Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 270492, 13 pages
doi:10.1155/2009/270492
Research Article
Asymptotic Expansions of the Wavelet Transform
for Large and Small Values of b
R. S. Pathak and Ashish Pathak
Department of Mathematics, Banaras Hindu University, Varanasi 221 005, India
Correspondence should be addressed to R. S. Pathak, ramshankarpathak@yahoo.co.in
Received 27 July 2009; Accepted 9 September 2009
Recommended by A. Zayed
Asymptotic expansions of the wavelet transform for large and small values of the translation
parameter b 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 b when asymptotic expansions of
the function and the wavelet near origin are given.
Copyright q 2009 R. S. Pathak and A. 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.
1. Introduction
The wavelet transform of f with respect to the wavelet ψ is defined by
Wψ f b, a a−1/2
∞
−∞
ftψ
t−b
a
dt,
b ∈ R, a > 0,
1.1
provided that the integral exists 1. Using Fourier transform it can also be expressed as
√ ∞
a
ψaω
eibω fω
dω,
Wψ f b, a 2π −∞
1.2
where
fω
∞
−∞
e−ixω fx dx.
1.3
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Asymptotic expansion with explicit error term for Mellin convolution
Iλ ∞
fthλtdt,
1.4
0
as λ → ∞, was obtained by Wong 2, pages 740–756. Let us recall basic results from Wong
2, which will be used in the present investigation.
Assume that
ft ∼
∞
as tsα−1 ,
as t −→ 0,
s0
n−1
1.5
as tsα−1 fn t,
s0
where 0 < α ≤ 1 and
ht ∼ eict
∞
bs t−s−β ,
as t −→ ∞,
1.6
s0
where c is real and 0 < β ≤ 1.
Also assume that
ft O t−ρ1 ,
as t −→ ∞,
1.7
as t −→ 0,
1.8
where β ρ1 > 1.
ht Otρ2 ,
where α ρ2 > 0.
Asymptotic expansion of 1.4 is given by the following 2, Theorem 3, page 752.
Theorem 1.1. Assume that if n t is continuous on 0, ∞, where n is a nonnegative integer;
iift has an expansion of the form 1.5, and the expansion can be differentiated n times; iii as
t → ∞, f j t is Ot−1−η for j 0, 1, . . . , n and for some η > 0; ivfn t has the meaning as given
in 1.5; vht satisfies 1.8 and 1.6 with c / 0. Then we have
Iλ n−1
s0
as Mh; s αλ−s−α δn λ,
1.9
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3
where
Mh; s α ∞
tsα−1 htdt,
1.10
0
and the remainder satisfies
δn λ −1n
λn
∞
0
n
fn th−n λt dt.
1.11
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 λ:
∞
fteiλt dt 0
n−1
as eiπsα/2 Γs αλ−s−α s0
n ∞
i
n
fn teiλt dt.
λ
0
1.12
The asymptotic expansion of the wavelet transform 1.2 for large values of dilation
parameter a has already been obtained in 3.
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 b. In Section 2 we assume that fω
and ψω
possess asymptotic expansions of the form 1.5 as ω → 0 and derive asymptotic
expansion of Wψ fb, a as b → ∞ 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 fω
and ψω
are known as ω → ∞ and derive asymptotic
expansion of Wψ fb, a as b → 0, using Theorem 6.1 due to Wong 4, Theorem 14, page
323. In Section 7 we assume the asymptotic expansions of ft and ψt as t → 0 and
derive asymptotic expansions of fω
and ψω
as ω → ∞, using 1.12. These asymptotic
expansions of fω
and ψω
give rise to the asymptotic expansion of Wψ fb, a as b → 0,
using Theorem 6.1.
2. Asymptotic Expansion for Large b
Let us rewrite 1.2 in the following form:
√ ∞
∞
a
eibω ψaω
fωdω
e−ibω ψ−aω
f−ωdω
2π
0
0
Wψ f b, a Wψ− f b, a say ,
Wψ f b, a 2.1
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where for definiteness we take b ∈ R and a ∈ R . Now, we consider
Wψ f
√ ∞
a
eibω ψaω
fωdω.
b, a 2π 0
2.2
Assume that
ψω
∼
∞
as ωsα−1 ,
as ω −→ 0,
2.3
as ω −→ 0,
2.4
s0
then for arbitrary but fixed a ∈ R , we have
ψaω
∼
∞
as ωsα−1 ,
s0
where as as asα−1 .
