(3s.) v. 36 3 (2018): 27–39. ISSN-00378712 in press doi:10.5269/bspm.v36i3.31221
Bol. Soc. Paran. Mat. c
SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm
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.
Key Words: Asymptotic expansion, Wavelet transform, Fourier transform, Mellin transform. Contents 1 Introduction
27
2 Asymptotic expansion of wavelet transform for large a
28
3 Application 35 3.1 Asymptotic Expansion of Mexican Hat Wavelet Transform . . . . . . 35 3.2 Asymptotic Expansion of Morlet Wavelet Transform . . . . . . . . . 37 1. Introduction The wavelet transform of g with respect to the wavelet φ is defined by (Wφ g)(b, a) = a−1/2
Z
∞
g(t) φ
−∞
t−b a
dt, b ∈ R, a > 0.
(1.1)
provided the integral exists [1]. Using Fourier transform it can also be expressed as Z a1/2 ∞ ibω ˆ (1.2) e gˆ(ω)φ(aω)dω, (Wφ g)(b, a) = 2π −∞ where gˆ(ω) =
Z
∞
e−itω g(t)dt.
−∞
The Asymptotic expansion for the integral of the form Z ∞ I(ξ) := eiτ t f (t)h(ξt)dt, τ ∈ R, τ 6= 0+, as ξ → 0+,
(1.3)
0
2010 Mathematics Subject Classification: 44A05;41A60. Submitted March 03, 2016. Published September 09, 2016
27
Typeset by BSP style. M c Soc. Paran. de Mat.
28
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 ˆ setting f (t) = gˆ(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 f (t) =
n−1 X
cr tηr + fn (t),
r=0
as t → 0+,
(1.4)
and that as t → ∞, h(t) =
n−1 X
di t−ρi + hn (t),
(1.5)
i=0
where η r < η r+1 , ∀r ≥ 0 and ρi < ρi+1 , ∀ i ≥ 0. Also assume that f (t) = O(t−α ), t → ∞, α ∈ R,
(1.6)
h(t) = O(t−β ), t → 0+, β ∈ R,
(1.7)
and
with α, β, η 0 and ρ0 satisfying the relations: β − η 0 < 1 < α + ρ0 and −η 0 < α and β < ρ0 . 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 ˆ possess asymptotic expansion of the form (1.4) and (1.5) as ω → 0+ gˆ(ω) and φ(ω) and ω → ∞ and derive the asymptotic expansion of (Wφ g) (b, a) as a → ∞+. In section 3, we obtain asymptotic expansion of Mexican Hat and Morlet wavelet transform. 2. Asymptotic expansion of wavelet transform for large a Let us rewrite (1.2) in the form: √ Z ∞ a ˆ eibω φ(aω)ˆ g(ω)dω (Wφ g) (b, a) = 2π 0 Z ∞ −ibω ˆ + e φ(−aω)ˆ g(−ω)dω 0 = Wφ+ g (b, a) + Wφ− g (b, a),
(2.1) (2.2)
where Wφ+ g (b, a) =
√ Z ∞ a ˆ eibω φ(aω)ˆ g (ω)dω, 2π 0
(2.3)
Asymptotic Expansion of Continuous Wavelet Transform
Wφ− g (b, a) =
√ Z ∞ a ˆ e−ibω φ(−aω)ˆ g (−ω)dω. 2π 0
29
(2.4)
ˆ are locally integrable in (−∞, ∞) and Assume that gˆ(ω), φ(ω) gˆ(ω) =
n−1 X
cr ω ηr + gˆn (ω), as ω → 0+,
r=0
(2.5)
and that as ω → ∞, ˆ φ(ω) =
n−1 X
ˆ (ω), di ω −ρi + φ n
(2.6)
i=0
where ρi < ρi+1 , ∀ i ≥ 0 and η r < η r+1 , ∀ r ≥ 0. Also assume that gˆ(ω) = O(ω −α ),
ω → ∞, α ∈ R, α + ρ0 > 1,
(2.7)
ˆ φ(ω) = O(ω −β ),
ω → 0+, β ∈ R, β − η 0 < 1.
