1. Introduction
The continuous wavelet transform of a function h with respect to the wavelet
is defined as
(1)
provided the integral exists [1] . The asymptotic expansion for Mellin convolution
(2)
was proposed by López [2] , under dyadic conditions on g and h. Let us remind earlier results from [2] , which will be used in present study. We assume that
and
have asymptotic expansions of the form:
(3)
and
(4)
Also assume that
(5)
and
(6)
with the parameters
,
,
and
satisfying the following conditions:
(7)
The asymptotic expansion of (2) at the origin is given by the following Theorem ( [2] , pp. 631, 633, 634).
Theorem 1 Assume that (i)
and
are locally integrable on
; (ii)
and
have expansions of the form (3), (5) and (4), (6) respectively and (iii)
,
,
and
satisfy (7), then the asym- ptotic expansion of (2) as
are given by
Case I: For any
and
with
, we have
(8)
Case II: For any
and
with
, we have
(9)
Case III: For any
and
with
, we have
(10)
By using Wong technique, the asymptotic expansions of wavelet transform (1) for large and small values of dilation parameters and translation, parameters were obtained by Pathak and Pathak 2009 [3] -[5] .
The main aim of the present paper is to derive asymptotic expansion of the wavelet transform for large value of a, by using Theorem 1. We also obtain asymptotic expansions for the special transforms corresponding to Shannon wavelet, Morlet wavelet and Mexican hat wavelet.
2. Asymptotic Expansion of the Wavelet Transform for Large Value of a
In this section, we obtain asymptotic expansion of the wavelet transform (1), when
.
Now, let us rewrite (1) in the form:
(11)
where,
and b is assumed to be a fixed real number.
Setting
and assume that
and
are locally integrable on
. Further as-
sume that
and
have asymptotic expansions of the form
(12)
(13)
Also assume that
(14)
and
(15)
with the parameters
,
,
and
satisfy the following condition
(16)
Then by using, Theorem 1, we obtain asymptotic expansion of
for large value of a.
Case I: When
and
with
, we have
(17)
Case II: When
and
with
, we have
(18)
Case III: When
and
with
, we have
(19)
Similarly, we can also obtain the asymptotic expansion of
as
.
3. Application
In this section, we apply the previous result and obtain the asymptotic expansions of Shannon wavelet transform, Morlet Wavelet transform and Mexican hat wavelet transform.
3.1. Asymptotic Expansion of the Shannon Wavelet Transform
Let us consider
to be Shannon wavelet and it is given by [1]
. Since,
is lo-
cally integrable on
and has the asymptotic expansion:
(20)
with
(21)
Consider,
is locally integrable on
and satisfies (13) and (15) with parameters
(22)
Now, by using (17), (18) and (19) respectively and by means of formula ([6] , p. 321, (41)), then the asymptotic expansions of Shannon wavelet transform are given by
Case I: When
and
, we get
(23)
Case II: When
and
, we get
(24)
Case III: When
and
, we get
(25)
3.2. Asymptotic Expansion of the Morlet Wavelet Transform
We choose
to be Morlet wavelet and it is given by [1]
. Since,
is locally integrable on
and has the asymptotic expansion as
(26)
with
![]()
Let
is locally integrable on
and satisfy (13) and (15) with parameters (22). Now by using (17), (18) and (19) respectively and by formula ([6] , pp. 318, 320, (10,30)),then the asymptotic expansions of Morlet wavelet transform are given by
Case I: When
and
, we have
(27)
Case II: When
and
, we have
(28)
Case III: When
and
, we have
(29)
3.3. Asymptotic Expansion of the Mexican Hat Wavelet Transform
We choose
to be Mexican hat wavelet
[1] . Since
is locally integrable on ![]()
and has the asymptotic expansion:
(30)
with
![]()
As
is locally integrable on
and satisfies (13) and (15) with parameters (22). Now by using (17), (18) and (19) respectively, we can obtain the asymptotic expansion of Mexican hat wavelet transform by using formula ([6] , p. 313, (13))
Case I: When
and
, we have
(31)
Case II: When
and
, we have
(32)
Case III: When
and
, we have
(33)
Acknowledgements
The authors are thankful to Prof. R. S. Pathak, DST Center for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi-221005, India, for his valuable suggestion for the improvement of the article. We thank the referee for their comments. The research of the first author was supported by U.G.C-BSR start-up grant No. F.30-12/2014 (BSR).