1yjujcscαjdy = Γ(n+12)πn+12Fα(f)(u)limε→0∫ε≤|y|≤1εyj|y|n+1e−iy⋅~udy.

By polar transformation formula and Lemma 5.1.15 in g1 , we get

 Fα(Rαjf)(u)= −iΓ(n+12)πn+12Fα(f)(u)limε→0∫sn−1∫ε≤r≤1εsin(r~u⋅θ)rrn+1rn−1drθjdθ = −iΓ(n+12)πn+12Fα(f)(u)∫sn−1∫∞0sin(r~u⋅θ)drrθjdθ = −iΓ(n+12)2πn−12Fα(f)(u)∫sn−1sgn(~u⋅θ)θjdθ = −i~uj|~u| Fα(f)(u),

which completes the proof of the theorem. ∎

###### Lemma 2.2.

(FRFT inversion theorem)  (see kr ) Suppose . Then

 f(x)=∫RnFα(f)(u)K−α(u,x)du,  a.e. x∈Rn.

By Theorem 2.1 and Lemma 2.2, the th fractional Riesz transform of order can be rewritten as

 (Rαjf)(x)=[F−α(−i~uj|~u|(Fαf)(u))](x).

Denote by . We see that the fractional Riesz transform of a function can be decomposed into three simpler operators, according to the diagram of Fig. 2.1:

1. FRFT of order , ;

2. multiplication by a fractional multiplier, ;

3. FRFT of order ,

Take a 2-dimensional fractional Riesz transform as an example. It can be seen from Theorem 2.1 that the fractional Riesz transform of order is a phase-shift converter that multiplies the positive portion in -order fractional Fourier domain of signal by , that is, shifts the phase by , while the negative portion of is shifted by . It is also a amplitude reducer that multiplies the amplitude in -order fractional Fourier domain of signal by . As shown in Fig. 2.2.

Next, we establish the boundedness of the fractional Riesz transform.

###### Theorem 2.3.

For all , there exists a positive constant such that

 ∥∥Rαj(f)∥∥Lp≤C∥f∥Lp,

for all in .

###### Proof.

From the boundedness of the Riesz transform in ldy with Theorem 2.1.4, it follows that

 ∥∥Rαj(f)∥∥Lp= (∫Rn∣∣cne−α(x)∫Rnyj|y|n+1f(x−y)eα(x−y)dy∣∣pdx)1p = ∥∥Rj(feα)∥∥Lp ≤ C∥f∥Lp,

for all in . ∎

According to the Theorem 2.1, we can obtain an identity property of the fractional Riesz transform.

###### Theorem 2.4.

The fractional Riesz transforms satisty

 n∑j=1(Rαj)2=−I,   on L2(Rn),

where is the identity operator.

###### Proof.

Use the FRFT and the identity to obtain

 Fα(n∑j=1(Rαj)2f)(u)= n∑j=1(−i~uj|~u|)2Fα(f)(u) = −Fα(f)(u),

for any in . ∎

### 2.2 The boundedness of singular integral operators with chirp function in rotation invariant spaces

The fractional Riesz transform, analogous to Riesz transform, is a singular integral operator. Following the definition of the fractional Riesz transform, we can naturally define more general singular integral operators, which kernels equipped with chirp functions.

 Tα(f)(x)=p.v.∫Rne−α(x)K(x,y)eα(y)f(y)dy=p.v.∫RnKα(x,y)f(y)dy.

When , can be regarded as the classical singular integral operators:

 T(f)(x)=p.v.∫RnK(x,y)f(y)dy.

Then we consider the boundedness of on rotation invariant Banach spaces.

###### Definition 2.5.

Suppose is a Banach space. Let’s call the rotation invariant spaces, if

 ∥eαf∥X=∥f∥X,

for any , where with .

When satisfies suitable conditions, the boundedness of and in rotation invariant space is equivalent.

###### Theorem 2.6.

If is a rotation invariant space, is bounded from to if and only if is bounded from to .

###### Proof.

Let and . We have that

 ∥Tα(f)∥X=∥T(eαf)∥X≤C∥eαf∥X=C∥f∥X.

Conversely, , we get

 ∥T(f)∥X=∥e−αT(f)∥X=∥Tα(e−αf)∥X≤C∥e−αf∥X=C∥f∥X.

Hence the theorem follows. ∎

## 3 Chirp Hardy spaces

In this section, we naturally consider the boundedness of singular integral operator with chirp function in non-rotation invariant space, such as Hardy space. Hardy spaces measure the smoothness within distributions, which become more singular as decreases. Hardy spaces can be regarded as a substitute for when .