(2.8)
and
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. ˆ Theorem 2.1. . Assume that (i) gˆ(w) and φ(w) are locally integrable on (0, ∞), ˆ (ii) gˆ(w) satisfies (2.5) and (2.7),(iii) φ(w) satisfies (2.6) and (2.8). Then, for any n, m ∈ N such that ρn−1 − η m < 1 < ρn − η m−1 , Z
∞
e
ibω
ˆ gˆ(ω)φ(ω)dω
=
0
n−1 X r=0
×
ibω/a ˆ cr M [φ(ω)e ; 1 + ηr ] −
a−ηr −1 +
m−1 X i=0
I(i)−1
− ×
R(r)−1
X j=0
X p=0
dp Bp (1 + η r ; a; b)
g(ω)eib ; 1 − ρi ] di M [ˆ
cj Aj (1 − ρi ; b)
a−ρi + Rn,m (a),
Bp (x; a; b) Aj (x; b)
x−ρp b = e Γ(x − ρp ) , a Γ(x + η j ) = eiπ(x+ηj )/2 . bx+ηj iπ(x−ρp )/2
(2.9)
30
A. Pathak, P. Yadav, M. M. Dixit
R(r) is the index r for which ρi − η r ≤ 1 < ρi − η r−1 and I(i) is the index i for which ρi−1 − η r < 1 ≤ ρi − η r . If ρi − η r = 1 for some pair (i, r) then in (2.9), the sum of the terms R(r)−1 X ibω/a ˆ cr M [φ(ω)e ; 1 + ηr ] − dp Bp (1 + η r ; a; b) a−ηr −1 p=0
I(i)−1 X ibw cj Aj (1 − ρi ; b) a−ρi , g (ω)e ; 1 − ρi ] − di M [ˆ
+
(2.10)
j=0
must be replaced by a
−ρi
I(i)−1 X ibω cj Aj (1 − ρi ; b) − cr loga g(ω)e ; 1 − ρi ] − di M [ˆ j=0
π ibω/a ˆ −cr di [i − logb − γ] + cr lim M [φ(ω)e ; z + 1 + ηr ] z→0 2 r−1 X di − dp Bp (z + 1 + η r ; a; b)) − , z p=0
(2.11)
and the remainder term Rn,m (a) as a → ∞+ is given by O(a−ρi + a−ηr −1 ), when − η r 6= −ρi + 1, Rn,m (a) = . O(a−ρi log a), when − η r = −ρi + 1.
(2.12)
ˆ (ω) + φ ˆ ˆ Proof: Define gˆ0 (ω) = gˆ(ω) and φ 0 α−1 (ω) = φ(ω). For any i there exists r such that ρi−1 − η r ≤ 1 < ρi − η r−1 . For the given (i, r), the following integral exists: Z
∞
0
ˆ (aω). eibω gˆr (ω) φ i
(2.13)
(a). For a given (i, r) satisfying ρi−1 − η r ≤ 1 < ρi − η r−1 , do the following: If ρi − η r < 1 go to (b). If ρi − η r > 1 go to (c). If ρi − η r = 1 go to (d). ˆ (aω) = di (aω)−ρi + φ ˆ (aω) in (2.13) and (iii) of Lemma(2) of [5], we (b). Use φ i i get Z
∞ 0
ˆ (aω) eibω gˆr (ω) φ i
I(i)−1 X cj Aj (1 − ρi ; b) g (ω)eibω ; 1 − ρi ] − = di a−ρi M [ˆ j=0
+
Z
0
∞
ˆ (aω)dω, eibω gˆr (ω) φ i+1
(2.14)
31
Asymptotic Expansion of Continuous Wavelet Transform