Next, assume that
fω
∼
∞
br ωrβ−1 ,
as ω −→ 0.
2.5
r0
Then
ψaω
fω
∼
∞
as ωsα−1
s0
∞
br ωrβ−1 ωαβ−2
r0
∞
cr ω r ,
2.6
r0
where
cr a0 br · · · ar b0 r
am br−m m0
r
am amα−1 br−m .
2.7
m0
Now, for fixed a ∈ R , write
fω
∼
φω : ψaω
∞
cr ωrγ−1 ,
2.8
r0
where γ α β − 1.
Let us set
φω n−1
cr ωrγ−1 φn ω,
as ω −→ 0,
2.9
r0
and assume that i φn t is continuous on 0, ∞, where n is a nonnegative integer; ii
the expansion 2.9 can be differentiated n times; iii as ω → ∞, φj ω Oω−1−η for
j 0, 1, . . . , n and for some η > 0.
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5
Then, by 1.12, for 1 < α β ≤ 2,
∞
e
ibω
ψaω
fωdω
0
∞
eibω φωdω
0
n−1
cr eiπrγ/2 Γ r γ b−r−γ r0
n ∞
i
n
eibω φn ωdω.
b
0
2.10
Similarly, we get
∞
e−ibω ψ−aω
−
f−ωdω
0
n−1
cr eiπrγ/2 Γ r γ b−r−γ r0
n ∞
i
n
e−ibω φn −ωdω.
b
0
2.11
Notice that the series expansions in 2.10 and 2.11 are the same but opposite in sign.
Therefore, we find asymptotic expansion of Wψ fb, a only. From 2.2 and 2.10, we have
√ n ∞
iπ rαβ−1/2 −r−α−β1
a n−1
i
ibω n
Wψ f b, a cr Γ r α β − 1 e
b
e φn ωdω
2π r0
b
0
√ r
a n−1 am br−m aαm−1 Γ r α β − 1 b−r−α−β1
2π r0 m0
n ∞
i
iπrαβ−1/2
ibω n
×e
e φn ωdω .
b
0
2.12
3. Mexican Hat Wavelet Transform
In this section we choose ψ to be Mexican hat wavelet and derive asymptotic expansion of
the corresponding wavelet transform. The Mexican hat wavelet is defined by
2
ψt 1 − t2 e−t /2 ,
3.1
√
2
2πω2 e−ω /2 .
3.2
then from 1, page 372,
ψω
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Now, in view of 2.5, we have
ψaω
φω : fω
∼
∞
√
2 2
br ωrβ−1 2πa2 ω2 e−a ω /2
r0
√
2
2πa
∞
br ω
rβ−1
ω
r0
√
2
2πa ω
2
2πa ω
∞
a2
−
2
s0
β1
∞
∞ br
β1
s
a2
−
2
∞ r/2
br−2j
s
ω2s
s!
ω2sr
s!
a2
−
2
j!
r0 j0
∞
r0 s0
√
2
3.3
j
ωr
cr ωrβ1 ,
r0
where
cr √
2
2πa
r/2
j0
br−2j
j!
a2
−
2
j
,
3.4
where r/2 stands for the greatest positive integer ≤ r/2. To ensure that Wψ fb, a exists
2
for large values of b we also impose the condition that fu
Oeσu for some real number
σ > 0 as u → ∞. Also, from 3.3 and 2.8 we conclude that in the present case α 3.
Therefore, from 2.12, using 3.4 we get
⎧
j
√ ⎨
n−1 r/2
br−2j a2
a √
2
−
Γ r β 2 b−r−β−2
2πa
Wψ f b, a 2π ⎩
2
j!
r0 j0
n ∞
i
n
iπrβ2/2
×e
eibω φn ωdω .
b
0
3.5
4. Morlet Wavelet Transform
In this section we choose
2
4.1
ψt eiω0 t−t /2 .
Then from 1, page 373,
ψω
√
2
2πe−ω−ω0 /2 .
4.2
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Now,
ψaω
√
√
2πe−ωo /2 eaω0 ω e−a ω
2
2πe
2
−ωo2 /2
√
2πe−ωo /2
2
r!
∞
∞ 2πe
∞
−ωo2 /2
−
∞ r/2
a2
−
2
s0
r0 s0
√
/2
∞
∞
aω0 ωr r0
2
a2
2
s
r0 j0
a2
−
2
s
s
ω2s
s!
aω0 r ωr2s
r!s!
s
4.3
aω0 r−2j ωr
j! r − 2j !
