Journal of Mathematics

Volume 2013 (2013), Article ID 105848, 9 pages

http://dx.doi.org/10.1155/2013/105848

Research Article

The Fractional S-Transform on Spaces of Type

Sunil Kumar Singh

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

Received 26 November 2012; Accepted 26 February 2013

Academic Editor: Peng-Yeng Yin

Copyright © 2013 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.

Certain properties of the S-transform are given and composition of S-transform is defined. We study the fractional S-transform on spaces of type . It is shown that the fractional S-transform is a continuous linear map of the space of type defined on into the space of type defined on .

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 of is defined as [4]

where the window is assumed to satisfy the following:

The most usual window is the Gaussian one

where is the frequency, is the time variable, and is a scaling factor that controls the number of oscillations in the window.

Then, (1) can be rewritten as a convolution

Applying the convolution property for the Fourier transform, we obtain

where is the inverse Fourier transform. For the Gaussian window case (3),

Thus we can write the S-transform in the following form:

Also, if and are the Fourier transform and S-transform of , respectively, then

so that

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. If , then .

Proof. We have by definition

Thus

Clearly,

Theorem 2. If , then .

Theorem 3. If , then .

Theorem 4. If , then is continuous with respect to the variables and .

Proof. By the definition

Hence the S-transform is continuous with respect to the first variables . Similarly we can show that it is also continuous with respect to the second variables .

2.1. Composition of S-Transforms

Let and be two S-transforms defined as

provided the integrals exist. Then their composition is defined by

where , provided the integral exists. The composition is an S-transform of three variables mapping from into and analogous to the composition of the wavelet transform [8, pages 52-53].

Theorem 5. If , then is continuous with respect to the variables , and .

3. Fractional Fourier Transform

The fractional Fourier transform (FRFT) has played an important role in signal processing. The th order FRFT of a signal is defined as [9]

where the transform kernel is defined as

where ,  ,  , is the complex unit, is an integer, and is the fractional Fourier frequency (FRFfr). The inverse FRFT of (16) is

We can write (16) as

Replacing by , we have

Putting , we have

where .

4. The Fractional S-Transform

The fractional S-transform (FRST) is a generalization of the S-transform. The th order continuous fractional S-transform of is defined as [10]

where the window is

and satisfies the condition

Inverse fractional S-transform is defined by

Note that the fractional S-transform depends on a parameter and can be interpreted as a rotation by an angle in the time-frequency plane. An FRST with corresponds to the S-transform, and an FRST with corresponds to the zero operator. The parameters and can be used to adjust the window function space.

Let

Since

By using (23) and (28) in (27) we obtain

By using the technique of [10], we obtain

and by using (28) we can write

Also, the FRST can be also defined as operations on the fractional Fourier domain:

By using (19) and (30) we can write (32) as

5. The Fractional S-Transform on Spaces of Type

The spaces of type play an important role in the theory of linear partial differential equations as intermediate spaces between the spaces of the and the analytic functions. The Fourier transform has been studied on the spaces of type by Friedman [11] and Gel’fand and Shilov [12]. Pathak and Singh [13] studied the wavelet transform on the spaces of type . Also, Singh [6] studied the S-transform on the spaces of type . In this section we study the fractional S-transform (22) on these spaces.

Let us recall the definitions of these spaces.

Definition 6. The space consists of all infinitely differentiable functions   , satisfying the inequalities

where the constants and depend on the function . For , the expression is considered to be equal to 1.

Definition 7. The space consists of all infinitely differentiable functions   , satisfying the inequalities

where the constants and depend on the function .

Definition 8. The space consists of all infinitely differentiable functions   , satisfying the inequalities

where the constants , and depend on the function .

The spaces of type are closely interrelated by means of the Fourier transformation; namely, the formulas

hold.

We will make use of the following inequalities in our investigation:

In what follows we will also need the following similar test function space defined on , the space of type .

Definition 9. The space ,  , is defined to be the space of all functions such that for all ,

where the constants , and depend on .

We show that the fractional S-transform is a continuous linear map of the space of type into the space of type defined mentioned previously.

Theorem 10. For , the fractional S-transform is a continuous linear map of into where and .

Proof. Let and . Then, after differentiation and integration by parts, we obtain

By using the convergence factor , we obtain

Using the property (38) we obtain

Use the properties (35) and (39) to obtain

Hence, we obtain the estimate

where and are positive constants.

Theorem 11. For , the fractional S-transform is a continuous linear map of into where and .

Theorem 12. For , the fractional S-transform is a continuous linear map of into where and .

Proof. By inequality (43), we have

Next, using the properties (36) and (39) we have

Next, by using the technique of Theorem 10, we obtain

Hence, we obtain the estimate

where and 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].

Acknowledgment

The author would like to express their sincere thanks to Professor R. S. Pathak for his help and encouragement.

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