iπ(1−ρi +ηj )
Γ(1−ρ +η )
i j 2 where Aj (1 − ρi ; b) = e . b1−ρi +ηj Go to (a) with i replaced by i + 1. (c). use gˆr (ω) = cr ω ηr + gˆr+1 (ω) in (2.13) and (iii) of lemma (3) of [5],we get Z ∞ ibω/a ˆ (aω) = cr a−ηr −1 M [φ(ω)e ˆ ; 1 + ηr ] eibω gˆr (ω) φ i
0
R(r)−1
X
− +
Z
p=0 ∞
0
where, Bp (1 + η r ; a; b) = e
iπ(1+η r −ρp ) 2
dp Bp (1 + η r ; a; b)
ˆ (aω)dω, eibω gˆr+1 (ω) φ i
(2.15)
(1+ηr −ρp ) . Γ(1 + η r − ρp ) ab
Go to (a) with r replaced by r + 1. ˆ (aω) = di (aw)−ρi + φ ˆ (aω) and then gˆr (ω) = cr ω ηr + gˆr+1 (ω) in (d). use first φ i i+1 (2.13) , we get Z ∞ Z ∞ ˆ (aω) = ˆ (aω)]dω eibω gˆr (ω) φ eibω [di (aω)−ρi gˆr (ω) + cr ωη r φ i
0
Z0 ∞
+
0
i+1
ˆ (aω)dω. eibω gˆr (ω) φ i+1
Define the function −ρi −ρi ηr ˆ Ei,r (z, ω) = ω di a ω gˆr (ω) + cr ω φi+1 (aω) z ∈ C. z
Then Z ∞ Z ˆ (aω)dω = eibω gˆr (ω) φ i 0
∞
eibω Ei,r (0, ω)dω +
0 ηr
Z
∞ 0
ˆ (aω)dω. eibω gˆr (ω) φ (2.16) i+1
η r +1
On the one hand gˆr (ω) = cr ω + O(ω ); when ω → 0+. On the other hand ˆ (ω) = −di ω−ρi + O(ω −ρi −1 ); when ω → 0+. Hence,Ei,r (z, .) ∈ L1 [0, ∞) for φ i+1 Max (ρi − ηr + 1, ρi − 1 − ηr ) − 1 < Re(z) < M in(ρi − η r − 1, ρi + 1 − ηr ) − 1. Choose two numbers z0 and z1 , satisfying Max (ρi − η r + 1, ρi − 1 − η r ) − 1 < z0 < 0 and 0 < z1 < M in(ρi − ηr − 1, ρi + 1 − ηr ) − 1. Then we have that for z0 < Re(z) < z 1: |Ei,r (ω, z)| ≤ Hi,r (ω) = |Ei,r (ω, z0 )| for ω ∈ [0, 1] and |Ei,r (ω, z1 )| for ω ∈ [1, ∞) and that Hi,r (ω) ∈ [0, ∞). using the dominated convergence theorem, we get Z ∞ Z ∞ eibω Ei,r (0, ω)dω = lim eibω Ei,r (z, ω)dω z→0 0 0 Z ∞ −ρi = a lim [di ω z−ρi eibω gˆr (ω) z→0
0
+cr a−z ωz+η r e
ibω a
ˆ (ω)]dω. φ i+1
(2.17)
32
A. Pathak, P. Yadav, M. M. Dixit
ˆ From ρi = 1+η r and from Lemma 2 and 3 of [5], we have that M [φ(ω); z+1+ηr ] and M [ˆ g(ω); z + 1 − ρi ] have a common strip of analyticity: a − ηr − 1 < Re(z) < η r + b. we have that a − η 0 − 1 < 0 < η 0 + b and then the point z = 0 belongs to that strip of analytic. Therefore,
Z
∞
0
ˆ (aω)dω eibω gˆr (ω)φ i
=
di M [ˆ gr (ω)eibω ; z + 1 − ρi ]
a−ρi lim
z→0
I(i)−1
