# The S-Transform of Distributions

Authors: Sunil Kumar Singh
Journal: The Scientific World Journal
 (2014)

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## Abstract

Parseval’s formula and inversion formula for the S-transform are given. A relation between the S-transform and pseudodifferential operators is obtained. The S-transform is studied on the spaces𝒮(ℝn) and 𝒮′(ℝn).


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## Body

## 1. Introduction

TheS-transform was first used by Stockwell et al. [1] in 1996. If ω(t,ξ) is a window function, then the continuous S-transform of ϕ(t) with respect to ω is defined as [2]
(1)(Sωϕ)(τ,ξ)=∫ℝn‍ϕ(t)ω(τ-t,ξ)e-i2π〈t,ξ〉dt,forx,ξ∈ℝn.
In signal analysis, at least in dimension n=1, ℝ2n is called the time-frequency plane, and in physics ℝ2n is called the phase space.Equation (1) can be rewritten as a convolution as
(2)(Sωϕ)(τ,ξ)=(ϕ(·)e-i2π〈·,ξ〉*ω(·,ξ))(τ).
Now, we recall the definitions of Fourier transform on ℝn.Definition 1.
Ifϕ(t) is defined on ℝn, then the Fourier transform of ϕ is given by
(3)ℱ[ϕ(t)](ξ)=∫ℝn‍ϕ(t)e-i2π〈t,ξ〉dt,
where 〈t,ξ〉=∑j=1n‍tjξj is the usual inner product on ℝn.Definition 2.
Ifϕ(t,x) is defined on ℝ2n, then the partial Fourier transform of ϕ(t,x) with respect to the first coordinate is given by
(4)ℱ1[ϕ(t,x)](ξ,x)=∫ℝn‍ϕ(t,x)e-i2π〈t,ξ〉dt,
and the partial Fourier transform of ϕ(t,x) with respect to the second coordinate is given by
(5)ℱ2[ϕ(t,x)](t,η)=∫ℝn‍ϕ(t,x)e-i2π〈x,η〉dx.
Applying the convolution property for the Fourier transform in (2), we obtain
(6)(Sωϕ)(τ,ξ)=ℱ1-1[ℱ1(ϕ)(α+ξ)ℱ1(ω)(α,ξ)](τ,ξ),
where ℱ1-1 is the inverse Fourier transform.Now, we define the translation, modulation, and involution operators, respectively, by(7)Tτϕ(t)=ϕ(t-τ)(translation)Mξϕ(t)=ei2π〈ξ,t〉ϕ(t)(modulation)ℐϕ(t)=ϕ(-t)¯(involution),
where t,τ,ξ∈ℝn.Definition 3 (the Dirac delta).
The Dirac delta function is defined by(8)∬ℝnϕ(t,x)δ(t,x)dtdx=ϕ(0,0).Definition 4 (tempered distribution).
A functionϕ∈C∞(ℝn) is said to be rapidly decreasing if
(9)γα,β(ϕ)=supx∈ℝn|xαDβϕ(x)|<∞,
for all pairs of multi-indices α,β∈ℕ0n. The space of all rapidly decreasing functions on ℝn is denoted by 𝒮(ℝn) or simply 𝒮. Elements in the dual space 𝒮′ of 𝒮 are called tempered distribution.


