Moment Asymptotic Expansions of the Wavelet Transforms

arXiv:1405.0146v1 [math.FA] 1 May 2014

MOMENT ASYMPTOTIC EXPANSIONS OF THE WAVELET TRANSFORMS R S PATHAK* AND ASHISH PATHAK** *DST CENTER FOR INTERDISCIPLINARY MATHEMATICA SCIENCES BANARAS HINDU UNIVERSITY , VARANASI, INDIA-221005 **DEPARTMENT OF MATHEMATICS AND STATISTICS DR. HARISINGH GOUR CENTRAL UNIVERSITY SAGAR-470003, INDIA. Abstract. Using distribution theory we present the moment asymptotic expansion of continuous wavelet transform in different distributional spaces for large and small values of dilation parameter a. We also obtain asymptotic expansions for certain wavelet transform.

1. Introduction In past few decades their were many mathematician who has done great work in the field of asymptotic expansion like Wong 1979 [10] using Mellin transform technique has obtained asymptotic expansion of classical integral transform and after that Pathak & Pathak 2009 [3, 4, 5, 6] has found the asymptotic expansion of continuous wavelet transform for large and small values of dilation and translation parameters. Estrada & Kanwal 1990 [7] has obtained the asymptotic expansion of generalized functions on different spaces of test functions. In present paper using Estrada & Kanwal technique we have obtained the asymptotic expansion of 2000 Mathematics Subject Classification. 42C40; 34E05. Key words and phrases. Asymptotic expansion, Wavelet transform,Distribution. ∗E-mail: [email protected] ∗∗ E-mail: pathak [email protected]. 1

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R.S.PATHAK AND ASHISH PATHAK

wavelet transform in different distributional spaces. The continuous wavelet transform of f with respect to wavelet ψ is defined by 1 (Wψ f ) (a, b) = √ a

Z

∞

f (x)ψ

−∞

x−b dx, b ∈ R, a > 0, a

(1.1)

provided the integral exists [3] Now, from (1.1) we get √

∞

b dx (Wψ f ) (a, b) = a f (x)ψ x − a −∞ √ b = a f (ax), ψ x − a Z

(1.2)

This paper is arranged in following manner. In section second, third , fourth and fifth we drive the asymptotic expansion in the distributional spaces E ′ (R), ′ P ′ (R),Oγ′ (R),Oc′ (R) and OM (R) respectively, studied in [7]

2. The moment asymptotic expansion of (Wψ f ) (a, b) as a → ∞ in the space E ′ (R) for given b The space E (R) is the space of all smooth functions on R and it’s dual space E ′ (R) , the space of distribution with compact support. If ψ ∈ E (R), then ψ x − ab ∈ E (R). So consider the seminorms Case 1 For b ≥ 0

b b α

ψ x − b = Max |D ψ x − |: −M 0,

1

ψ x − b =O as a → ∞

a aq α,M

(2.4)

Proof 1. For b ≥ 0. For ψ ∈ Xq we can find a constant K such that ψ x − b ≤ K x − a

Therefore, if a > M we obtain

q b b , − 1 < x < b + 1. a a

(2.5)

b

x − b x − b

: −M 0 and b ∈ R. So kψ (x) k γ, β,

b a

is also seminorm on P(R) for γ > 0, β ∈ N and for a given

b ∈ R. Therefore these seminorm generate the topology of the space P(R). If Xq = {ψ ∈ P(R) : D α ψ(0) = 0, f or α < q}. Therefore for any γ > 0 we can find a constant C such that q b b γ b ≤ C x − e γ|x| ψ x − 2 e |a| a a

if a > 1

−γ|x|

e and thus

x − b q − γ|x| γ | b | C1 x − b ≤C ψ a ≤ 2 e a e a aq

ψ x

a

γ, 0,

=O b a

1 aq

as a → ∞, ψ ∈ Xq .