−
X
cj Aj (z + 1 − ρi ; b) − dj e
j=0
+cr a
−z
R(r)−1
X
− +
Z
p=0 ∞
0
iπz 2
Γ(z) bz
ibω ˆ a ;z + 1 + η ] M [φ(ω)e r
dp Bp (z + 1 + η r ; a; b)
ˆ (aω)dω. eibω gˆr (ω)φ i+1
(2.18)
Using a−z = 1 − z log(a) + O(z 2 ); when z → 0 and ˆ M [φ(ω)e
ibω a
; z + 1 + ηr ] = =
Z
∞
e
0
ibω a
ˆ (ω)ω z+ηr dω gˆr (ω)φ i
di + 0(1); when z → 0. z
(2.19)
We find that the above expression can be rewritten as
Z
0
∞
e
ibω
ˆ (aω)dω gˆr (ω)φ i
= a
−ρi
I(i)−1 X ibω cj Aj (1 − ρi ; b) gr (ω)e ; 1 − ρi ] − di M [ˆ j=0
ibω ˆ a ;z + 1 + η ] − cr log a + cr lim M [φ(ω)e r
z→0
R(r)−1
−
X p=0
− cr di [
dp Bp (z + 1 + η r ; a; b) −
di z
Z ∞ iπ ˆ (aω)dω. − log b − γ] + eibω gˆr (ω)φ i+1 2 0 (2.20)
Asymptotic Expansion of Continuous Wavelet Transform
33
Hence, (Wφ+ g)(b, a)
= =
Z ∞ a1/2 ibω ˆ e gˆ(ω)φ(aω)dω 2π 0 n−1 1 X ibω ˆ a ;1 + η ] cr M [φ(ω)e r 2π r=0 R(r)−1
− +
X p=0
m−1 X i=0
g (ω)eibω ; 1 − ρi ] di M [ˆ
I(i)−1
−
X j=0
dp Bp (1 + η r ; a; b) a−ηr −1/2
−ρi +1/2 cj Aj (1 − ρi ; b) a + Rn,m .
(2.21)
If ρi − η r = 1 for some pair (i, r) then in (2.21), the sum of the terms R(r)−1 X ibω/a ˆ cr M [φ(ω)e ; 1 + ηr ] − dp Bp (1 + η r ; a; b) a−ηr −1/2 p=0
g(ω)e + di M [ˆ
I(i)−1 ibω
; 1 − ρi ] −
X j=0
cj Aj (1 − ρi ; b) a−ρi +1/2 ,
(2.22)
must be replaced by I(i)−1 X cj Aj (1 − ρi ; b) − cr loga g(ω)eibω ; 1 − ρi ] − a−ρi +1/2 di M [ˆ j=0
π ibω/a ˆ − logb − γ] + cr lim M [φ(ω)e ; z + 1 + ηr ] z→0 2 r−1 X di − dp Bp (z + 1 + η r ; a; b) − , z p=0 −cr di [i
(2.23)
and the remainder term Rn,m (a) as a → ∞+ is given by Rn,m (a) =
O(a−ρi +1/2 + a−ηr −1/2 ), when − η r 6= −ρi + 1, . −ρi +1/2 O(a log a), when − η r = −ρi + 1.
(2.24)
ˆ and f (ω) = gˆ(−ω), we get the asymptotic Similarly by setting h(ω) = φ(−ω) − expansion of Wφ g (b, a), as a → ∞+ ,
34
A. Pathak, P. Yadav, M. M. Dixit
Z ∞ a1/2 ˆ e−ibω gˆ(−ω)φ(−aω)dω 2π 0 n−1 −ibω 1 X ˆ a ; 1 + ηr ] cr (−1)ηr M [φ(−ω)e 2π r=0
Wφ− g (b, a) = =
R(r)−1
X
−
p=0
(−1)−ρp dp Bp (1 + η r ; a; b)
× a−ηr −1/2 + I(i)−1
X
−
j=0
m−1 X i=0
g(−ω)e−ibω ; 1 − ρi ] (−1)−ρi di M [ˆ
(−1)ηj cj Aj (1 − ρi ; b)
× a−ρi +1/2 + Rn,m .
