Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b

Abstract

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.

1. Introduction

The wavelet transform of 𝑓 with respect to the wavelet ψ is defined by

(𝑊_𝜓𝑓)(𝑏,𝑎)=𝑎^−1/2𝑓(𝑡)𝜓(𝑡−𝑏𝑎)𝑑𝑡,𝑏∈𝐑,𝑎>0, (1.1) provided that the integral exists [1]. Using Fourier transform it can also be expressed as

(𝑊_𝜓√𝑓)(𝑏,𝑎)=𝑎𝑒2𝜋^𝑖𝑏𝜔𝑓(𝜔)𝜓(𝑎𝜔)𝑑𝜔, (1.2) where

𝑓(𝜔)=𝑒^{−𝑖𝑥𝜔}𝑓(𝑥)𝑑𝑥. (1.3)

Asymptotic expansion with explicit error term for Mellin convolution

𝐼(𝜆)=𝑓(𝑡)ℎ(𝜆𝑡)𝑑𝑡, (1.4) 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

𝑓(𝑡)∼^∞_𝑠=0𝑎_𝑠𝑡^{𝑠+𝛼−1},as=𝑡→0,_{𝑛−1𝑠=0}𝑎_𝑠𝑡^{𝑠+𝛼−1}+𝑓_𝑛(𝑡), (1.5) where 0<𝛼≤1 and

ℎ(𝑡)∼𝑒^ict∞_s=0𝑏_𝑠𝑡^{−𝑠−𝛽},as𝑡→+∞, (1.6) where 𝑐 is real and 0<𝛽≤1.

Also assume that

𝑓(𝑡)=𝑂(𝑡^−𝜌1),as𝑡→+∞, (1.7) where 𝛽+𝜌₁>1.

ℎ(𝑡)=𝑂(𝑡^𝜌2),as𝑡→0, (1.8) where 𝛼+𝜌₂>0.

Asymptotic expansion of (1.4) is given by the following [2, Theorem  3, page 752].

Theorem 1.1. Assume that (i)𝑓^(𝑛)(𝑡) is continuous on (0,∞), where 𝑛 is a nonnegative integer; (ii)𝑓(𝑡) has an expansion of the form (1.5), and the expansion can be differentiated 𝑛 times; (iii) as 𝑡→∞,𝑓^(𝑗)(𝑡) is 𝑂(𝑡^−1−𝜂) for 𝑗=0,1,…,𝑛 and for some 𝜂>0; (iv)𝑓_𝑛(𝑡) has the meaning as given in (1.5); (v)ℎ(𝑡) satisfies (1.8) and (1.6) with 𝑐≠0. Then we have 𝐼(𝜆)=^𝑛−1_𝑠=0𝑎_𝑠𝑀[ℎ;𝑠+𝛼]𝜆^{−𝑠−𝛼}+𝛿_𝑛(𝜆), (1.9) where 𝑀[ℎ;𝑠+𝛼]=𝑡^{𝑠+𝛼−1}ℎ(𝑡)𝑑𝑡, (1.10) and the remainder satisfies 𝛿_𝑛(𝜆)=(−1)^𝑛𝜆^𝑛(𝑡)h⁽⁻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: 𝑓(𝑡)𝑒^𝑖𝜆𝑡𝑑𝑡=^𝑛−1_𝑠=0𝑎_𝑠𝑒^{𝑖𝜋(𝑠+𝛼)/2}Γ(𝑠+𝛼)𝜆^{−𝑠−𝛼}𝑖+(𝜆)^𝑛(𝑡)𝑒^𝑖𝜆𝑡𝑑𝑡. (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 𝑓(𝜔) and 𝜓(𝜔) possess asymptotic expansions of the form (1.5) as 𝜔→0+ and derive asymptotic expansion of (𝑓)(𝑏,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 𝑓(𝜔) and 𝜓(𝜔) are known as 𝜔→∞ and derive asymptotic expansion of (𝑓)(𝑏,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 𝑓(𝑡) and 𝜓(𝑡) as 𝑡→0+ and derive asymptotic expansions of 𝑓(𝜔) and 𝜓(𝜔) as 𝜔→∞, using (1.12). These asymptotic expansions of 𝑓(𝜔) and 𝜓(𝜔) give rise to the asymptotic expansion of (𝑓)(𝑏,𝑎) as 𝑏→0+, using Theorem 6.1.