Ar ω r ,
r0
where
Ar √
2πe
−ωo2 /2
r/2
j0
a2
−
2
aω0 r−2j
.
j! r − 2j !
4.4
Hence
ψaω
φω : fω
∼
∞
br ωrβ−1
∞
r0
Ap ω p
p0
ωβ−1
∞
br ωr
r0
∞
Ap ω p
p0
⎛
⎞
∞
r
⎝ Ap br−p ⎠ωrβ−1
r0
∞
4.5
p0
cr ωrβ−1 ,
r0
where
cr r
p0
Ap br−p .
4.6
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Also, from 2.8 and 4.5 it follows that α 1. Therefore, from 2.12, using 4.6 we get
Wψ f
⎧
√ ⎨
r
a n−1 Ap br−p Γ r β b−r−β eiπrβ/2
b, a 2π ⎩ r0 p0
n ∞
i
n
ibω
φn ωe dω .
b
0
4.7
5. Haar Wavelet Transform
The Haar wavelet is defined by
⎧
⎪
1,
⎪
⎪
⎨
ψt −1,
⎪
⎪
⎪
⎩
0,
0 ≤ t < 1/2,
1/2 ≤ t < 1,
5.1
otherwise,
whose Fourier transform 1, page 368 is
i 2e−iω/2 − 1 − e−iω
.
ψω
ω
5.2
Therefore, Haar wavelet transform on half-line is given by
∞
1 eiaω 2eiaω/2
i
ibω Wψ f b, a √
−
dω
e fω
ω
ω
ω
a2π 0
∞
∞
∞
i
ibω fω
ibωa fω
ibωa/2 fω
√
dω dω − 2 e
dω .
e
e
ω
ω
ω
a2π
0
0
0
5.3
For fω
possessing asymptotic behavior 2.5, we have
∞
fω
∼
ar ωrβ−2 .
ω
r0
5.4
Then, from 5.3 and 5.4 using formula 2, page 753
M eit ; z eizπ/2 Γz,
5.5
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we get, for β ≥ 1 and b → ∞,
Wψ f b, a
√
∞
i ar Γ r β − 1 eiπrβ−1/2 × b−r−β1 b a−r−β1 − 2b a/2−r−β1
a2π r0
∞
i ar Γ r β − 1 eiπrβ−1/2
a2π r0
∞
∞
−r − β 1 −s −r−β−s1
−r − β 1 a −s −r−β−s1
−r−β1
a b
.
× b
−2
b
2
s
s
s0
s0
√
5.6
6. Asymptotic Expansion for Small b
In this section we assume that asymptotic expansions of fω
and ψω
as ω → ∞ are known
and then derive asymptotic expansion of Wψ fb, a as b → 0 for fixed a > 0, using the
following 4, Therorem 14, page 323.
Theorem 6.1. Let f be a locally integrable function on 0, ∞ and let ft possess an asymptotic
expansion of the form
ft ∼
n−1
a∗s t−s−α fn t,
as t −→ ∞,
6.1
s0
where 0 < α < 1. Then for small values of ω,
∞
0
fteitω dt e−iαπ/2
n−1
−is−1 a∗s Γ1 − s − αωsα−1 −
s0
n
cs∗ −iωs−1 Rn ω,
6.2
s1
where
−1s
M f; s ,
s − 1!
∞
Rn ω −iωn eiωt fn,n tdt
cs∗ 6.3
0
with
−1n
fn,n t n − 1!
∞
t
τ − tn−1 fn τdτ.
6.4
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Let
fω
∼
∞
bs ω−s−α
as ω −→ ∞,
s0
ψω
∼
∞
6.5
as ω
−r−β
as ω −→ ∞.
r0
Then, writing ar a−r−α ar , we have
ψaω
fω
∼
∞
ar ω−r−α
r0
∞
bs ω−s−β ω−α−β
∞
cr ω−r ,
6.6
am a−m−α br−m .
6.7
s0
r0
where
cr a0 br · · · ar b0 r
am br−m m0
r
m0
Now, for fixed a ∈ R , write
fω
φω : ψaω
∼
∞
cr ω−r−α−β ,
ω −→ ∞
6.8
r0
n−1
cr ω−r−α−β φn ω,
r0
where 0 < α β < 1.