(2.25)
If ρi − η r = 1 for some pair (i, r) then in (2.25), the sum of the terms −ibω ηr ˆ a ; 1 + ηr ] cr (−1) M [φ(−ω)e R(r)−1
− +
X
−ρp
(−1)
p=0
dp Bp (1 + η r ; a; b) a−ηr −1/2
(2.26)
I(i)−1 X (−1)ηj cj Aj (1 − ρi ; b) a−ρi +1/2 , g(−ω)e−ibω ; 1 − ρi ] − (−1)−ρi di M [ˆ j=0
(2.27) must be replaced by a
−ρi +1/2
−ρi g (−ω)e−ibω ; 1 − ρi ] (−1) di M [ˆ
I(i)−1
−
X j=0
(−1)ηj cj Aj (1 − ρi ; b) − (−1)ηr cr loga
−ibω/a ˆ +(−1)ηr cr lim M [φ(−ω)e ; z + 1 + ηr ] z→0
−
r−1 X p=0
(−1)−ρp dp Bp (z + 1 + η r ; a; b) − (−1)−ρi
−(−1)ηr −ρi cr di [i
π − logb − γ] . 2
di z
(2.28)
35
Asymptotic Expansion of Continuous Wavelet Transform
Therefore by using (2.21) and (2.25) in (2.2), we get the required asymptotic expansion of wavelet transform for large a→ ∞+ is 1 (Wφ g) (b, a) = 2π
n−1 X r=0
R(r)−1
− +
X p=0 m−1 X i=0
−
j=0
1 η r −ρp −iπ(1+η r −ρp ) a−ηr − 2 dp Bp (1 + η r ; a; b) 1 + (−1) e
g(ω)eibω ; 1 − ρi ] + (−1)−ρi M [ˆ g (−ω)e−ibω ; 1 − ρi ] di M [ˆ
I(i)−1
X
ibω/a −ibω/a ˆ ˆ ; 1 + η r ] + (−1)ηr M [φ(−ω)e ; 1 + ηr ] cr M [φ(ω)e
1 η j −ρi −iπ(1+η j −ρi ) a−ρi + 2 cj Aj (1 − ρi ; b) 1 + (−1) e
+ Rn,m (a) .
(2.29)
If ρi − η r = 1 for some pair (i, r) then in (2.29), the corresponding sum of (2.22) and (2.26) must be replace by 1 g(ω)eibω ; 1 − ρi ] + (−1)−ρi M [ˆ g(−ω)e−ibω ; 1 − ρi ] a−ρi + 2 di M [ˆ I(i)−1
−
X j=0
cj Aj (1 − ρi ; b) 1 + (−1)ηj −ρi e−iπ(1+ηj −ρi )
ibω/a −ibω/a ˆ ˆ +cr lim M [φ(ω)e ; z + 1 + η r ] + (−1)ηr M [φ(−ω)e ; z + 1 + ηr ] z→0
−
r−1 X p=0
dp Bp (z + 1 + η r ; a; b) 1 + (−1)ηr −ρp e−iπ(z+1+η r −ρp ) .
(2.30) ✷
3. Application Using the aforesaid technique, we find the asymptotic expansions of Mexican Hat wavelet and Morlet wavelet transform. 3.1. Asymptotic Expansion of Mexican Hat Wavelet Transform −t2
We consider φ(t) = (1 − t2 ) e 2 to be a Mexican Hat wavelet.Since Fourier √ −ω2 ˆ transform of Mexican Hat φ(ω) = 2π ω2 e 2 is locally integrable in (−∞, ∞) ˆ as ω → ∞+ [1] and has an asymptotic expansion of φ(ω) ∞
X ¯ˆ φ(ω) = dr ω −2r+1 , r=0
(3.1)
36
A. Pathak, P. Yadav, M. M. Dixit
iπ
where, dr = ar e 2 (2r−1) Γ(2r − 1) and ar =
(−1)r+1 r (2r−1) . r! 2r−1
ˆ φ(ω) = O(1) as ω → 0 + .