2. Asymptotic Expansion for Large 𝑏

Let us rewrite (1.2) in the following form:

(𝑊_𝜓√𝑓)(𝑏,𝑎)=𝑎2𝜋{𝑒^𝑖𝑏𝜔𝜓(𝑎𝜔)𝑓(𝜔)𝑑𝜔+𝑒^{−𝑖𝑏𝜔}𝜓(−𝑎𝜔)𝑓(−𝜔)𝑑𝜔}=(𝑓)(𝑏,𝑎)+(𝑓)(𝑏,𝑎)(say), (2.1) where for definiteness we take 𝑏∈𝐑₊ and 𝑎∈𝐑₊. Now, we consider √(𝑓)(𝑏,𝑎)=𝑎𝑒2𝜋^𝑖𝑏𝜔𝜓(𝑎𝜔)𝑓(𝜔)𝑑𝜔. (2.2) Assume that

𝜓(𝜔)∼^∞_𝑠=0𝑎_𝑠𝜔^{𝑠+𝛼−1},as𝜔→0, (2.3) then for arbitrary but fixed 𝑎∈𝐑₊, we have

𝜓(𝑎𝜔)∼^∞_𝑠=0𝜔^{𝑠+𝛼−1},as𝜔→0, (2.4) where =𝑎_𝑠𝑎^{𝑠+𝛼−1}.

Next, assume that

𝑓(𝜔)∼^∞_𝑟=0𝑏_𝑟𝜔^{𝑟+𝛽−1},as𝜔→0. (2.5) Then

𝜓(𝑎𝜔)𝑓(𝜔)∼^∞_𝑠=0𝜔^{∞𝑠+𝛼−1}_𝑟=0𝑏_𝑟𝜔^{𝑟+𝛽−1}=𝜔^{∞𝛼+𝛽−2}_𝑟=0𝑐_𝑟𝜔^𝑟, (2.6) where 𝑐_𝑟=𝑏_𝑟+⋯+𝑏₀=^𝑟_𝑚=0𝑏_𝑟−𝑚=^𝑟_𝑚=0𝑎_𝑚𝑎^{𝑚+𝛼−1}𝑏_𝑟−𝑚. (2.7) Now, for fixed 𝑎∈𝐑₊, write

𝜙(𝜔)∶=𝜓(𝑎𝜔)𝑓(𝜔)∼^∞_𝑟=0𝑐_𝑟𝜔^{𝑟+𝛾−1}, (2.8) where 𝛾=𝛼+𝛽−1.

Let us set

𝜙(𝜔)=^𝑛−1_𝑟=0𝑐_𝑟𝜔^{𝑟+𝛾−1}+𝜙_𝑛(𝜔),as𝜔→0, (2.9) and assume that (i) 𝜙^(𝑛)(𝑡) is continuous on (0,∞), where 𝑛 is a nonnegative integer; (ii) the expansion (2.9) can be differentiated 𝑛 times; (iii) as 𝜔→∞,𝜙^(𝑗)(𝜔)=𝑂(𝜔^−1−𝜂) for 𝑗=0,1,…,𝑛 and for some 𝜂>0.

Then, by (1.12), for 1<𝛼+𝛽≤2,

𝑒^𝑖𝑏𝜔𝜓(𝑎𝜔)𝑓(𝜔)𝑑𝜔=𝑒^𝑖𝑏𝜔=𝜙(𝜔)𝑑𝜔_{𝑛−1𝑟=0}𝑐_𝑟𝑒^{𝑖𝜋(𝑟+𝛾)/2}Γ(𝑟+𝛾)𝑏^{−𝑟−𝛾}𝑖+(𝑏)^𝑛𝑒^𝑖𝑏𝜔(𝜔)𝑑𝜔. (2.10) Similarly, we get

𝑒^{−𝑖𝑏𝜔}𝜓(−𝑎𝜔)𝑓(−𝜔)𝑑𝜔=−^𝑛−1_𝑟=0𝑐_𝑟𝑒^{𝑖𝜋(𝑟+𝛾)/2}Γ(𝑟+𝛾)𝑏^{−𝑟−𝛾}𝑖+(𝑏)^𝑛𝑒^{−𝑖𝑏𝜔}(−𝜔)𝑑𝜔. (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 (𝑓)(𝑏,𝑎) only. From (2.2) and (2.10), we have √(𝑓)(𝑏,𝑎)=𝑎{2𝜋^𝑛−1_𝑟=0𝑐_𝑟Γ(𝑟+𝛼+𝛽−1)𝑒^{𝑖𝜋(𝑟+𝛼+𝛽−1)/2}𝑏^{−𝑟−𝛼−𝛽+1}𝑖+(𝑏)^𝑛𝑒^𝑖𝑏𝜔=√(𝜔)𝑑𝜔}𝑎{2𝜋^𝑛−1𝑟_𝑟=0𝑚=0𝑎_𝑚𝑏_𝑟−𝑚𝑎^{𝛼+𝑚−1}Γ(𝑟+𝛼+𝛽−1)𝑏^{−𝑟−𝛼−𝛽+1}×𝑒^{𝑖𝜋(𝑟+𝛼+𝛽−1)/2}𝑖+(𝑏)^𝑛𝑒^𝑖𝑏𝜔(𝜔)𝑑𝜔}. (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

𝜓(𝑡)=(1−𝑡²)𝑒^−𝑡2/2, (3.1) then from [1, page 372],

𝜓(𝜔)=√2𝜋𝜔²𝑒^−𝜔2/2. (3.2) Now, in view of (2.5), we have

𝜙(𝜔)∶=𝑓(𝜔)∼𝜓(𝑎𝜔)_∞𝑟=0𝑏_𝑟𝜔^{𝑟+𝛽−1}√2𝜋𝑎²𝜔²𝑒^{−𝑎2𝜔2/2}=√2𝜋𝑎^2∞_𝑟=0𝑏_𝑟𝜔^{𝑟+𝛽−1}𝜔^2∞_𝑠=0𝑎(−²2)^𝑠𝜔^2𝑠=√𝑠!2𝜋𝑎²𝜔^∞𝛽+1∞_𝑟=0𝑠=0𝑏_𝑟𝑎(−²2)^𝑠𝜔^2𝑠+𝑟=√𝑠!2𝜋𝑎²𝜔^∞𝛽+1_𝑟=0^[𝑟/2]_𝑗=0𝑏_𝑟−2𝑗𝑎𝑗!(−²2)^𝑗𝜔^𝑟=^∞_𝑟=0𝑐_𝑟𝜔^𝑟+𝛽+1, (3.3) where

𝑐_𝑟=√2𝜋𝑎^2[𝑟/2]_𝑗=0𝑏_𝑟−2𝑗𝑎𝑗!(−²2)^𝑗, (3.4) where [𝑟/2] stands for the greatest positive integer ≤𝑟/2. To ensure that (𝑓)(𝑏,𝑎) exists for large values of 𝑏 we also impose the condition that 𝑓(𝑢)=𝑂(𝑒^𝜎𝑢2) 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

√(𝑓)(𝑏,𝑎)=a{√2𝜋2𝜋𝑎^2𝑛−1_𝑟=0^[𝑟/2]_𝑗=0𝑎(−²2)^𝑗𝑏_𝑟−2𝑗𝑗!Γ(𝑟+𝛽+2)𝑏^{−𝑟−𝛽−2}×𝑒^{𝑖𝜋(𝑟+𝛽+2)/2}𝑖+(𝑏)^𝑛𝑒^𝑖𝑏𝜔(𝜔)𝑑𝜔}. (3.5)

4. Morlet Wavelet Transform

In this section we choose 𝜓(𝑡)=𝑒^{𝑖𝜔0𝑡−𝑡2/2}. (4.1) Then from [1, page 373], 𝜓(𝜔)=√2𝜋𝑒^{−(𝜔−𝜔0)2/2}. (4.2) Now, 𝜓(𝑎𝜔)=√2𝜋𝑒^−/2𝑒^{𝑎𝜔0𝜔}𝑒^{−𝑎2𝜔2/2}=√2𝜋𝑒^∞−/2_𝑟=0(𝑎𝜔₀𝜔)^𝑟𝑟!^∞_𝑠=0𝑎(−²2)^𝑠𝜔^2𝑠=√𝑠!2𝜋𝑒^∞−/2∞_𝑟=0𝑠=0𝑎(−²2)^𝑠(𝑎𝜔₀)^𝑟𝜔^𝑟+2𝑠=√𝑟!𝑠!2𝜋𝑒^∞−/2_𝑟=0^[𝑟/2]_𝑗=0𝑎(−²2)^𝑠(𝑎𝜔₀)^𝑟−2𝑗𝜔^𝑟=𝑗!(𝑟−2𝑗)!_∞𝑟=0𝐴_𝑟𝜔^𝑟, (4.3) where

𝐴_𝑟=√2𝜋𝑒^{−/2[𝑟/2]}_𝑗=0𝑎(−²2)^𝑠(𝑎𝜔₀)^𝑟−2𝑗.𝑗!(𝑟−2𝑗)! (4.4) Hence

𝜙(𝜔)∶=𝑓(𝜔)∼𝜓(𝑎𝜔)_∞𝑟=0𝑏_𝑟𝜔^{∞𝑟+𝛽−1}_𝑝=0𝐴_𝑝𝜔^𝑝=𝜔^∞𝛽−1_𝑟=0𝑏_𝑟𝜔^𝑟∞_𝑝=0𝐴_𝑝𝜔^𝑝=^∞_𝑟=0(^𝑟_𝑝=0𝐴_𝑝𝑏_𝑟−𝑝)𝜔^{𝑟+𝛽−1}=^∞_𝑟=0𝑐_𝑟𝜔^{𝑟+𝛽−1}, (4.5) where

𝑐_𝑟=^𝑟_𝑝=0𝐴_𝑝𝑏_𝑟−𝑝. (4.6) Also, from (2.8) and (4.5) it follows that 𝛼=1. Therefore, from (2.12), using (4.6) we get √(𝑓)(𝑏,𝑎)=𝑎{2𝜋^𝑛−1𝑟_𝑟=0𝑝=0𝐴_𝑝𝑏_𝑟−𝑝Γ(𝑟+𝛽)𝑏^{−𝑟−𝛽}𝑒^{𝑖𝜋(𝑟+𝛽)/2}𝑖+(𝑏)^𝑛(𝜔)𝑒^𝑖𝑏𝜔𝑑𝜔}. (4.7)

5. Haar Wavelet Transform

The Haar wavelet is defined by

𝜓(𝑡)=1,0≤𝑡<1/2,−1,1/2≤t<1,0,otherwise, (5.1) whose Fourier transform [1, page 368] is 𝜓(𝜔)=𝑖(2𝑒^{−𝑖𝜔/2}−1−𝑒^−𝑖𝜔)𝜔. (5.2) Therefore, Haar wavelet transform on half-line is given by

𝑖(𝑓)(𝑏,𝑎)=√𝑒𝑎2𝜋^𝑖𝑏𝜔1𝑓(𝜔)(𝜔+𝑒^𝑖𝑎𝜔𝜔−2𝑒^{𝑖𝑎𝜔/2}𝜔=𝑖)𝑑𝜔√𝑎2𝜋{𝑒^𝑖𝑏𝜔𝑓(𝜔)𝜔𝑑𝜔+𝑒^{𝑖𝑏(𝜔+𝑎)}𝑓(𝜔)𝜔𝑑𝜔−2𝑒^{𝑖𝑏(𝜔+𝑎/2)}𝑓(𝜔)𝜔𝑑𝜔}. (5.3) For 𝑓(𝜔) possessing asymptotic behavior (2.5), we have

𝑓(𝜔)𝜔∼^∞_𝑟=0𝑎_𝑟𝜔^{𝑟+𝛽−2}. (5.4) Then, from (5.3) and (5.4) using formula [2, page 753]

𝑀[𝑒^𝑖𝑡;𝑧]=𝑒^{𝑖𝑧𝜋/2}Γ(𝑧), (5.5) we get, for 𝛽≥1 and 𝑏→∞,

=𝑖(𝑓)(𝑏,𝑎)√𝑎2𝜋^∞_𝑟=0𝑎_𝑟Γ(𝑟+𝛽−1)𝑒^{𝑖𝜋(𝑟+𝛽−1)/2}×(𝑏^{−𝑟−𝛽+1}+(𝑏+𝑎)^{−𝑟−𝛽+1}−2(𝑏+𝑎/2)^{−𝑟−𝛽+1})=𝑖√𝑎2𝜋^∞_𝑟=0𝑎_𝑟Γ(𝑟+𝛽−1)𝑒^{𝑖𝜋(𝑟+𝛽−1)/2}×{𝑏^{−𝑟−𝛽+1}+^∞_𝑠=0𝑠𝑎−𝑟−𝛽+1_−𝑠𝑏^{−𝑟−𝛽−𝑠+1}−2^∞_𝑠=0𝑠(−𝑟−𝛽+1𝑎2)^−𝑠𝑏^{−𝑟−𝛽−𝑠+1}}. (5.6)

6. Asymptotic Expansion for Small 𝑏

In this section we assume that asymptotic expansions of 𝑓(𝜔) and 𝜓(𝜔) as 𝜔→∞ are known and then derive asymptotic expansion of (𝑓)(𝑏,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 𝑓(𝑡) possess an asymptotic expansion of the form 𝑓(𝑡)∼^𝑛−1_𝑠=0𝑡^{−𝑠−𝛼}+𝑓_𝑛(𝑡),as𝑡→∞, (6.1) where 0<𝛼<1. Then for small values of 𝜔, 𝑓(t)𝑒^𝑖𝑡𝜔dt=𝑒^{−𝑖𝛼𝜋/2𝑛−1}_𝑠=0(−𝑖)^𝑠−1Γ(1−𝑠−𝛼)𝜔^{𝑠+𝛼−1}−^𝑛_𝑠=1(−𝑖𝜔)^𝑠−1+𝑅_𝑛(𝜔), (6.2) where =(−1)^𝑠𝑅(𝑠−1)!𝑀[𝑓;𝑠],_𝑛(𝜔)=(−𝑖𝜔)^𝑛𝑒^𝑖𝜔𝑡𝑓_𝑛,𝑛(𝑡)𝑑𝑡 (6.3) with 𝑓_𝑛,𝑛(𝑡)=(−1)^𝑛(𝑛−1)!(𝜏−𝑡)^𝑛−1𝑓_𝑛(𝜏)𝑑𝜏. (6.4) Let 𝑓(𝜔)∼^∞_𝑠=0𝑏_𝑠𝜔^{−𝑠−𝛼}as𝜔→∞,𝜓(𝜔)∼^∞_𝑟=0𝑎_𝑠𝜔^{−𝑟−𝛽}as𝜔→∞. (6.5) Then, writing =𝑎^{−𝑟−𝛼}𝑎_𝑟, we have 𝜓(𝑎𝜔)𝑓(𝜔)∼^∞_𝑟=0𝜔^{∞−𝑟−𝛼}_𝑠=0𝑏_𝑠𝜔^{−𝑠−𝛽}=𝜔^{∞−𝛼−𝛽}_𝑟=0𝑐_𝑟𝜔^−𝑟, (6.6) where 𝑐_𝑟=𝑏_𝑟+⋯+𝑏₀=^𝑟_𝑚=0𝑏_𝑟−𝑚=^𝑟_𝑚=0𝑎_𝑚𝑎^{−𝑚−𝛼}𝑏_𝑟−𝑚. (6.7)

Now, for fixed 𝑎∈𝑅₊, write

𝜙(𝜔)∶=∼𝜓(𝑎𝜔)𝑓(𝜔)_∞𝑟=0𝑐_𝑟𝜔^{−𝑟−𝛼−𝛽}=,𝜔→∞_{𝑛−1𝑟=0}𝑐_𝑟𝜔^{−𝑟−𝛼−𝛽}+𝜙_𝑛(𝜔), (6.8) 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: (𝑓)(𝑏,𝑎)=𝑒^{−𝑖𝜋(𝛼+𝛽)/2𝑛−1}_𝑠=0(−𝑖)^𝑠−1𝑐_𝑠Γ(1−𝑠−𝛼−𝛽)𝑏^{𝑠+𝛼+𝛽−1}−^𝑛_𝑠=1(−𝑖𝑏)^𝑠−1+(𝑏), (6.9) where

=(−1)^𝑠(𝑠−1)!𝑀[𝜙;𝑠],(𝜔)=(−𝑖𝑏)^𝑛𝑒^𝑖𝑏𝑡𝜙_𝑛,𝑛(𝑡)𝑑𝑡 (6.10) with

𝜙_𝑛,𝑛(𝑡)=(−1)^𝑛(𝑛−1)!(𝜏−𝑡)^𝑛−1𝜙_𝑛(𝜏)𝑑𝜏. (6.11)

7. Asymptotic Expansion for Small 𝑏 Continued

In this section we assume that asymptotic expansions of 𝑓 and 𝜓 are known, instead of 𝑓(𝜔) and 𝜓(𝜔) as in previous sections. Then as in [2, page 753] we get asymptotic expansions of 𝑓(𝜔) and 𝜓(𝜔) 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 (𝑓)(𝑏,𝑎) as 𝑏→∞. However, in this case, following [4, pages 321–323] we can obtain asymptotic expansion of (𝑓)(𝑏,𝑎)as 𝑏→0+.

Let

𝑓(𝑡)∼^∞_𝑠=0𝑡^{𝑠+𝛼′−1},as𝑡→0,0<𝛼^′<1,𝜓(𝑡)∼^∞_𝑟=0𝑡^{𝑟+𝛽′−1},𝑡→0. (7.1) Now, using (1.12)

𝑓(𝜔)=𝑒^{−𝑖𝑡𝜔}∼𝑓(𝑡)𝑑𝑡_∞𝑠=0𝑒^{−𝑖𝜋(𝑠+𝛼′)/2}Γ(𝑠+𝛼^′)𝜔^{−𝑠−𝛼′},as𝜔→+∞. (7.2) Similarly,

𝜓(𝜔)=𝑒^{−𝑖𝑡𝜔}∼𝜓(𝑡)𝑑𝑡_∞𝑟=0𝑒^{−𝑖𝜋(𝑟+𝛽′)/2}Γ(𝑟+𝛽^′)𝜔^{−𝑟−𝛽′}as𝜔→+∞. (7.3) Then

𝜙(𝜔)∶=𝑓(𝜔)𝜓(𝑎𝜔)∼^∞_𝑟=0𝑑_𝑟𝜔^{−𝑟−𝛼′−𝛽′}, (7.4) where

𝑑_𝑟=𝑒^{−𝑖𝜋(−𝑟+𝛼′−𝛽′)𝑟}_𝑚=0𝑎^{−(𝑟−𝑚)−𝛽′}𝑒^{−𝑖𝜋2𝑚}Γ(𝛼^′+𝑚)Γ(𝛽^′+𝑟−𝑚). (7.5) Let us set

𝜙(𝜔)=^𝑛−1_𝑟=0𝑑_𝑟𝜔^{−𝑟−𝛼′−𝛽′}+𝜙_𝑛(𝜔), (7.6) where 0<𝛼^′+𝛽^′<1.

Assume that 𝜙(𝜔)=f(̂𝜔)𝜓(𝑎𝜔) integrable is locally on (0,∞). Then applying (6.2) to (2.2) with 𝜙(𝜔) given by (7.6), finally we get

(𝑓)(𝑏,𝑎)=𝑒^{−𝑖𝜋(𝛼′+𝛽′)/2𝑛−1}_𝑠=0(−𝑖)^𝑠−1𝑑_𝑠Γ(1−𝑠−𝛼^′−𝛽^′)𝑏^{𝑠+𝛼′+𝛽′−1}−^𝑛_𝑠=1𝐷_𝑠(−𝑖𝑏)^𝑠−1+(𝑏), (7.7) where

𝐷_𝑠=(−1)^𝑠(𝑠−1)!𝑀[𝜙;𝑠],(𝑏)=(−𝑖𝑏)^𝑛𝑒^𝑖𝑏𝑡𝜙_𝑛,𝑛(𝑡)𝑑𝑡 (7.8) with

𝜙_𝑛,𝑛(𝑡)=(−1)^𝑛(𝑛−1)!(𝜏−𝑡)^𝑛−1𝜙_𝑛(𝜏)𝑑𝜏. (7.9)

Remark 7.1. We observe that if we assume the asymptotic expansions 𝑓(𝑡) and 𝜓(𝑡) as 𝑡→∞ and derive asymptotic expansions of 𝑓(𝜔) and 𝜓(𝑡) as 𝜔→∞ , then formula (1.2) gives asymptotic expansion of (𝑓)(𝑏,𝑎) as 𝑏→0+ for fixed 𝑎>0. The aforesaid technique does not yield asymptotic expansion of (𝑓)(𝑏,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 (𝑓)(𝑏,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.