# The Fractional S-Transform on Spaces of TypeS

Authors: Sunil Kumar Singh
Journal: Journal of Mathematics
 (2013)

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## Abstract

Certain properties of the S-transform are given and
composition of S-transform is defined. We study the fractional S-transform on
spaces of typeS. It is shown that the fractional S-transform is a continuous
linear map of the space of type S defined on ℝ into the space of type S~ defined on ℝ×(ℝ∖{0}).


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## Body

## 1. Introduction

In this section we give the definition and some basic properties of the S-transform. The S-transform was first introduced by Stockwell et al. [1] in 1996 as invertible time-frequency spectral localization technique. The S-transform is an extension to the ideas of continuous wavelet transform and is based on a moving and scalable localizing Gaussian window and has characteristics superior to both of the Fourier transform and the wavelet transform [2–5].The one-dimensional continuous S-transform ofu(t) is defined as [4]
(1)(Su)(τ,f)=S(u(t))(τ,f)=∫ℝ‍u(t)ω(τ-t,f)e-i2πftdt,
where the window ω is assumed to satisfy the following:
(2)∫ℝ‍ω(t,f)dt=1∀f∈ℝ∖{0}.
The most usual window ω is the Gaussian one
(3)ω(t,f)=|f|k2πe-(f2t2/2k2),k>0,
where f is the frequency, t is the time variable, and k is a scaling factor that controls the number of oscillations in the window.Then, (1) can be rewritten as a convolution
(4)(Su)(τ,f)=(u(·)e-i2πf·*ω(·,f))(τ).
Applying the convolution property for the Fourier transform, we obtain
(5)(Su)(τ,f)=ℱ-1{u^(·+f)ω^(·,f)}(τ),
where ℱ-1 is the inverse Fourier transform. For the Gaussian window case (3),
(6)ω^(α,f)=e-2(πkα/f)2.
Thus we can write the S-transform in the following form:
(7)(Su)(τ,f)=∫ℝ‍u^(α+f)e-2(πkα/f)2ei2πατdα.
Also, if u^(f) and (Su)(τ,f) are the Fourier transform and S-transform of u, respectively, then
(8)u^(f)=∫ℝ‍(Su)(τ,f)dτ,
so that
(9)u(t)=ℱ-1(∫ℝ‍(Su)(τ,·)dτ)(t).


## 2. Some Important Properties of S-Transform

Some basic properties of S-transform can be found in [6, 7] and certain properties of S-transform are obtained in this section.Theorem 1.
Ifu(t)∈L1(ℝ), then lim|τ|→∞|(Su)(τ,f)|=0.Proof.
We have by definition(10)(Su)(τ,f)=∫ℝ‍u^(α+f)e-2(πkα/f)2ei2πατdα=-∫ℝ‍u^(α+f)e-2(πkα/f)2ei2π(α+1/2τ)τdα=-∫ℝ‍u^(x+f-12τ)e-2[πk(x-1/2τ)/f]2ei2πτxdx.
Thus
(11)(Su)(τ,f)=12[∫ℝ‍u^(α+f)e-2(πkα/f)2ei2πατdα-∫ℝ‍u^(α+f-12τ)e-2[πk(α-1/2τ)/f]2ei2πταdα].
Clearly,(12)lim|τ|→∞|(Su)(τ,f)|≤12lim|τ|→∞∫ℝ‍|u^(α+f)-u^(α+f-12τ)|dα=0.Theorem 2.
Ifu(t)∈L1(ℝ), then lim|f|→0|(Su)(τ,f)|=0.Theorem 3.
Ife-2(πkα/f)2u^(α+f)∈L2(ℝ), then ∥(Su)(τ,f)∥L2≤∥u∥L2.Theorem 4.
Ifu(t)∈L1(ℝ), then (Su)(τ,f) is continuous with respect to the variables τ and f.Proof.
By the definition(13)|(Su)(τ+h,f)-(Su)(τ,f)|=|∫ℝ‍u^(α+f)ω^(α,f)(ei2παh-1)ei2πατdα|≤∫ℝ‍|u^(α+f)||ei2παh-1|dα,⟶0ash⟶0.
Hence the S-transform is continuous with respect to the first variables τ. Similarly we can show that it is also continuous with respect to the second variables f.


