2. Asymptotic Expansion for Large b

Volume 2009, Article ID 270492,13pages 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,[email protected] 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 parameterbare 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 ofbwhen asymptotic expansions of the function and the wavelet near origin are given.

Copyrightq2009 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 offwith respect to the waveletψis defined by Wψf

b, a a^−1/2 _∞

−∞ftψ t−b

a

dt, b∈R, a >0, 1.1

provided that the integral exists1. Using Fourier transform it can also be expressed as Wψf

b, a

√a 2π

_∞

−∞e^ibωfωψaω dω, 1.2

where

fω _∞

−∞e^−ixωfxdx. 1.3

Asymptotic expansion with explicit error term for Mellin convolution

Iλ _∞

0

fthλtdt, 1.4

asλ → ∞, was obtained by Wong2, pages 740–756. Let us recall basic results from Wong 2, which will be used in the present investigation.

Assume that

ft∼^∞

s0

ast^sα−1, ast−→0,

ⁿ⁻¹

s0ast^sα−1fnt,

1.5

where 0< α≤1 and

ht∼e^ict ∞ s0

bst^−s−β, ast−→∞, 1.6

wherecis real and 0< β≤1.

Also assume that

ft O t^−ρ1

, ast−→∞, 1.7

whereβρ1>1.

ht Ot^ρ2, ast−→0, 1.8

whereαρ2>0.

Asymptotic expansion of1.4is given by the following2, Theorem 3, page 752.

Theorem 1.1. Assume that ifⁿt is continuous on0,∞,where nis a nonnegative integer;

iifthas an expansion of the form1.5, and the expansion can be differentiatedntimes;iiias t → ∞, f^jtisOt^−1−ηforj 0,1, . . . , nand for someη >0;ivfnthas the meaning as given in1.5;vhtsatisfies1.8and1.6withc /0. Then we have

Iλ ⁿ⁻¹

s0

asMh;sαλ^−s−αδnλ, 1.9

where

Mh;sα _∞

0

t^sα−1htdt, 1.10

and the remainder satisfies

δnλ −1ⁿ λⁿ

_∞

0

fnⁿth⁻ⁿλtdt. 1.11

As an application of the above theorem, Wong2, page 753has derived the following asymptotic expansion for the Fourier transform for large values ofλ:

_∞

0

fte^iλtdtⁿ⁻¹

s0

ase^iπsα/2Γsαλ^−s−α i

λ _n∞

0

fnⁿte^iλtdt. 1.12

The asymptotic expansion of the wavelet transform 1.2 for large values of dilation parameterahas already been obtained in3.

The aim of the present paper is to derive asymptotic expansion of the wavelet transform given by1.2for large and small values ofb. InSection 2we assume thatfω andψω possess asymptotic expansions of the form1.5asω → 0and derive asymptotic expansion ofW_ψfb, aasb → ∞using formula1.12. Asymptotic expansions of certain special forms of the wavelet transform are obtained in Sections3–5. InSection 6we assume that asymptotic expansions offω andψω are known asω → ∞and derive asymptotic expansion ofW_ψfb, aasb → 0,usingTheorem 6.1due to Wong4, Theorem 14, page 323. In Section 7 we assume the asymptotic expansions of ft and ψt ast → 0and derive asymptotic expansions offω andψω asω → ∞, using1.12. These asymptotic expansions offω andψωgive rise to the asymptotic expansion ofW_ψfb, aasb → 0, usingTheorem 6.1.

