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The Scientific World Journal

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

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

## The S-Transform of Distributions

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

- R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform,”
*IEEE Transactions on Signal Processing*, vol. 44, no. 4, pp. 998–1001, 1996. View at Publisher · View at Google Scholar · View at Scopus - S. Ventosa, C. Simon, M. Schimmel, J. J. Danobeitia, and A. Manuel, “The S-transform from a wavelet point of view,”
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*Integral Transforms and Special Functions*, vol. 23, no. 12, pp. 891–899, 2012. - S. K. Singh, “The fractional S-Transform of tempered ultradistibutions,”
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*Foundations of Time-Frequency Analysis*, Birkhäauser, Boston, Mass, USA, 2001. - S. Zaidman,
*Distributions and Pseudo-Differential Operators*, Logman, Essex, UK, 1991. - R. S. Pathak,
*A Course in Distribution Theory and Applications*, Narosa Publishing House, New Delhi, India, 2009.