### 3.1 Chirp Hardy spaces and chirp BMO spaces

We recall the real variable characterization and atom characterization of Hardy spaces.

###### Definition 3.1.

(see g1 ) Let be a bounded tempered distribution on and let . We say that lies in the Hardy spaces if the Poisson maximal function

 M(f;P)(x)=supt>0|(Pt∗f)(x)|

lies in . If this is the case, we set

 ∥f∥Hp=∥M(f;P)∥Lp.

Before introducing atom characterization of Hardy spaces, we recall the definition of atoms.

###### Definition 3.2.

(see c ; l ) If and , then satisfying the above conditions are said to be admissible triples, where represents an integer function. If the real valued function satisfies the following conditions

(1) and supp, where is a cube centered on ;

(2) ;

(3) , for any ;

then is called the atom centered in .

###### Definition 3.3.

(see c ; l ) Suppose are admissible triples. The atom Hardy spaces is defined as

 Hp,q,satom(Rn):={f∈S′(Rn):f(x)=∑jλjaj(x), where aj is (p,q,s)−atom, ∞∑j=1|λj|p<∞},

and the norm is defined as follows

###### Remark 3.1.

Suppose . We have

 ∥f∥Hp= ∥∥∥supt>0|(Pt∗f)|∥∥∥Lp=∥∥∥supt>0∣∣∣∫RnPt(⋅−y)f(y)dy∣∣∣∥∥∥Lp, ∥eαf∥Hp=

We can clearly see that depends on , that is,

 ∥f∥Hp≠∥eαf∥Hp.

Hardy space is not a rotation invariant space.

Now let us consider the boundedness of singular integral operators with chirp function in Hardy space when the kernel satisfies certain size and smoothness conditions. Let’s recall the definition of the -Calderón-Zygmund operator.

###### Definition 3.4.

(see l1 ) Let is a bounded linear operator. We say is a -Calderón-Zygmund operator, if is bounded on and is a continuous function on satisfies

(1)

(2)

(3) For and supp supp , one has

 (Tf,g)=∫K(x,y)f(y)g(x)dydx.
###### Lemma 3.5.

(see l1 ) Suppose is a -Calderón-Zygmund operator and its conjugate operator . Then can be extended to bounded operator from to , where , .

By standard calculations, we have the following estimates for

:

 |Kα(x,y)−Kα(x,z)|≤ |K(x,y)−K(x,z)|+Lα(y,z)|K(x,z)||y−z|,

where , for . It is know that for , satisfies -Calderón-Zygmund operator kernel condition . It is obvious that satisfies in Definition 3.4, but in Definition 3.4 is not guaranteed.

We define a new class of Hardy space with chirp funtions as follows.

###### Definition 3.6.

Let be a bounded tempered distribution on and let . We say that lies in the chirp Hardy spaces for with , if the Poisson maximal function with chirp function

 Mα(f;P)(x)=supt>0|(Pt∗(eαf))(x)|

lies in . If this is the case, we set

 ∥f∥Hpα=∥Mα(f;P)∥Lp.
###### Lemma 3.7.

(see g2 ) Hardy space is a complete space.

###### Theorem 3.8.

Chirp Hardy space is a complete space.

###### Proof.

Let is a cauchy sequence in . Then is cauchy sequence in . By Lemma 3.7, there exists a such that

 limk→∞∥eαfk−¯f∥Hp=0.

The above identity is rewritten as

 limk→∞∥fk−e−α¯f∥Hpα=0.

Since , we obtain , which completes the proof of the theorem. ∎

It is known that the dual space of is the space. In order to study dual space of chirp Hardy space, we define a new space with chirp function as follows.

###### Definition 3.9.

Suppose that is a locally integrable function on . Define chirp space

 BMOα(Rn)={f:∥f∥BMOα<∞}.

Let

 ∥f∥BMOα=supQ1|Q|∫Q|eα(x)f(x)−AvgQ(eαf)|dx,

where the supremum is taken over all cubes in and with .

###### Lemma 3.10.

(see g2 ) BMO space is a complete space.

###### Theorem 3.11.

Chirp BMO space is a complete space.

The proof is the same as Theorem 3.8, which can be proved by Lemma 3.10.

### 3.2 Dual space of chirp Hardy space

We discuss the dual spaces of chirp Hardy spaces . When , we have the following theorem.

###### Theorem 3.12.

. That is,
(1) For any ,

 L(f):=∫Rnf(x)g(x)dx

is a bounded linear functional on