An Asymptotic Expansion of Continuous Wavelet Transform for Large Dilation Parameter

Ashish Pathak, Prabhat Yadav, M. M. Dixit abstract: In this paper , we derive asymptotic expansion of the wavelet transform for large values of the dilation parameter a by using Lopez and Pagola technique. Asymptotic expansion of Mexican Hat wavelet and Morlet wavelet transform are obtained as a special cases.


Introduction
The wavelet transform of g with respect to the wavelet φ is defined by provided the integral exists [1].Using Fourier transform it can also be expressed as where The Asymptotic expansion for the integral of the form A. Pathak, P. Yadav, M. M. Dixit was obtained by Lopez and Pagola [ [5], Theorem 2,3] under certain conditions on f and h.Then the asymptotic expansion of (1.2) for large a can be obtained by setting f (t) = ĝ(t) for fixed b ∈ R and h(t) = φ(t).Here we assume that f (t) and h(t) has an expansion of the form and that as t → ∞, where η r < η r+1 , ∀r ≥ 0 and ρ i < ρ i+1 , ∀ i ≥ 0. Also assume that and with α, β, η 0 and ρ 0 satisfying the relations: β − η 0 < 1 < α + ρ 0 and −η 0 < α and β < ρ 0 .The aim of the present paper is to derive asymptotic expansion of the wavelet transform (1.2) for large value of dilation parameter a .In section 2, we assume that ĝ(ω) and φ(ω) possess asymptotic expansion of the form (1.4) and (1.5) as ω → 0+ and ω → ∞ and derive the asymptotic expansion of (W φ g) (b, a) as a → ∞+.In section 3, we obtain asymptotic expansion of Mexican Hat and Morlet wavelet transform.

Asymptotic expansion of wavelet transform for large a
Let us rewrite (1.2) in the form: where (2.4) Assume that ĝ(ω), φ(ω) are locally integrable in (−∞, ∞) and and that as ω → ∞, where Now by using [ [5],Theorem 2,3], we can also prove the theorem for asymptotic expansion of wavelet transform for large value of dilation parameter a.
(2.13) (a).For a given (i, r) satisfying where Go to (a) with i replaced by i + 1.
Go to (a) with r replaced by r + 1.
Define the function Then we have that for z 0 < Re(z) < z 1 : and that H i,r (ω) ∈ [0, ∞).using the dominated convergence theorem, we get +c r a −z ω z+η r e ibω a φi+1 (ω)]dω. (2.17) From ρ i = 1+η r and from Lemma 2 and 3 of [5], we have that M [ φ(ω); z+1+η r ] and we have that a − η 0 − 1 < 0 < η 0 + b and then the point z = 0 belongs to that strip of analytic.Therefore, We find that the above expression can be rewritten as e ibω ĝr (ω) φi+1 (aω)dω. (2.20) Hence, If ρ i − η r = 1 for some pair (i, r) then in (2.21), the sum of the terms must be replaced by and the remainder term R n,m (a) as a → ∞+ is given by

Application
Using the aforesaid technique, we find the asymptotic expansions of Mexican Hat wavelet and Morlet wavelet transform.
where Morlet wavelet transform for a → ∞+ as where R(r) is the index r for which (r + 1) − η r ≤ 1 < (r + 1) − η r − 1 and I(i) is the index i for which r − η r < 1 ≤ (r + 1) − η r .If (r + 1) − η r = 1 for some pair (i, r) then in (3.9), the corresponding sum of the terms must be replace by