## 2. Some Important Properties of S-Transform

Some properties ofS-transform can be found in [3–8] and certain properties of S-transform are obtained in this section. By definition, we have
(10)(Sωϕ)(τ,ξ)=∫ℝn‍ω(τ-t,ξ)ϕ(t)e-i2π〈t,ξ〉dt=∫ℝn‍ω(-x,ξ)ϕ(τ+x)e-i2π〈τ+x,ξ〉dx=∫ℝn‍ω(-x,ξ)M-ξϕ(τ+x)dx=∫ℝn‍ω(-x,ξ)T-τM-ξϕ(x)dx.
Thus, the S-transform Sω appears as a superposition of time-frequency shifts as follows:
(11)Sω:=∫ℝn‍ω(-x,ξ)T-τM-ξdx.Example 5.
Ifω(t,ξ)=m(ξ), that is, independent of t, then
(12)(Sωϕ)(τ,ξ)=∫ℝn‍m(ξ)ϕ(t)e-i2π〈t,ξ〉dt=m(ξ)(ℱϕ)(ξ).
So Sω is a multiplication operator. In particular, if ω(t,ξ)=1, then (Sωϕ)(τ,ξ)=(ℱϕ)(ξ).Example 6.
Ifω(t,ξ)=m(t), then
(13)(Sωϕ)(τ,ξ)=∫ℝn‍m(τ-t)ϕ(t)e-i2π〈t,ξ〉dt=∫ℝn‍T-τm(-t)ϕ(t)e-i2π〈t,ξ〉dt=∫ℝn‍T-τℐm¯(t)ϕ(t)e-i2π〈t,ξ〉dt=ℱ(ϕT-τℐm¯)(ξ).Theorem 7 (Parseval's formula).
Letω1 and ω2 be the window functions such that
(14)∫ℝn‍ω1(τ-t,ξ)ω2(τ-x,η)¯dτ=δ(x-t,ξ-η).
Let ϕ1,ϕ2∈L2(ℝn) and let (Sω1ϕ1) and (Sω2ϕ2) be the S-transforms of ϕ1 and ϕ2, respectively. Then
(15)∬ℝn(Sω1ϕ1)(τ,ξ)(Sω2ϕ2)(τ,ξ)¯dτdξ=∫ℝn‍ϕ1(t)ϕ2(t)¯dt.
This immediately implies the Plancherel formula(16)∥Sωϕ∥L2(ℝn×ℝn)=∥ϕ∥L2(ℝn).Proof.
Consider(17)∬ℝn(Sω1ϕ1)(τ,ξ)(Sω2ϕ2)(τ,ξ)¯dτdξ=∬ℝn(∫ℝn‍ϕ1(t)ω1(τ-t,ξ)e-i2π〈t,ξ〉dt)×(∫ℝn‍ϕ2(x)ω2(τ-x,ξ)e-i2π〈x,ξ〉dx)¯dτdξ=∭ℝnϕ1(t)ϕ2¯(x)ei2π〈(x-t),ξ〉×(∫ℝn‍ω1(τ-t,ξ)ω2(τ-x,ξ)¯dτ)dξdtdx=∭ℝnϕ1(t)ϕ2¯(x)ei2π〈(x-t),ξ〉δ(x-t,0)dξdtdx=∫ℝn‍ϕ1(t)ϕ2(t)¯dt.Theorem 8 (inversion formula).
Ifϕ∈L2(ℝn) and window function ω satisfy the condition (14) of the previous theorem, then
(18)ϕ(t)=∬ℝn(Sωϕ)(τ,ξ)ω(τ-t,ξ)¯ei2π〈t,ξ〉dτdξ.Proof.
By the previous theorem we can write(19)〈ϕ1,ϕ2〉=∬ℝn(Sωϕ1)(τ,ξ)(Sωϕ2)(τ,ξ)¯dτdξ=∬ℝn(Sωϕ1)(τ,ξ)×(∫ℝn‍ϕ2(x)ω(τ-x,ξ)e-i2π〈x,ξ〉dx)¯dτdξ=∫ℝn‍(∬ℝn(Sωϕ1)(τ,ξ)×ω(τ-x,ξ)¯ei2π〈x,ξ〉dτdξ∬ℝn(Sωϕ1)(τ,ξ))ϕ2¯(x)dx.
Hence(20)ϕ1(t)=∬ℝn(Sωϕ1)(τ,ξ)ω(τ-t,ξ)¯ei2π〈t,ξ〉dτdξ.Definition 9.
Letω be a window function and Sω is the S-transform. Then the transform Sω* defined by
(21)〈Sωϕ,ψ〉=〈ϕ,Sω*ψ〉
is called the adjoint of Sω. If ϕ∈L2(ℝn) and ψ∈L2(ℝn×ℝn), then (21) implies that
(22)(Sω*ψ)(t)=∬ℝnψ(τ,ξ)ω(τ-t,ξ)¯ei2π〈t,ξ〉dτdξ,
where t,τ,ξ∈ℝn.Theorem 10 (Parseval's formula forSω*).
Letω1 and ω2 be the window functions that satisfy the condition (14). If ψ1,ψ2∈L2(ℝn×ℝn), then
(23)∫ℝn‍(Sω1*ψ1)(t)(Sω2*ψ2)(t)¯dt=∬ℝnψ1(τ,ξ)ψ2(τ,ξ)¯dτdξ,
and the Plancherel formula is
(24)∥Sω*ψ∥L2(ℝn)=∥ψ∥L2(ℝn×ℝn).Proof.
Consider(25)∫ℝn‍(Sω1*ψ1)(t)(Sω2*ψ2)(t)¯dt=∫ℝn‍(∬ℝnψ1(τ,ξ)ω1(τ-t,ξ)¯ei2π〈t,ξ〉dτdξ)×(∬ℝnψ2(α,η)ω2(α-t,η)¯ei2π〈t,η〉dαdη)¯dt=∫∭ℝnψ1(τ,ξ)ψ2(α,η)¯ei2π〈t,ξ-η〉×(∫ℝn‍ω1(τ-t,ξ)¯ω2(α-t,η)dt)dτdξdαdη=∫∭ℝnψ1(τ,ξ)ψ2(α,η)¯ei2π〈t,ξ-η〉×δ(τ-α,ξ-η)dτdξdαdη=∬ℝnψ1(τ,ξ)ψ2(τ,ξ)¯dτdξ.
This proves the theorem.Theorem 11.
If the window functionω satisfies the condition (14), then
(26)SωSω*=I=Sω*Sω,
where I is the identity operator.Proof.
By definition(27)Sω*[Sω1ψ](t)=∬ℝn(Sω1ψ)(τ,ξ)ω(τ-t,ξ)¯ei2π〈t,ξ〉dτdξ=Sω-1[Sω1ψ](t).
Thus
(28)Sω*=Sω-1.
This proves the theorem.Definition 12 (pseudodifferential operator).
Letσ be a (measurable) function or a tempered distribution on ℝn. Then the operator
(29)Kσϕ(t)=∫ℝn‍σ(t,ξ)ϕ^(ξ)ei2π〈t,ξ〉dξ
is called the pseudodifferential operator.The pseudodifferential operator plays an important role in the theory of partial differential equations. The pseudodifferential operator has been studied on function and distribution spaces by many authors. Details of the concept can be found in [9, 10].


