The Scientific World Journal

Volume 2014 (2014), Article ID 623294, 4 pages

http://dx.doi.org/10.1155/2014/623294

Research Article

The S-Transform of Distributions

Sunil Kumar Singh

Department of Mathematics, Rajiv Gandhi University, Doimukh, Arunachal Pradesh 791112, India

Received 30 August 2013; Accepted 10 October 2013; Published 2 January 2014

Academic Editors: B. Carpentieri and A. Ibeas

Copyright © 2014 Sunil Kumar Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

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