(3.2)
Now, assume gˆ(ω) satisfies (2.5) and (2.7) with η 0 > 1. Then by using (2.29) and by means of formula [ [2],(10,30),pp.318,320], we get the asymptotic expansion of Mexican Hat wavelet transform at a → ∞+ is n−1 X
×
ηr √ 1√ 2 − 1 + (−1)ηr bi cr 21+ 2 π − a r=0 (3 + η r ) (4 + η r ) (4 + η r ) 3 b2 i2 Γ[ ]1F 1 , , 2 + 1 + (−1)ηr Γ[ ] 2 2 2 2a 2 R(r)−1 X iπ (3 + η r ) 1 b2 i2 − ap e 2 (2p−1) Γ[2p − 1] 1F 1 , , 2 2 2 2a p=0
×
m−1 X 1 Bp (1 + η r ; a; b) 1 − (−1)−ηr e−iπη r a−ηr − 2 + ai
×
iπ e 2 (2i−1) Γ[2i − 1] M [ˆ g(ω)eibω ; 2r] − M [ˆ g (−ω)e−ibω ; 2r]
Wφ g (b, a) =
1 2π
×
i=0
I(i)−1
X
−
j=0
2r− 12 η j −iπη j + Rn,m (a) , a cj Aj (2r; b) 1 − (−1) e (3.3)
where R(r) is the index r for which (1 − 2r) − η r ≤ 1 < (1 − 2r) − η r − 1 and I(i) is the index i for which −2r − η r < 1 ≤ (1 − 2r) − η r . If (−2r + 1) − η r = 1 for some pair (i, r) then, in (3.3), the corresponding sum of the terms must be replace by iπ 1 g (ω)eibω ; 2r] − M [ˆ g(−ω)e−ibω ; 2r] a2r− 2 ai e 2 (2i−1) Γ[2i − 1] M [ˆ I(i)−1
−
+ × ×
X
cj Aj (2r; b)
j=0
1 2
2+z+η r
1 − (−1)ηj e−iπηj
√ π 1 + (−1)ηr z→0 (3 + z + η r ) 1 b2 i2 1√ 1 Γ[ 2 − 1 + (−1)ηr 3 + z + ηr , , 2 − ] 1F 1 2 2 2 2a a 2 2 3 b i 1 (4 + z + η r ) 4 + z + ηr , , 2 ] 1F 1 bi Γ[ 2 2 2 2a
cr lim 2
37
Asymptotic Expansion of Continuous Wavelet Transform
−
r−1 X
+
m−1 X
ap e
p=0
i=0
iπ 2 (2p−1)
1 Γ[2p − 1] Bp (1 + η r ; a; b) 1 − (−1)−ηr e−iπηr a−ηr − 2
iπ ai e 2 (2i−1) Γ[2i − 1] M [ˆ g(ω)eibω ; 2r] − M [ˆ g(−ω)e−ibω ; 2r]
I(i)−1
−
X
1 1 − (−1)ηj e−iπηj a2r− 2 ,
cj Aj (2r; b)
j=0
(3.4)
and the remainder term Rn,m (a) as a → ∞+ is given by O(a2r− 12 + a−ηr − 12 ), when − η r 6= 2r, . Rn,m (a) = 2r− 21 O(a log a), when − η r = 2r.
(3.5)
3.2. Asymptotic Expansion of Morlet Wavelet Transform 2
t ˆ = In this application, we consider Morlet wavelet φ(t)= eiωo t− 2 . Since φ(ω)
√ 2π e is locally integrable in (−∞, ∞) and has an asymptotic expansion as ω → ∞+ [4]. −(ω−ω o )2 2
ˆ φ(ω) =
∞ X
iπ
ar e 2
(r+1)
Γ[r + 1]ω −(r+1) ,
(3.6)
r=0
where [r/2]
X (−1)j (iω o )r−2j , (2)j j!(r − 2j)! j=0
(3.7)
ˆ φ(ω) = O(1) as ω → 0 + .
