International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 270492 | 13 pages | https://doi.org/10.1155/2009/270492

Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b

Academic Editor: A. Zayed
Received27 Jul 2009
Accepted09 Sep 2009
Published18 Oct 2009

Abstract

Asymptotic expansions of the wavelet transform for large and small values of the translation parameter are obtained using asymptotic expansions of the Fourier transforms of the function and the wavelet. Asymptotic expansions of Mexican hat wavelet transform, Morlet wavelet transform, and Haar wavelet transform are obtained as special cases. Asymptotic expansion of the wavelet transform has also been obtained for small values of when asymptotic expansions of the function and the wavelet near origin are given.

1. Introduction

The wavelet transform of with respect to the wavelet ψ is defined by

provided that the integral exists [1]. Using Fourier transform it can also be expressed as

where

Asymptotic expansion with explicit error term for Mellin convolution

as , was obtained by Wong [2, pages 740–756]. Let us recall basic results from Wong [2], which will be used in the present investigation.

Assume that

where and

where is real and .

Also assume that

where .

where .

Asymptotic expansion of (1.4) is given by the following [2, Theorem  3, page 752].

Theorem 1.1. Assume that (i) is continuous on where is a nonnegative integer; (ii) has an expansion of the form (1.5), and the expansion can be differentiated times; (iii) as is for and for some ; (iv) has the meaning as given in (1.5); (v) satisfies (1.8) and (1.6) with . Then we have where and the remainder satisfies

As an application of the above theorem, Wong [2, page 753] has derived the following asymptotic expansion for the Fourier transform for large values of : The asymptotic expansion of the wavelet transform (1.2) for large values of dilation parameter has already been obtained in [3].

The aim of the present paper is to derive asymptotic expansion of the wavelet transform given by (1.2) for large and small values of . In Section 2 we assume that and possess asymptotic expansions of the form (1.5) as and derive asymptotic expansion of as using formula (1.12). Asymptotic expansions of certain special forms of the wavelet transform are obtained in Sections 35. In Section 6 we assume that asymptotic expansions of and are known as and derive asymptotic expansion of as using Theorem 6.1 due to Wong [4, Theorem  14, page 323]. In Section 7 we assume the asymptotic expansions of and as and derive asymptotic expansions of and as , using (1.12). These asymptotic expansions of and give rise to the asymptotic expansion of as , using Theorem 6.1.

2. Asymptotic Expansion for Large

Let us rewrite (1.2) in the following form:

where for definiteness we take and . Now, we consider Assume that

then for arbitrary but fixed , we have

where .

Next, assume that

Then

where Now, for fixed , write

where .

Let us set

and assume that (i) is continuous on where is a nonnegative integer; (ii) the expansion (2.9) can be differentiated times; (iii) as for and for some .

Then, by (1.12), for ,

Similarly, we get

Notice that the series expansions in (2.10) and (2.11) are the same but opposite in sign. Therefore, we find asymptotic expansion of only. From (2.2) and (2.10), we have

3. Mexican Hat Wavelet Transform

In this section we choose to be Mexican hat wavelet and derive asymptotic expansion of the corresponding wavelet transform. The Mexican hat wavelet is defined by

then from [1, page 372],

Now, in view of (2.5), we have

where

where stands for the greatest positive integer . To ensure that exists for large values of we also impose the condition that for some real number as . Also, from (3.3) and (2.8) we conclude that in the present case . Therefore, from (2.12), using (3.4) we get

4. Morlet Wavelet Transform

In this section we choose Then from [1, page 373], Now, where

Hence

where

Also, from (2.8) and (4.5) it follows that . Therefore, from (2.12), using (4.6) we get

5. Haar Wavelet Transform

The Haar wavelet is defined by

whose Fourier transform [1, page 368] is Therefore, Haar wavelet transform on half-line is given by

For possessing asymptotic behavior (2.5), we have

Then, from (5.3) and (5.4) using formula [2, page 753]

we get, for and ,

6. Asymptotic Expansion for Small

In this section we assume that asymptotic expansions of and as are known and then derive asymptotic expansion of as for fixed , using the following [4, Therorem  14, page 323].

Theorem 6.1. Let be a locally integrable function on and let possess an asymptotic expansion of the form where . Then for small values of , where with Let Then, writing , we have where

Now, for fixed , write

where .

Then, using (2.2), (6.2), and (6.8), we find asymptotic expansion of wavelet transform for small value of : where

with

7. Asymptotic Expansion for Small Continued

In this section we assume that asymptotic expansions of and are known, instead of and as in previous sections. Then as in [2, page 753] we get asymptotic expansions of and as . On the other hand, in (2.3) and (2.5) their behaviors near the origin were known, that yielded the asymptotic expansion of as . However, in this case, following [4, pages 321–323] we can obtain asymptotic expansion of as .

Let

Now, using (1.12)

Similarly,

Then

where

Let us set

where .

Assume that integrable is locally on . Then applying (6.2) to (2.2) with given by (7.6), finally we get

where

with

Remark 7.1. We observe that if we assume the asymptotic expansions and as and derive asymptotic expansions of and as , then formula (1.2) gives asymptotic expansion of as for fixed . The aforesaid technique does not yield asymptotic expansion of as using (1.2). However, if one uses the form (1.1) of the wavelet transform and applies Li and Wong-technique involving a theory of noncommutative convolution [5], asymptotic expansion of as can be obtained. This gives rise to a complicated form of the asymptotic expansion and needs separate treatment [6].

Acknowledgments

The authors are thankful to the referees for their valuable comments. The work of the first author was supported by U.G.C. Emeritus Fellowship.

References

  1. L. Debnath, Wavelet Transforms and Their Applications, Birkhäuser, Boston, Mass, USA, 2002. View at: Zentralblatt MATH | MathSciNet
  2. R. Wong, “Explicit error terms for asymptotic expansions of Mellin convolutions,” Journal of Mathematical Analysis and Applications, vol. 72, no. 2, pp. 740–756, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. R. S. Pathak and A. Pathak, “Asymptotic expansion for the wavelet transform with error term,” communicated. View at: Google Scholar
  4. R. Wong, Asymptotic Approximations of Integrals, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1989. View at: MathSciNet
  5. X. Li and R. Wong, “Error bounds for asymptotic expansions of Laplace convolutions,” SIAM Journal on Mathematical Analysis, vol. 25, no. 6, pp. 1537–1553, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. R. S. Pathak, “Asymptotic expansion of the wavelet transform,” in Industrial and Applied Mathematics, A. H. Siddiqui and K. Ahmad, Eds., pp. 43–50, Narosa, New Delhi, India, 1998. View at: Google Scholar

Copyright © 2009 R. S. Pathak and Ashish Pathak. 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.


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