Hence using above equation we get

1 x

ψ as a → ∞. =O

a γ, β, b aq a

(3.1)

(3.2)

MOMENT ASYMPTOTIC EXPANSIONS OF THE WAVELET TRANSFORMS

7

Using (3.2) we obtain the following theorem Theorem 3.1. Let ψ ∈ P(R), f ∈ P ′ (R) and µα = hf, xα i be its moment sequence. Then for a fixed b the asymptotic expansion of wavelet transform is √

X ∞ µα D α ψ(−b/a) b a f (ax), ψ x − ∼ as a → ∞. a α! aα+1/2 α=0

(3.3)

Proof 3. Similarly as Theorem 2.2 x2

Example 3.2. Let ψ(x) = (1 − x2 )e− 2 ∈ P(R) is Mexican-Hat wavelet and f (x) ∈ P ′ (R). Therefore by Theorem 3.3 moment asymptotic expansion of continuous Mexican-Hat wavelet transform for large a in P ′ (R) is given by √

2 X ∞ µ D α [(1 − x2 )e− x2 ] α b x=− ab ∼ as a → ∞. a f (ax), ψ x − a α! aα+1/2 α=0

Case 2. In this case we consider wavelet ψ(x) ∈ P ′ (R) and f (x) ∈ P(R). Then the wavelet transform (1.1) can we rewrite as 1 x (Wψ f ) (a, b) = √ ψ , f (x + b) a a Similarly as Theorem 3.1 we can also obtain the following theorem Theorem 3.3. Let ψ ∈ P ′ (R), f ∈ P(R) and µα = hψ, xα i be its moment sequence. Then for a fixed b the asymptotic expansion of wavelet transform is X ∞ µα D α f (b)aα+1/2 1 x √ ψ , f (x + b) ∼ as a → 0. a α! a α=0

(3.4)

Example 3.4. In this example again we consider the Mexican-Hat wavelet which is less then exponential growth , so by applying Theorem 3.3 and using formula [30,pp.320, 1], we get the asymptotic expansion of wavelet transform for small

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values of a X ∞ 1 x 1 2α + 1 D 2α f (b)a2α+1/2 (2α−1) √ ψ , f (x + b) ∼ −2 2 as a → 0. Γ a a 2 (2α)! α=0 4. The moment asymptotic expansion of (Wψ f ) (a, b) as a → ∞ in the space Oγ′ (R) for given b A test function ψ belongs to Oγ (R), if it is smooth and D α ψ(x) = O(|x|γ ) as x → ∞ for every α ∈ N and γ ∈ R. The family of seminorms kψ(x)kα,γ = sup{ργ (|x|)|D α ψ(x)| : x ∈ R} where  

 1, 0 ≤ |x| ≤ 1  ργ (|x|) = ,  |x|−γ , |x| > 1 

(4.1)

generates a topology for Oγ (R). Now with the help of the translation version of ψ(x), we can define the seminorms on Oγ (R) as α b kψ(x)kα,γ,b/a = sup ργ (|x|) D ψ x − :x∈R a b α b = sup ργ x − D ψ x − : x ∈ R ∇(x, b/a) a a

b

∇(x, b/a)

= ψ x − a α,γ ργ (|x|) where ∇(x, b/a) = sup ρ (|x− b |) : x ∈ R . γ

a

So, for γ > 0.    1, f or 0 ≤ |x| ≤ 1 and 0 ≤ |x − b/a| ≤ 1   γ      1 + |b/a| , f or |x| > 1 and |x − b/a| > 1 1−|b/a| ∇(x, b/a) ≤    (1 + | ab |)γ , f or 0 ≤ |x| ≤ 1 and |x − b/a| > 1       1, f or |x| > 1 and |x − b/a| ≤ 1

                  

.


Similarly for γ < 0. we have    1, f or 0 ≤ |x| ≤ 1 and 0 ≤ |x − b/a| ≤ 1 −γ ∇(x, b/a) ≤   1 + |b/a| , otherwise Thus sup

ργ (|x|) ργ (|x− ab |)

Therefore kψ(x)kα,

:x∈R γ, b/a

≤

1 + |b/a|

|γ|

  

9

.