### 2.1. Composition of S-Transforms

LetS1 and S2 be two S-transforms defined as
(14)(S1u)(τ1,f1)=∫ℝ‍u^(α+f1)ω1^(α,f1)ei2πατ1dα,(S2u)(τ2,f2)=∫ℝ‍u^(α+f2)ω2^(α,f2)ei2πατ2dα,
provided the integrals exist. Then their composition S1∘S2 is defined by
(15)(S*u)(τ1,f1,f2)=((S1∘S2)u)(τ1,f1,f2)=∫ℝ‍ℱ{(S2u)(τ2,f2)}(α+f1)ω1^(α,f1)ei2πατ1dα=∫ℝ‍u^(α+f1+f2)ω2^(α+f1,f2)ω1^(α,f1)ei2πατ1dα=∫ℝ‍u^(α+f1+f2)ω^(α,f1,f2)ei2πατ1dα,
where ω^(α,f1,f2)=ω2^(α+f1,f2)ω1^(α,f1), provided the integral exists. The composition S1∘S2 is an S-transform of three variables mapping from ℝ into ℝ×(ℝ∖{0})×(ℝ∖{0}) and analogous to the composition of the wavelet transform [8, pages 52-53].Theorem 5.
Ifu(t)∈L1(ℝ), then (S*u)(τ,f1,f2) is continuous with respect to the variables τ, f1 and f2.


## 2.1. Composition of S-Transforms

LetS1 and S2 be two S-transforms defined as
(14)(S1u)(τ1,f1)=∫ℝ‍u^(α+f1)ω1^(α,f1)ei2πατ1dα,(S2u)(τ2,f2)=∫ℝ‍u^(α+f2)ω2^(α,f2)ei2πατ2dα,
provided the integrals exist. Then their composition S1∘S2 is defined by
(15)(S*u)(τ1,f1,f2)=((S1∘S2)u)(τ1,f1,f2)=∫ℝ‍ℱ{(S2u)(τ2,f2)}(α+f1)ω1^(α,f1)ei2πατ1dα=∫ℝ‍u^(α+f1+f2)ω2^(α+f1,f2)ω1^(α,f1)ei2πατ1dα=∫ℝ‍u^(α+f1+f2)ω^(α,f1,f2)ei2πατ1dα,
where ω^(α,f1,f2)=ω2^(α+f1,f2)ω1^(α,f1), provided the integral exists. The composition S1∘S2 is an S-transform of three variables mapping from ℝ into ℝ×(ℝ∖{0})×(ℝ∖{0}) and analogous to the composition of the wavelet transform [8, pages 52-53].Theorem 5.
Ifu(t)∈L1(ℝ), then (S*u)(τ,f1,f2) is continuous with respect to the variables τ, f1 and f2.


## 3. Fractional Fourier Transform

The fractional Fourier transform (FRFT) has played an important role in signal processing. Theath order FRFT of a signal u(t) is defined as [9]
(16)Fua(f)=∫ℝ‍u(t)Ka(t,f)dt,
where the transform kernel Ka(t,f) is defined as
(17)Ka(t,f)={Aθeiπ(f2cotθ-2ftcscθ+t2cotθ),ifθ≠nπδ(t-f),ifθ=2nπδ(t+f),ifθ+π=2nπ,
where Aθ=1-icotθ,  θ=aπ/2,  a∈[0,4), i is the complex unit, n is an integer, and f is the fractional Fourier frequency (FRFfr). The inverse FRFT of (16) is
(18)u(t)=∫ℝ‍Fua(f)Ka(t,f)¯df.
We can write (16) as
(19)Fua(f)=∫ℝ‍u(t)Aθeiπ(f2cotθ-2ftcscθ+t2cotθ)dt=Aθeiπf2cotθ∫ℝ‍u(t)e-i2πftcscθeiπt2cotθdt=Aθeiπf2cotθ[eiπt2cotθu(t)]^(fcscθ).
Replacing u(t) by e-iπt2cotθϕ(t), we have
(20)Fua(f)=Aθeiπf2cotθ[ϕ(t)]^(fcscθ).
Putting f=ξsinθ, we have
(21)Fua(ξsinθ)=Aθeiπξ2sin2θ/2ϕ^(ξ),ϕ^(ξ)=1Aθe-iπξ2sin2θ/2Fua(ξsinθ),
where u(t)=e-iπt2cotθϕ(t).