Then, using 2.2, 6.2, and 6.8, we find asymptotic expansion of wavelet transform
for small value of b:
n−1
n
Ds∗ −ibs−1 R∗n b,
Wψ f b, a e−iπ αβ/2
−is−1 cs Γ 1 − s − α − β bsαβ−1 −
s0
s1
6.9
where
−1s
M φ; s ,
s − 1!
∞
n
∗
Rn ω −ib
eibt φn,n tdt
Ds∗ 0
6.10
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with
φn,n t ∞
−1n
n − 1!
τ − tn−1 φn τdτ.
6.11
t
7. Asymptotic Expansion for Small b Continued
In this section we assume that asymptotic expansions of f and ψ are known, instead of fω
and ψω
as in previous sections. Then as in 2, page 753 we get asymptotic expansions of
fω
and ψω
as ω → ∞. On the other hand, in 2.3 and 2.5 their behaviors near the origin
were known, that yielded the asymptotic expansion of Wψ fb, a as b → ∞. However, in
this case, following 4, pages 321–323 we can obtain asymptotic expansion of Wψ fb, aas
b → 0.
Let
ft ∼
∞
bs tsα −1 ,
as t −→ 0, 0 < α < 1,
s0
ψt ∼
∞
7.1
ar trβ −1 ,
t −→ 0.
r0
Now, using 1.12
∞
fω
e−itω ftdt
0
∞
∼
bs e−iπsα /2 Γ
sα ω
7.2
−s−α
,
as ω −→ ∞.
s0
Similarly,
∞
ψω
e−itω ψtdt
0
∞
∼
ar e−iπ rβ /2 Γ
rβ ω
7.3
−r−β
as ω −→ ∞.
r0
Then
ψaω
φω : fω
∼
∞
dr ω−r−α −β ,
7.4
r0
where
dr e−iπ −rα −β r
m0
−r−m−β bm
a
ar−m e−iπ2m Γ α m Γ β r − m .
7.5
12
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Let us set
φω n−1
dr ω−r−α −β φn ω,
7.6
r0
where 0 < α β < 1.
Assume that φω fωψaω
integrable is locally on 0, ∞. Then applying 6.2 to
2.2 with φω given by 7.6, finally we get
n
n−1
Wψ f b, a e−iπα β /2
Ds −ibs−1 R∗n b,
−is−1 ds Γ 1 − s − α − β bsα β −1 −
s0
s1
7.7
where
−1s
M φ ;s ,
s − 1!
∞
n
∗
Rn b −ib
eibt φn,n tdt
Ds 7.8
0
with
−1n
φn,n t n − 1!
∞
τ − tn−1 φn τdτ.
7.9
t
Remark 7.1. We observe that if we assume the asymptotic expansions ft and ψt as t → ∞
and derive asymptotic expansions of fω
and ψt
as ω → ∞ , then formula 1.2 gives
asymptotic expansion of Wψ fb, a as b → 0 for fixed a > 0. The aforesaid technique
does not yield asymptotic expansion of Wψ fb, a as b → ∞ 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 5, asymptotic expansion of Wψ fb, a as b → ∞
can be obtained. This gives rise to a complicated form of the asymptotic expansion and needs
separate treatment 6.
Acknowledgments
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.
References
1 L. Debnath, Wavelet Transforms and Their Applications, Birkhäuser, Boston, Mass, USA, 2002.
2 R. Wong, “Explicit error terms for asymptotic expansions of Mellin convolutions,” Journal of
Mathematical Analysis and Applications, vol. 72, no. 2, pp. 740–756, 1979.
International Journal of Mathematics and Mathematical Sciences
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3 R. S. Pathak and A. Pathak, “Asymptotic expansion for the wavelet transform with error term,”
communicated.
4 R. Wong, Asymptotic Approximations of Integrals, Computer Science and Scientific Computing,
Academic Press, Boston, Mass, USA, 1989.
5 X. Li and R. Wong, “Error bounds for asymptotic expansions of Laplace convolutions,” SIAM Journal
on Mathematical Analysis, vol. 25, no. 6, pp. 1537–1553, 1994.
6 R. S. Pathak, “Asymptotic expansion of the wavelet transform,” in Industrial and Applied Mathematics,
A. H. Siddiqui and K. Ahmad, Eds., pp. 43–50, Narosa, New Delhi, India, 1998.