The main objective of this paper is to define the mother wavelet on local
fields and study the continuous wavelet transform (CWT) and some of their basic
properties. its inversion formula, the Parseval relation and associated
convolution are also studied

Pathak, Ashish

Boletim da Sociedade Paranaense de Matemática

English

arXiv

The main objective of this paper is to define the mother wavelet on local fields and study the continuous wavelet transform (CWT) and some of their basic properties. its inversion formula, the Parseval relation and associated convolution are also studied.</jats:p

Ashish Pathak

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Bol. Soc. Paran. Mat. (3s.) v. 34 2 (2016): 113–121.c©SPM –ISSN-2175-1188 on line ISSN-00378712 in pressSPM: www.spm.uem.br/bspm doi:10.5269/bspm.v34i2.27340Continuous Wavelet Transform on Local Fields ∗Ashish Pathakabstract: The main objective of this paper is to define the mother wavelet onlocal fields and study the continuous wavelet transform (CWT) and some of its basicproperties. Its inversion formula, the Parseval relation and associated convolutionare also studied.Key Words: Continuous Wavelet Transform , Local field, Fourier transform.Contents1 Introduction 1132 Continuous Wavelet transform on local fields 1143 Basic Properties of CWT on local fields 1164 Main Properties of the CWT 1184.1 Associated convolution for CWT on local fields . . . . . . . . . . . . 1191. IntroductionA local field means an algebraic field and a topological space with the topologi-cal properties of locally compact, non-discrete, complete and totally disconnected,denoted by K [8]. The additive and multiplicative groups of K are denoted by K+and K∗, respectively. We may choose a Haar measure dx for K+. If α 6= 0 (α ∈ K),then d(αx) is also a Haar measure. Let d(αx) = |α|dx and call |α| the absolutevalue or valuation of α. Let |0| = 0. The absolute value has the following proper-ties:(i) |x| ≥ 0 and |x| = 0 if and only if x = 0;(ii) |xy| = |x||y|;(iii) |x+ y| ≤ max(|x|, |y|).The last one of these properties is called the ultrametric inequality. The setD = {x ∈ K : |x| ≤ 1} is called the ring of integers in K. It is the uniquemaximal compact subring of K. Define P = {x ∈ K : |x| < 1}. The set P is calledthe prime ideal in K. The prime ideal in K is the unique maximal ideal in D. It isprincipal and prime.If K is a local field, then there is a nontrivial, unitary, continuous character χ onK+ and K∗ is self-dual.χ is fixed character on K+ that is trivial on D but is nontrivial on P−1 . It follows∗ The work is supported by U.G.C start-up grant No:F.30-12(2014)/(BSR).2000 Mathematics Subject Classification: 65T60; 43A25; 11F85.113Typeset by BSPMstyle.c© Soc. Paran. de Mat.114 Ashish Pathakthat χ is constant on cosets of D and that if y ∈ Pk , then χy(χy(x)) = χ(xy) isconstant on cosets of P−kDefinition 1.1. The Fourier transform of f ∈ L1(K) is denoted by fˆ(ξ) anddefined by [9]fˆ(ξ) =∫Kf(x)χξ(x)dx =∫Kf(x)χ(−ξx)dx, ξ ∈ K, (1.1)and the inverse Fourier transform byf(x) =∫Kfˆ(ξ)χx(ξ)dx, x ∈ K. (1.2)Some important properties of the Fourier transform can be proved easily :(i) ||fˆ ||L∞(K) ≤ ||f ||L1(K).(ii) If f ∈ L1(K), then fˆ is uniformly continuous.(iii) Parseval formula:If f ∈ L1(K) ∩ L2(K), then ||fˆ ||L2(K) = ||f ||L2(K)(iv) If the convolution of f and g is defined as(f ∗ g)(t) =∫Kf(x)g(t− x)dx, (1.3)thenF ((f ∗ g)) = F (f).F (g). (1.4)The article is divided in four sections. In section 2, we propose the definition ofmother wavelet and define the continuous wavelet transform (CWT). In section 3,we discuss some basic properties of CWT. In section 4, we prove the Plancherelformula , inversion formula and also define the convolution associated with CWT.2. Continuous Wavelet transform on local fieldsSimilar to L2(R) [1,3,5], we define the wavelet on local fields and define thecontinuous wavelet transform.Definition 2.1. Admissible wavelet on local fieldsThe function ψ(x) ∈ L2(K) is said to be an admissible wavelet on local fields ifψ(x) satisfies the following admissibility condition:cψ =∫K|ψˆ(ξ)|2|ξ| dξ <∞, (2.1)where ψˆ is the Fourier transform of ψ.Continuous Wavelet Transform on Local Fields 115Remark 2.2. If |ψˆ(ξ)| is continuous near ξ = 0, then the existence of integral(2.1) guarantees that , ψˆ(0) = 0. Since the Fourier transform of mother waveletψ ∈ L1(K) ∩ L2(K) is bounded and uniformly continuous, we have0 = ψˆ(0) =∫Kψ(x)χ0(x)dx=∫Kψ(x)dx.This means that the integral of mother wavelet is zero:Theorem 2.3. If ψ is a mother wavelet and φ ∈ L1(K), then the convolutionfunction ψ ∗ φ is a mother wavelet.Proof: Since∫K|(ψ ∗ φ)(x)|2dx =∫K∣∣∣ ∫Kψ(x− y)φ(y)dy∣∣∣2dx≤∫K(∫K|ψ(x− y)||φ(y)| 12 ||φ(y)| 12 dy|)2dx≤∫K(∫K|ψ(x− y)||φ(y)|dy∫K|φ(y)|dy)dx=∫K|φ(y)|dy∫K∫K|ψ(x − y)|2|φ(y)|dydx=(∫K|φ(y)|dy)2 ∫K|ψ(x)|2dx= ||φ||2L1(K)||ψ||2L2(K).Therefore (ψ ∗ φ)(x) ∈ L2(K). Moreovercψ∗φ =∫K| ˆψ ