Research Article | Open Access
R. S. Pathak, Ashish Pathak, "Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b", International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 270492, 13 pages, 2009. https://doi.org/10.1155/2009/270492
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b
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.
The wavelet transform of with respect to the wavelet ψ is defined by
provided that the integral exists . Using Fourier transform it can also be expressed as
Asymptotic expansion with explicit error term for Mellin convolution
where is real and .
Also assume that
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 .
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 3–5. 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
Next, assume that
where Now, for fixed , write
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
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 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
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
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
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 .
Now, using (1.12)
Let us set
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 , asymptotic expansion of as can be obtained. This gives rise to a complicated form of the asymptotic expansion and needs separate treatment .
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.
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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.