Moment Asymptotic Expansions of the Wavelet Transforms

Using distribution theory we present the moment asymptotic expansion of continuous wavelet transform in different distributional spaces for large and small values of dilation parameter $a$. We also obtain asymptotic expansions for certain wavelet transform.


Introduction
In past few decades their were many mathematician who has done great work in the field of asymptotic expansion like Wong 1979 [10] using Mellin transform technique has obtained asymptotic expansion of classical integral transform and after that Pathak & Pathak 2009 [3, 4, 5, 6] has found the asymptotic expansion of continuous wavelet transform for large and small values of dilation and translation parameters. Estrada & Kanwal 1990 [7] has obtained the asymptotic expansion of generalized functions on different spaces of test functions. In present paper using Estrada & Kanwal technique we have obtained the asymptotic expansion of wavelet transform in different distributional spaces.
The continuous wavelet transform of f with respect to wavelet ψ is defined by provided the integral exists [3] Now, from (1.1) we get This paper is arranged in following manner. In section second, third , fourth and fifth we drive the asymptotic expansion in the distributional spaces E ′ (R), respectively, studied in [7] 2. The moment asymptotic expansion of (W ψ f ) (a, b) as a → ∞ in the space E ′ (R) for given b The space E (R) is the space of all smooth functions on R and it's dual space E ′ (R) , the space of distribution with compact support. If ψ ∈ E (R), then . So consider the seminorms for α ∈ N and M > 0, these seminorm generate the topology of E (R). If q = 0, 1, 2, 3, ..., we set Lemma 2.1. Let ψ ∈ X q , then for every α ∈ N and M > 0, If α ≤ q and ψ ∈ X q then D α ψ ∈ X q−α and thus Similarly by using (2.2) we can prove that Now, by using Lemma 2.1 we obtain the following theorem and µ α = f, x α be its moment sequence . Then for a fixed b the moment asymptotic expansion of wavelet transform is where the existence of L, q and M is guaranteed by the continuity of f . Hence we get the required asymptotic expansion 2.7. The Mexican-Hat wavelet is given by [3] ψ Now, using Theorem2.2 we get the asymptotic expansion of Mexican-Hat wavelet 3. The moment asymptotic expansion of (W ψ f ) (a, b) for large and small values of a in the space P ′ (R) for a given b Case 1. Let ψ ∈ P(R).
We now consider the moment asymptotic expansion in the space P ′ (R) of distributions of "less than exponential growth". The space P(R) consist of those smooth functions φ(x) that satisfy lim x→∞ e −γ|x| D β φ(x) = 0 f or γ > 0 and each β ∈ N, with seminorms Let wavelet ψ(x) ∈ P(R). Then where for a given γ > 0 and b ∈ R.
Therefore for any γ > 0 we can find a constant C such that Using (3.2) we obtain the following theorem Theorem 3.1. Let ψ ∈ P(R), f ∈ P ′ (R) and µ α = f, x α be its moment sequence. Then for a fixed b the asymptotic expansion of wavelet transform is f (x) ∈ P ′ (R). Therefore by Theorem 3.3 moment asymptotic expansion of continuous Mexican-Hat wavelet transform for large a in P ′ (R) is given by Case 2. In this case we consider wavelet ψ(x) ∈ P ′ (R) and f (x) ∈ P(R).
Then the wavelet transform (1.1) can we rewrite as Similarly as Theorem 3.1 we can also obtain the following theorem Theorem 3.3. Let ψ ∈ P ′ (R), f ∈ P(R) and µ α = ψ, x α be its moment sequence. Then for a fixed b the asymptotic expansion of wavelet transform is x → ∞ for every α ∈ N and γ ∈ R. The family of seminorms generates a topology for O γ (R). Now with the help of the translation version of ψ(x), we can define the seminorms on O γ (R) as Similarly for γ < 0. we have Thus sup ργ (|x|) Therefore ψ(x) α, γ, b/a are also seminorm on O γ (R).These seminorm generate the topology of the space ψ(x) ∈ O γ (R).If So for any γ we can find a constant C such that If a > 1 Hence using above equation we get Similarly as Theorem 3.1 we can obtain the following theorem ] − 1 and µ α = f, x α be its moment sequence. Then for a fixed b ∈ R the asymptotic expansion of wavelet transform is , we obtain the asymptotic expansion of wavelet transform in the space O ′ c (R) and µ α = f, x α be its moment sequence. Then for a fixed b ∈ R the asymptotic expansion of wavelet transform