## 4. The Fractional S-Transform

The fractional S-transform (FRST) is a generalization of the S-transform. Theath order continuous fractional S-transform of u(t) is defined as [10]
(22)FRSTua(τ,f)=∫ℝ‍u(t)g(τ-t,f)Ka(t,f)dt,
where the window g is
(23)g(t,f)=|fcscθ|pk2πe-t2(fcscθ)2p/2k2,k,p>0
and satisfies the condition
(24)∫ℝ‍g(t,f)dt=1∀f∈ℝ∖{0}.
Inverse fractional S-transform is defined by
(25)u(t)=∫ℝ[∫ℝ‍FRSTua(τ,f)dτ]Ka(t,f)¯df.
Note that the fractional S-transform depends on a parameter θ and can be interpreted as a rotation by an angle θ in the time-frequency plane. An FRST with θ=π/2 corresponds to the S-transform, and an FRST with θ=0 corresponds to the zero operator. The parameters p and k can be used to adjust the window function space.Let(26)h(t,τ,f)=g(τ-t,f)Ka(t,f),(27)Ha(τ,f,f1)=∫ℝ‍h(t,τ,f)Ka(t,f1)¯=∫ℝ‍g(τ-t,f)Ka(t,f)Ka(t,f1)¯dt.
Since
(28)Ka(t,f)Ka(t,f1)¯=AθA¯θeiπ[(f2-f12)cotθ-2(f-f1)tcscθ].
By using (23) and (28) in (27) we obtain
(29)Ha(τ,f,f1)=∫ℝ‍|fcscθ|pk2πe-(τ-t)2(fcscθ)2p/2k2×AθA¯θeiπ[(f2-f12)cotθ-2(f-f1)tcscθ]dt=|fcscθ|pk2πAθA¯θeiπ[(f2-f12)cotθ]×∫ℝ‍e-(τ-t)2(fcscθ)2p/2k2e-i2π(f-f1)tcscθdt.
By using the technique of [10], we obtain
(30)Ha(τ,f,f1)=AθA¯θeiπ[(f2-f12)cotθ]e-i2π(f-f1)τcscθ×e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2p,
and by using (28) we can write
(31)Ha(τ,f,f1)=e-[2π2k2(f-f1)2(cscθ)2/(fcscθ)2p]Ka(t,f)Ka(t,f1)¯.
Also, the FRST can be also defined as operations on the fractional Fourier domain:
(32)FRSTua(τ,f)=∫ℝ‍[∫ℝ‍Fua(f)Ka(t,f)¯df]×g(τ-t,f)Ka(t,f)dt=∫ℝ‍Fua(f1)Ha(τ,f,f1)df1.
By using (19) and (30) we can write (32) as
(33)FRSTua(τ,f)=∫ℝ‍Aθeiπf12cotθ[eiπt2cotθu(t)]^×(f1cscθ)AθA¯θeiπ[(f2-f12)cotθ]×e-i2π(f-f1)τcscθ×e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2pdf1=Aθ|Aθ|2∫ℝ‍eiπf12cotθ[eiπt2cotθu(t)]^×(f1cscθ)eiπ[(f2-f12)cotθ]×e-i2π(f-f1)τcscθ×e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2pdf1=Aθ|Aθ|2eiπf2cotθ×∫ℝ‍[eiπt2cotθu(t)]^(f1cscθ)×e-i2π(f-f1)τcscθ×e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2pdf1.