2. Asymptotic Expansion for Large b

Let us rewrite1.2in the following form:

Wψf b, a

√a 2π

∞ 0

e^ibωψaω fωdω _∞

0

e^−ibωψ−aω f−ωdω

W_ψf

b, a W_ψ⁻f

b, a say

,

2.1

where for definiteness we takeb∈Randa∈R. Now, we consider

W_ψf b, a

√a 2π

_∞

0

e^ibωψaω fωdω. 2.2

Assume that

ψω∼^∞

s0asω^sα−1, asω−→0, 2.3

then for arbitrary but fixeda∈R, we have

ψaω∼^∞

s0

a_sω^sα−1, asω−→0, 2.4

wherea_s asa^sα−1. Next, assume that

fω ∼^∞

r0

brω^rβ−1, as ω−→0. 2.5

Then

ψaωfω ∼^∞

s0

a_s ω^sα−1 ∞ r0

brω^rβ−1 ω^αβ−2 ∞ r0

crω^r, 2.6

where

cr a₀br· · ·a_rb0^r

m0

a_mb_r−m^r

m0

ama^mα−1b_r−m. 2.7

Now, for fixeda∈R, write

φω:ψaω fω ∼^∞

r0

crω^rγ−1, 2.8

whereγαβ−1.

Let us set

φω ⁿ⁻¹

r0

crω^rγ−1φnω, asω−→0, 2.9

and assume that i φⁿt is continuous on 0,∞, where nis a nonnegative integer; ii the expansion2.9can be differentiated ntimes;iii asω → ∞, φ^jω Oω^−1−η for j 0,1, . . . , nand for someη >0.

Then, by1.12, for 1< αβ≤2, _∞

0

e^ibωψaω fωdω _∞

0

e^ibωφωdω

ⁿ⁻¹

r0

cre^iπrγ/2Γ rγ

b^−r−γ i

b _n∞

0

e^ibωφⁿn ωdω.

2.10

Similarly, we get

_∞

0

e^−ibωψ−aω f−ωdω −ⁿ⁻¹

r0

cre^iπrγ/2Γ rγ

b^−r−γ i

b _n∞

0

e^−ibωφⁿn −ωdω.

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 ofW_ψfb, aonly. From2.2and2.10, we have

W_ψf

b, a

√a 2π

_n−1

r0crΓ

rαβ−1

e^{iπrαβ−1/2}b^{−r−α−β1} i

b _n∞

0

e^ibωφⁿn ωdω

√a 2π

_n−1

r0

r m0

ambr−ma^αm−1Γ

rαβ−1

b^{−r−α−β1}

×e^{iπrαβ−1/2} i

b _n∞

0

e^ibωφⁿn ωdω

.

2.12

3. Mexican Hat Wavelet Transform

In this section we chooseψ to be Mexican hat wavelet and derive asymptotic expansion of the corresponding wavelet transform. The Mexican hat wavelet is defined by

ψt

1−t²

e^−t2/2, 3.1

then from1, page 372,

ψω √

2πω²e^−ω2/2. 3.2

Now, in view of2.5, we have

φω:fωψaω

∼^∞

r0

brω^rβ−1√

2πa²ω²e^−a2ω2/2

√ 2πa²

∞ r0

brω^rβ−1ω² ∞ s0

−a² 2

s

ω^2s s!

√

2πa²ω^β1 ∞ r0

∞ s0

br

−a² 2

s

ω^2sr s!

√

2πa²ω^β1 ∞ r0

r/2

j0

b_r−2j j!

−a² 2

_j ω^r

^∞

r0

crω^rβ1,

3.3

where

cr √ 2πa²

r/2

j0

b_r−2j j!

−a² 2

_j

, 3.4

wherer/2stands for the greatest positive integer≤r/2. To ensure thatW_ψfb, aexists for large values ofbwe also impose the condition thatfu Oe^σu2for some real number σ > 0 as u → ∞. Also, from 3.3and 2.8 we conclude that in the present caseα 3.

Therefore, from2.12, using3.4we get

W_ψf b, a

√a 2π

⎧⎨

⎩

√2πa² n−1 r0

r/2

j0

−a² 2

_j br−2j

j! Γ

rβ2 b^{−r−β−2}

×e^iπrβ2/2 i

b n_∞

0

e^ibωφⁿn ωdω

.

3.5

4. Morlet Wavelet Transform

In this section we choose

ψt e^iω0t−t2/2. 4.1

Then from1, page 373,

ψω √

2πe^{−ω−ω02/2}. 4.2

Now,

ψaω √

2πe^−ω2o/2e^aω0ωe^−a2ω2/2 √

2πe^−ω2o/2 ∞ r0

aω0ω^r r!

∞ s0

−a² 2

_s ω^2s

s!

√

2πe^−ω2o/2 ∞ r0

∞ s0

−a² 2

_s

aω0^rω^r2s r!s!

√

2πe^−ω2o/2 ∞ r0

r/2

j0

−a² 2

_s

aω0^r−2jω^r j!

r−2j

! ^∞

r0

Arω^r,

4.3

where

Ar √

2πe^−ω2o/2

r/2

j0

−a² 2

_s

aω0^r−2j j!

r−2j

!. 4.4

Hence

φω:fωψaω

∼^∞

r0

brω^rβ−1 ∞ p0

Apω^p

ω^β−1 ∞ r0

brω^r ∞ p0

Apω^p

^∞

r0

⎛

⎝^r

p0

Apbr−p

⎞

⎠ω^rβ−1

^∞

r0

crω^rβ−1,

4.5

where

cr ^r

p0

Apb_r−p. 4.6

Also, from2.8and4.5it follows thatα1. Therefore, from2.12, using4.6we get

W_ψf

b, a

√a 2π

⎧⎨

⎩

n−1

r0

r p0

Apbr−pΓ rβ

b^−r−βe^iπrβ/2

i

b _n∞

0

φnⁿωe^ibωdω

.

4.7

5. Haar Wavelet Transform

The Haar wavelet is defined by

ψt

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1, 0≤t <1/2,

−1, 1/2≤t<1, 0, otherwise,

5.1

whose Fourier transform1, page 368is

ψω i

2e^−iω/2−1−e^−iω

ω . 5.2

Therefore, Haar wavelet transform on half-line is given by W_ψf

b, a i

√a2π _∞

0

e^ibωfω 1

ω e^iaω

ω − 2e^iaω/2 ω

dω

√i a2π

_∞

0

e^ibωfω ω dω

_∞

0

e^ibωafω ω dω−2

_∞

0

e^ibωa/2fω ω dω

. 5.3

Forfω possessing asymptotic behavior2.5, we have fω

ω ∼^∞

r0

arω^rβ−2. 5.4

Then, from5.3and5.4using formula2, page 753

M e^it;z

e^izπ/2Γz, 5.5

we get, forβ≥1 andb → ∞,

W_ψf b, a √ i

a2π ∞ r0

arΓ

rβ−1

e^iπrβ−1/2×

b^−r−β1 ba^−r−β1−2ba/2^−r−β1

i

√a2π ∞ r0

arΓ

rβ−1

e^iπrβ−1/2

×

b^{−r−β1∞}

s0

−r−β1 s

a^−sb^{−r−β−s1}−2 ∞ s0

−r−β1 s

a 2

_−s

b^{−r−β−s1}

. 5.6

6. Asymptotic Expansion for Small b

In this section we assume that asymptotic expansions offω andψω asω → ∞are known and then derive asymptotic expansion ofW_ψfb, aasb → 0for fixeda > 0, using the following4, Therorem 14, page 323.

Theorem 6.1. Letf be a locally integrable function on0,∞and letftpossess an asymptotic expansion of the form

ft∼ⁿ⁻¹

s0

a^∗_st^−s−αfnt, ast−→ ∞, 6.1

where 0< α <1. Then for small values ofω, _∞

0

fte^itωdte^−iαπ/2

n−1

s0

−i^s−1a^∗_sΓ1−s−αω^sα−1−ⁿ

s1

c^∗_s−iω^s−1Rnω, 6.2

where

c^∗_s −1^s s−1!M

f;s , Rnω −iωⁿ

_∞

0

e^iωtfn,ntdt

6.3

with

fn,nt −1ⁿ n−1!

_∞

t

τ−tⁿ⁻¹fnτdτ. 6.4

Let

fω∼^∞

s0

bsω^−s−α asω−→ ∞,

ψω∼^∞

r0

asω^−r−β as ω−→ ∞.

6.5

Then, writinga_r a^−r−α ar, we have

ψaωfω ∼^∞

r0

a_rω^−r−α ∞ s0

bsω^−s−βω^−α−β ∞ r0

crω^−r, 6.6

where

cr a₀br· · ·a_rb0^r

m0

a_mbr−m^r

m0

ama^−m−αbr−m. 6.7

Now, for fixeda∈R, write

φω:ψaω fω

∼^∞

r0

crω^{−r−α−β}, ω−→ ∞

ⁿ⁻¹

r0

crω^{−r−α−β}φnω,

6.8

where 0< αβ <1.

Then, using2.2,6.2, and6.8, we find asymptotic expansion of wavelet transform for small value ofb:

W_ψf

b, a e^{−iπαβ/2n−1}

s0

−i^s−1csΓ

1−s−α−β

b^sαβ−1−ⁿ

s1

D^∗_s−ib^s−1R^∗_nb, 6.9

where

D_s^∗ −1^s s−1!M

φ;s , R^∗_nω −ibⁿ

_∞

0

e^ibtφn,ntdt

6.10

with

φn,nt −1ⁿ n−1!

_∞

t

τ−tⁿ⁻¹φnτdτ. 6.11

7. Asymptotic Expansion for Small b Continued

In this section we assume that asymptotic expansions offandψare known, instead offω andψω as in previous sections. Then as in2, page 753we get asymptotic expansions of fω andψω asω → ∞. On the other hand, in2.3and2.5their behaviors near the origin were known, that yielded the asymptotic expansion ofW_ψfb, aasb → ∞. However, in this case, following4, pages 321–323we can obtain asymptotic expansion ofW_ψfb, aas b → 0.

Let

ft∼^∞

s0b_st^sα−1, as t−→0, 0< α<1, ψt∼^∞

r0

a_rt^rβ−1, t−→0.

7.1

Now, using1.12

fω _∞

0

e^−itωftdt

∼^∞

s0

b_se^−iπsα/2Γ sα

ω^−s−α, asω−→∞.

7.2

Similarly,

ψω

_∞

0

e^−itωψtdt

∼^∞

r0

a_re^−iπrβ/2Γ rβ

ω^−r−β as ω−→∞.

7.3

Then

φω:fωψaω ∼^∞

r0

drω^{−r−α−β}, 7.4

where

dr e^{−iπ−rα−βr}

m0

b_ma^{−r−m−β}a_r−me^−iπ2mΓ αm

Γ

βr−m

. 7.5

Let us set

φω ⁿ⁻¹

r0

drω^{−r−α−β}φnω, 7.6

where 0< αβ<1.

Assume thatφω fωψaω integrable is locally on0,∞. Then applying6.2to 2.2withφωgiven by7.6, finally we get

W_ψf

b, a e^−iπαβ/2

n−1

s0

−i^s−1dsΓ

1−s−α−β

b^sαβ−1−ⁿ

s1

Ds−ib^s−1R^∗_nb, 7.7

where

Ds −1^s s−1!M

φ ;s , R^∗_nb −ibⁿ

_∞

0

e^ibtφn,ntdt

7.8

with

φn,nt −1ⁿ n−1!

_∞

t

τ−tⁿ⁻¹φnτdτ. 7.9

Remark 7.1. We observe that if we assume the asymptotic expansionsftandψtast → ∞ and derive asymptotic expansions offω andψt asω → ∞, then formula1.2gives asymptotic expansion ofW_ψfb, aas b → 0 for fixed a > 0. The aforesaid technique does not yield asymptotic expansion ofW_ψfb, aasb → ∞using1.2. However, if one uses the form1.1of the wavelet transform and applies Li and Wong-technique involving a theory of noncommutative convolution5, asymptotic expansion ofW_ψfb, aasb → ∞ can be obtained. This gives rise to a complicated form of the asymptotic expansion and needs separate treatment6.

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, Birkh¨auser, 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.

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.