### 2.1. Relation between the S-Transform and Pseudodifferential Operator

Here we give a direct relation betweenS-transform and pseudodifferential operator which will may be very useful in the study of S-transform of distribution spaces. The continuous S-transform of a function ϕ with respect to a window function ω is given by
(30)(Sωϕ)(τ,ξ)=∫ℝn‍ϕ(t)ω(τ-t,ξ)e-i2π〈t,ξ〉dt=∫ℝn‍ϕ(τ-x)ω(x,ξ)e-i2π〈(τ-x),ξ〉dx=e-i2π〈τ,ξ〉∫ℝn‍T-τϕ(-x)ω(x,ξ)ei2π〈x,ξ〉dx=e-i2π〈τ,ξ〉∫ℝn‍ℱ[ℱ(T-τϕ)](x)×ω(x,ξ)ei2π〈x,ξ〉dx=e-i2π〈τ,ξ〉Kσ[ℱ(T-τϕ)](ξ),
where σ(ξ,x)=ω(x,ξ).


## 2.1. Relation between the S-Transform and Pseudodifferential Operator

Here we give a direct relation betweenS-transform and pseudodifferential operator which will may be very useful in the study of S-transform of distribution spaces. The continuous S-transform of a function ϕ with respect to a window function ω is given by
(30)(Sωϕ)(τ,ξ)=∫ℝn‍ϕ(t)ω(τ-t,ξ)e-i2π〈t,ξ〉dt=∫ℝn‍ϕ(τ-x)ω(x,ξ)e-i2π〈(τ-x),ξ〉dx=e-i2π〈τ,ξ〉∫ℝn‍T-τϕ(-x)ω(x,ξ)ei2π〈x,ξ〉dx=e-i2π〈τ,ξ〉∫ℝn‍ℱ[ℱ(T-τϕ)](x)×ω(x,ξ)ei2π〈x,ξ〉dx=e-i2π〈τ,ξ〉Kσ[ℱ(T-τϕ)](ξ),
where σ(ξ,x)=ω(x,ξ).


## 3. The S-Transform of Distributions

In this section we will investigate theS-transform of tempered distribution by means of the Fourier transform.Theorem 13.
Ifω∈𝒮(ℝ2n), then Sω maps 𝒮(ℝn) into 𝒮(ℝ2n).Proof.
By (6) we have
(31)(Sωϕ)(τ,ξ)=ℱ1-1[ℱ1(ϕ)(α+ξ)ℱ1(ω)(α,ξ)](τ,ξ).
Thus, (Sωϕ)∈𝒮(ℝ2n), since the Fourier transform is continuous isomorphism from 𝒮(ℝn) to 𝒮(ℝn), and its inverse is also a continuous isomorphism from 𝒮(ℝn) to 𝒮(ℝn) (see [11], page 66-67).Theorem 14.
Ifω∈𝒮(ℝ2n), then Sω maps 𝒮′(ℝn) into 𝒮′(ℝ2n).Proof.
For anyf∈𝒮′(ℝn) and ψ∈𝒮(ℝ2n), we have
(32)〈(Sωf)(τ,ξ),ψ(τ,ξ)〉=∬ℝn(∫ℝn‍f(t)ω(τ-t,ξ)e-i2π〈t,ξ〉dt)ψ(τ,ξ)¯dτdξ(infactψ-=ψ,sinceψ∈𝒮)=∫ℝn‍f(t)(∬ℝnψ(τ,ξ)¯ω(τ-t,ξ)e-i2π〈t,ξ〉dτdξ)dt=〈f,ϕ〉,
where
(33)ϕ(t)=∬ℝnψ(τ,ξ)ω(τ-t,ξ)¯ei2π〈t,ξ〉dτdξ=∫ℝn‍(ψ*(ℐω))(t,ξ)ei2π〈t,ξ〉dξ=ℱ2-1(ψ*(ℐω))(t)∈𝒮(ℝn).
Thus, (Sωf)(τ,ξ)∈𝒮′(ℝ2n).Theorem 15.
Ifω∈𝒮′(ℝ2n), then Sω maps 𝒮(ℝn) into 𝒮′(ℝ2n).Proof.
Ifϕ∈𝒮(ℝn) and ψ∈𝒮(ℝ2n), then
(34)Uϕ,ψ(τ,ξ):=ℱ(ϕ)(α+ξ)¯ψ(α,ξ)∈𝒮(ℝ2n).
Thus for any ω∈𝒮′(ℝ2n), we have
(35)〈(ℱ1ω)(α,ξ),Uϕ,ψ(α,ξ)〉=∬ℝn(ℱ1ω)(α,ξ)(ℱϕ)(α+ξ)ψ(α,ξ)¯dαdξ=∬ℝn(ℱ1(Sωϕ))(α,ξ)ψ(α,ξ)¯dαdξ=〈(ℱ1(Sωϕ))(α,ξ),ψ(α,ξ)〉.
Thus ℱ1(Sωϕ)∈𝒮′(ℝ2n) and hence (Sωϕ)∈𝒮′(ℝ2n).



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