## 5. The Fractional S-Transform on Spaces of TypeS

The spaces of typeS play an important role in the theory of linear partial differential equations as intermediate spaces between the spaces of the C∞ and the analytic functions. The Fourier transform has been studied on the spaces of type S by Friedman [11] and Gel’fand and Shilov [12]. Pathak and Singh [13] studied the wavelet transform on the spaces of type S. Also, Singh [6] studied the S-transform on the spaces of type S. In this section we study the fractional S-transform (22) on these spaces.Let us recall the definitions of these spaces.Definition 6.
The spaceSα(α≥0) consists of all infinitely differentiable functions ϕ(x)  (-∞<x<∞), satisfying the inequalities
(34)γk,q(ϕ)≔supx∈ℝ|xkϕ(q)(x)|≤CqAkkkα(k,q=0,1,2,…),
where the constants A and Cq depend on the function ϕ. For k=0, the expression kkα is considered to be equal to 1.Definition 7.
The spaceSβ(β≥0) consists of all infinitely differentiable functions ϕ(x)  (-∞<x<∞), satisfying the inequalities
(35)γk,q(ϕ)≔supx∈ℝ|xkϕ(q)(x)|≤CkBqqqβ(k,q=0,1,2,…),
where the constants B and Ck depend on the function ϕ.Definition 8.
The spaceSαβ(α≥0,β≥0) consists of all infinitely differentiable functions ϕ(x)  (-∞<x<∞), satisfying the inequalities
(36)γk,q(ϕ)≔supx∈ℝ|xkϕ(q)(x)|≤CAkBqkkαqqβ(k,q=0,1,2,…),
where the constants A,B, and C depend on the function ϕ.The spaces of typeS are closely interrelated by means of the Fourier transformation; namely, the formulas
(37)S^α=Sα,S^β=Sβ,S^αβ=Sβα
hold.We will make use of the following inequalities in our investigation:(38)q!(q-k)!=k!(qk)≤k!∑k=0q‍(qk)=k!2q,(39)(m+q)(m+q)β≤mmβqqβemβeqβ[11,page265].
In what follows we will also need the following similar test function space defined on ℝ×(ℝ∖{0}), the space of type S~.Definition 9.
The spaceS~αβ(ℝ×(ℝ∖{0})),  α,β≥0, is defined to be the space of all functions ϕ∈C∞(ℝ×(ℝ∖{0})) such that for all s,t∈ℕ0,
(40)|(fcscθ)2psτs(1+|fcscθ|)t(2(fcscθ)2p+|fcscθ|+1)s(∂∂τ)tϕ(τ,f)|≤Cθ,s,tAsBtssαttβ,
where the constants A,B, and Cθ,s,t depend on ϕ.We show that the fractional S-transform is a continuous linear map of the space of typeS into the space of type S~ defined mentioned previously.Theorem 10.
For0<p≤1, the fractional S-transform is a continuous linear map of Sα(ℝ) into S~α~β~(ℝ×(ℝ∖{0})) where α~=(α+3) and β~=0.Proof.
Lets,t∈ℕ and I=τs(∂/∂τ)tFRSTua(τ,f). Then, after differentiation and integration by parts, we obtain