 

, = K < ∞, ∀ γ ∈ R

are also seminorm on Oγ (R).These seminorm generate

the topology of the space ψ(x) ∈ Oγ (R).If Xq = {ψ ∈ Oγ (R) : D α ψ(0) = 0, f or α < q}. So for any γ we can find a constant C such that b ≤ Cρ(|x|) x − ρ(|x|) ψ x − a

If a > 1

q b ∇(x, b/a). a

b M ≤ q ρ(|x|) ψ x − a a

Hence using above equation we get

x 1

ψ

=O as a → ∞.

a α, γ ,b/a aq

(4.2)

Similarly as Theorem 3.1 we can obtain the following theorem Theorem 4.1. Let ψ ∈ Oγ (R), f ∈ Oγ′ (R), N = [[γ]] − 1 and µα = hf, xα i be

its moment sequence. Then for a fixed b ∈ R the asymptotic expansion of wavelet transform is X N √ 1 µα D α ψ(−b/a) b = + O N +1/2 as a → ∞. (4.3) a f (ax), ψ x − a α!aα+1/2 a α=0 Since Oc′ (R) =

T

Oγ′ (R), we obtain the asymptotic expansion of wavelet trans-

form in the space Oc′ (R)

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Theorem 4.2. Let ψ ∈ Oc (R), f ∈ Oc′ (R) and µα = hf, xα i be its moment sequence. Then for a fixed b ∈ R the asymptotic expansion of wavelet transform is √

X ∞ 1 µα D α ψ(−b/a) b ∼ + O N +1/2 as a → ∞. (4.4) a f (ax), ψ x − α+1/2 a α!a a α=0

5. The moment asymptotic expansion of (Wψ f ) (a, b) as a → ∞ in the ′ space OM (R) for given b

The space OM (R) consist of all c∞ -function whose derivatives are bounded by polynomials ( of probably different degrees). Let ψ ∈ OM (R) then its translation version is also in OM (R). Then by using Theorem 9 [7] we can also derive the ′ asymptotic expansion of wavelet transform in OM (R) ′ Theorem 5.1. Let ψ ∈ OM (R), f ∈ OM (R) and µα = hf, xα i be its moment

sequence. Then for a fixed b ∈ R the asymptotic expansion of wavelet transform is hf (x), ψa,b (x)i ∼

∞ X µα (f )D αψ(− b ) a

α=0

α! aα+1/2

as a → ∞.

(5.1)

Proof. By using (1.7.1) [3] we can be write the wavelet transform √

√ b ˆ = aheibω fˆ(ω), ψ(aω)i a f (ax), ψ x − a

(5.2)

′ ˆ where ψ(x) ∈ OM (R) and f (x) ∈ OM (R) then its Fourier transforms ψ(ω) ∈

Oc′ (R) and fˆ(ω) ∈ Oc (R) respectively. Now by using Theorem 4.2 we get hf (x), ψa,b (x)i ∼

b ∞ α ˆ X ˆ µα (e−i a ω ψ(ω))D (f (0)

α=0

α! aα+1/2

as a → ∞.

(5.3)


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But by the properties of Fourier transform we have b −ibaω ˆ −i ab ω ˆ α −α α µα (e ψ(ω)) = he , D α (fˆ(ω))ω=0 = iα µα (f (x)) ψ(ω), ω i = i D ψ − a and hence hf (x), ψa,b (x)i ∼

∞ X µα (f )D α ψ(− b ) a

α=0

α! aα+1/2

as a → ∞.

(5.4)

Acknowledgment The work of second author is supported by U.G.C. start-up grant. References [1] Erde’lyi, A., W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York (1954). [2] I. Daubechies, Ten Lectures of Wavelets, SIAM, Philadelphia. 1992. [3] R . S. Pathak. The Wavelet transform. Atlantis Press/World Scientific, 2009. [4] R S Pathak and Ashish Pathak. Asymptotic expansion of Wavelet Transform for small value a, arXiv:submit/0949812 [math.FA] 4 Apr 2014. [5] R S Pathak and Ashish Pathak. Asymptotic expansion of Wavelet Transform with error term, arXiv:submit/0949812 [math.FA] 4 Apr 2014. [6] R S Pathak and Ashish Pathak. Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b, Int. Jou. of Math. and Mathematical Sciences, 2009, 13 page. [7] R.Estrada , R. P. Kanwal. A distributional theory for asymptotic expansions. Proc. Roy. Soc. London Ser. A 428 (1990), 399-430. [8] L. Schwartz. Th´ eorie des Distributions, Hermann, Paris, 1966. [9] J Horv´ ath,Topological Vector Spaces and Distributions. Vol. I Addison-Wesley, Reading, MA (1966) [10] R. Wong. Explicit error terms for asymptotic expansion of Mellin convolutions. J. Math. Anal. Appl. 72(1979), 740-756.