(3.8)
ar =
and
Now, assume gˆ(ω) satisfies (2.5) and (2.7) with η 0 > 1. Hence, using (2.29) and formula [ [2],(11, 31), pp.318, 320], we get the following asymptotic expansion of
38
A. Pathak, P. Yadav, M. M. Dixit
Morlet wavelet transform for a → ∞+ as n−1 2 −wo ηr 1 X (1 + η r ) ηr 2 2 (Wφ g) (b, a) = Γ[ 1 + (−1) ] cr 2 e 2π r=0 2 1√ (1 + η r ) 1 (bi + awo )2 ηr − , , × 1F 1 2 − 1 + (−1) 2 2 2a2 a (2 + η r ) (2 + η r ) 3 (bi + awo )2 × Γ[ (bi + awo ) ] 1F 1 , , 2 2 2 2a2 m−1 X iπ (i+1) 2 ai e Γ[i + 1] M [ˆ g(ω)eibω ; −r] + i=0
− (−1)−r M [ˆ g(−ω)e−ibω ; −r] (3.9) I(i)−1 X −r− 12 −η j −r −iπ(η j −r) a cj Aj (−r; b) 1 − (−1) e − + Rn,m , j=0
(3.10) 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 iπ −r− 12 (i+1) 2 a ai e Γ[i + 1] M [ˆ g (ω)eibω ; −r] − (−1)−r M [ˆ g (−ω)e−ibω ; −r] I(i)−1
− + × × −
X j=0
−ηj −r −iπ(η j −r) cj Aj (−r; b) 1 − (−1) e
2 wo √ 1 1 1 + z + ηr cr lim 2 2 (z+ηr ) e− 2 π 1 + (−1)ηr Γ z→0 2 1 (bi + awo )2 1√ 1 ηr − 1 + (−1) 1F 1 1 + z + ηr , , − 2 2 2 2a2 a bi + awo 2 1 3 1 Γ 2 + z + η r 1F 1 2 + z + ηr , , bi + aw o 2 2 2 2a2 R(r)−1 X iπ (p+1) η r −p −iπ(z+η r −p) 2 ap e , Γ[p + 1]Bp (z + η r + 1; a; b) 1 − (−1) e p=0
(3.11) and the remainder term Rn,m (a) as a → ∞+ is given by
O(a−r− 12 + a−ηr − 21 ), when − η r 6= −r, Rn,m (a) = . 1 O(a−r− 2 log a), when − η r = −r.
(3.12)
Asymptotic Expansion of Continuous Wavelet Transform
39
Acknowledgments The author is thankful to the referees for their valuable comments and this research work is supported by U.G.C start-up grant. References 1. R.S.Pathak; Wavelet Transforms world scientific ,(2009). 2. Erde’lyi, A., W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms , Vol. 1. McGraw-Hill, New York (1954). 3. Wong, R., Explicit error terms for asymptotic expansion of Mellin convolutions , J. Math. Anal. Appl. 72,740-756.(1979) 4. Wong, R., Asymptotic Approximations of Integrals, Acad. Press, New York (1989). 5. Jose L. Lopez , Pedro Pagola, Asymptotic expansions of Mellin convolution integrals: An oscillatory case, Journal of Computational and Applied Mathematics 233 , 1562 –1569,(2010). 6. R S Pathak and Ashish Pathak, Asymptotic expansion of Wavelet Transform for small value a , The Wavelet Transforms, World scientific, 164-168, (2009). 7. R S Pathak and Ashish Pathak, Asymptotic expansion of Wavelet Transform with error term, The Wavelet Transforms ,World scientific,154-164, (2009). 8. R S Pathak and Ashish Pathak, Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b, International Journal of Mathematics and Mathematical Sciences, 13 page, doi:10.1155/2009/270492. (2009)
Ashish Pathak (Corresponding Author) Department of Mathematics Institute of Science Banaras Hindu University Varanasi-221005, India. E-mail address:
[email protected], pathak
[email protected] and Prabhat Yadav, M M Dixit Department of Mathematics NERIST, Nirjuli- 791109, India. E-mail address:
[email protected] 
