The S-Transform of Distributions

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 spaces and .

1. Introduction

The -transform was first used by Stockwell et al. [1] in 1996. If is a window function, then the continuous -transform of with respect to is defined as [2] In signal analysis, at least in dimension , is called the time-frequency plane, and in physics is called the phase space.

Equation (1) can be rewritten as a convolution as Now, we recall the definitions of Fourier transform on .

Definition 1. If is defined on , then the Fourier transform of is given by where is the usual inner product on .

Definition 2. If is defined on , then the partial Fourier transform of with respect to the first coordinate is given by and the partial Fourier transform of with respect to the second coordinate is given by Applying the convolution property for the Fourier transform in (2), we obtain where is the inverse Fourier transform.

Now, we define the translation, modulation, and involution operators, respectively, by where .

Definition 3 (the Dirac delta). The Dirac delta function is defined by

Definition 4 (tempered distribution). A function is said to be rapidly decreasing if for all pairs of multi-indices . The space of all rapidly decreasing functions on is denoted by or simply . Elements in the dual space of are called tempered distribution.

2. Some Important Properties of S-Transform

Some properties of -transform can be found in [3–8] and certain properties of -transform are obtained in this section. By definition, we have Thus, the -transform appears as a superposition of time-frequency shifts as follows:

Example 5. If , that is, independent of , then So is a multiplication operator. In particular, if , then .

Example 6. If , then

Theorem 7 (Parseval's formula). Let and be the window functions such that Let and let and be the -transforms of and , respectively. Then
This immediately implies the Plancherel formula

Proof. Consider

Theorem 8 (inversion formula). If and window function satisfy the condition (14) of the previous theorem, then

Proof. By the previous theorem we can write
Hence

Definition 9. Let be a window function and is the -transform. Then the transform defined by is called the adjoint of . If and , then (21) implies that where .

Theorem 10 (Parseval's formula for ). Let and be the window functions that satisfy the condition (14). If , then and the Plancherel formula is

Proof. Consider This proves the theorem.

Theorem 11. If the window function satisfies the condition (14), then where is the identity operator.

Proof. By definition Thus This proves the theorem.

Definition 12 (pseudodifferential operator). Let be a (measurable) function or a tempered distribution on . Then the operator 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 between -transform and pseudodifferential operator which will may be very useful in the study of -transform of distribution spaces. The continuous -transform of a function with respect to a window function is given by where .

3. The S-Transform of Distributions

In this section we will investigate the -transform of tempered distribution by means of the Fourier transform.

Theorem 13. If , then maps into .

Proof. By (6) we have Thus, , since the Fourier transform is continuous isomorphism from to , and its inverse is also a continuous isomorphism from to (see [11], page 66-67).

Theorem 14. If , then maps into .

Proof. For any and , we have where Thus, .

Theorem 15. If , then maps into .

Proof. If and , then Thus for any , we have Thus and hence .

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgment

The author expresses his sincere thanks to Professor R. S. Pathak for his help and encouragement.