(41)I=τs(∂∂τ)tAθ|Aθ|2eiπf2cotθ×∫ℝ‍e-i2π(f-f1)τcscθ×e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2p×[eiπt2cotθu(t)]^(f1cscθ)df1=τsAθ|Aθ|2eiπf2cotθ×∫ℝ‍(-i2π(f-f1)cscθ)te-i2π(f-f1)τcscθ×e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2p×[eiπt2cotθu(t)]^(f1cscθ)df1=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t×∫ℝ‍1(-i2πcscθ)se-i2π(f-f1)τcscθ×Df1s(f-f1)te-2π2k2(f-f1)2(cscθ)2/(fcscθ)2p×[eiπt2cotθu(t)]^(f1cscθ)df1=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∫ℝ‍e-i2π(f-f1)τcscθ∑l=0s‍(sl)Df1s-l(f-f1)t×Df1l[[eiπt2cotθu(t)]^e-2π2k2(f-f1)2(cscθ)2/(fcscθ)2p×[eiπt2cotθu(t)]^(f1cscθ)]df1=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∑l=0s‍(sl)t!(-1)s-l(t-s+l)!×∫ℝ‍e-i2π(f-f1)τcscθ(f-f1)t-s+l×∑m=0l‍(lm)Df1m[e-2[πkcscθ(f-f1)/(fcscθ)p]2]×Df1l-m[eiπt2cotθu(t)]^(f1cscθ)df1=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∑l=0s‍(sl)t!(-1)s-l(t-s+l)!∑m=0l‍(lm)×∫ℝ‍e-i2π(f-f1)τcscθ(f-f1)t-s+l×(-2πkcscθ(fcscθ)p)me-2[πkcscθ(f-f1)/(fcscθ)p]2×∑n=0[m/2]‍(-1)nm!(4πkcscθ)m-2nn!(m-2n)!×(f-f1(fcscθ)p)(m-2n)×Df1l-m[eiπt2cotθu(t)]^(f1cscθ)df1=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∑l=0s‍(sl)t!(-1)s-l(t-s+l)!∑m=0l‍(lm)×(-2πkcscθ(fcscθ)p)m×∑n=0[m/2]‍(-1)nm!(4πkcscθ)m-2nn!(m-2n)!1(fcscθ)p(m-2n)×∫ℝ‍e-i2π(f-f1)τcscθ(f-f1)t-s+l+m-2n×e-2[πkcscθ(f-f1)/(fcscθ)p]2×Df1l-m[eiπt2cotθu(t)]^(f1cscθ)df1=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∑l=0s‍(sl)t!(-1)s-l(t-s+l)!∑m=0l‍(lm)×(-2πkcscθ(fcscθ)p)m×∑n=0[m/2]‍(-1)nm!(4πkcscθ)m-2nn!(m-2n)!1(fcscθ)p(m-2n)×∫ℝ‍e-i2π(f-ysinθ)τcscθ(f-ysinθ)t-s+l+m-2n×e-2[πkcscθ(f-ysinθ)/(fcscθ)p]2×(sinθ)-l+m+1Dyl-m[eiπt2cotθu(t)]^(y)dy=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∑l=0s‍(sl)t!(-1)s-l(t-s+l)!∑m=0l‍(lm)×(-2πkcscθ(fcscθ)p)m×∑n=0[m/2]‍(-1)nm!(4πkcscθ)m-2nn!(m-2n)!1(fcscθ)p(m-2n)×(sinθ)t-s+2m-2n+1×∫ℝ‍e-i2π(fcscθ-y)τ(fcscθ-y)t-s+l+m-2n×e-2[πk(fcscθ-y)/(fcscθ)p]2×Dyl-m[eiπt2cotθu(t)]^(y)dy=Aθ|Aθ|2eiπf2cotθ(-i2πcscθ)t-s×∑l=0s‍(sl)t!(t-s+l)!∑m=0l‍(lm)×(-2πkcscθ(fcscθ)p)m×∑n=0[m/2]‍(-1)nm!(4πkcscθ)m-2nn!(m-2n)!1(fcscθ)p(m-2n)×(sinθ)t-s+2m-2n+1×∫ℝ‍e-i2π(fcscθ-y)τ×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×(fcscθ)t-s+l+m-2n-r(-y)r×e-2[πk(fcscθ-y)/(fcscθ)p]2×Dyl-m[eiπt2cotθu(t)]^(y)dy.
By using the convergence factor 1+y2, we obtain
(42)|I|≤|Aθ|3(2π)t-s∑l=0s‍(sl)t!(t-s+l)!∑m=0l‍(lm)×|2πk(fcscθ)p|m×∑n=0[m/2]‍m!(4πk)m-2nn!(m-2n)!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×∫ℝ‍|(1+y2)yrDyl-m[eiπt2cotθu(t)]^(y)|×11+y2dy.
Using the property (38) we obtain
(43)|I|≤Cθ,s,t′∑l=0s‍(sl)(s-l)s-l2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×∫ℝ‍|[eiπt2cotθu(t)]^(1+y2)yrDyl-m×[eiπt2cotθu(t)]^(y)|11+y2dy.
Use the properties (35) and (39) to obtain
(44)|I|≤Cθ,s,t′∑l=0s‍(sl)ss2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×CrBl-m(l-m)(l-m)α≤Cθ,s,t′′ss2t∑l=0s‍(sl)∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-rBlllα≤Cθ,s,t′′A1sB1ts(α+3)s∑l=0s‍(sl)∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍1|fcscθ|p(m-2n)(1+|fcscθ|)t-s+l+m-2n=Cθ,s,t′′A1sB1ts(α+3)s∑l=0s‍(sl)∑m=0l‍(lm)×1|fcscθ|pm(1+|fcscθ|)t-s+l+m|fcscθ|pm×∑n=0[m/2]‍(|fcscθ|2p(1+|fcscθ|)2)n≤Cθ,s,t′′A1sB1ts(α+3)s∑l=0s‍(sl)×∑m=0l‍(lm)(1+|fcscθ|)t-s+l+m|fcscθ|2pm[m2]≤Cθ,s,tA1sB1ts(α+3)s(1+|fcscθ|)t×∑l=0s‍(sl)∑m=0l‍(lm)(1+|fcscθ|)m|fcscθ|2pm=Cθ,s,tA1sB1ts(α+3)s(1+|fcscθ|)t×∑l=0s‍(sl)(1+1+|fcscθ||fcscθ|2p)l=Cθ,s,tA1sB1ts(α+3)s(1+|fcscθ|)t×(2+1+|fcscθ||fcscθ|2p)s=Cθ,s,tA1sB1ts(α+3)s(1+|fcscθ|)t×(2|fcscθ|2p+|fcscθ|+1|fcscθ|2p)s.
Hence, we obtain the estimate
(45)|(fcscθ)2psτs(1+|fcscθ|)t(2(fcscθ)2p+|fcscθ|+1)s×(∂∂τ)tFRSTua(τ,f)(fcscθ)2psτs(1+|fcscθ|)t(2(fcscθ)2p+|fcscθ|+1)s|≤Cθ,s,tA1sB1ts(α+3)s,
where A1 and B1 are positive constants.Theorem 11.
For0<p≤1, the fractional S-transform is a continuous linear map of Sα(ℝ) into S~α~β~(ℝ×(ℝ∖{0})) where α~=(α+3) and β~=α.Theorem 12.
For0<p≤1, the fractional S-transform is a continuous linear map of Sαβ(ℝ) into S~α~β~(ℝ×(ℝ∖{0})) where α~=(α+3) and β~=β.Proof.
By inequality (43), we have
(46)|I|≤Cθ,s,t′∑l=0s‍(sl)(s-l)s-l2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×∫ℝ‍|[eiπt2cotθu(t)]^(1+y2)yrDyl-m×[eiπt2cotθu(t)]^(y)|11+y2dy.
Next, using the properties (36) and (39) we have
(47)|I|≤Cθ,s,t′∑l=0s‍(sl)ss2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×C(A1rrrβ+A1(r+2)(r+2)(r+2)β)×B1(l-m)(l-m)(l-m)α≤Cθ,s,t′∑l=0s‍(sl)ss2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×CA2r(r+2)(r+2)βB1lllα≤Cθ,s,t′∑l=0s‍(sl)ss2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r×CA2rrrβ22βerβe2βB1lllα≤Cθ,s,tAsBtss(α+3)ttβ×∑l=0s‍(sl)ss2t∑m=0l‍(lm)×1|fcscθ|pm×∑n=0[m/2]‍(4πk)m-2n(2n)2n2mn!1|fcscθ|p(m-2n)×∑r=0t-s+l+m-2n‍(t-s+l+m-2nr)×|fcscθ|t-s+l+m-2n-r.
Next, by using the technique of Theorem 10, we obtain
(48)|I|≤Cθ,s,tAsBtss(α+3)ttβ(1+|fcscθ|)t×(2|fcscθ|2p+|fcscθ|+1|fcscθ|2p)s.
Hence, we obtain the estimate
(49)|(fcscθ)2psτs(1+|fcscθ|)t(2(fcscθ)2p+|fcscθ|+1)s×(∂∂τ)tFRSTua(τ,f)(fcscθ)2psτs(1+|fcscθ|)t(2(fcscθ)2p+|fcscθ|+1)s|≤Cθ,s,tAsBtss(α+3)ttβ,
where A and B are positive constants.


## 6. Future Directions

The results of this paper will be useful in the investigation of time-frequency behaviour of test functions and distributions. We may use these results to obtain new results in modulation spaces, which may be defined by means of the fractional S-transform, by using the techniques developed in [14, Chapter 11–13].



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*Source: 105848-2013-04-10.xml*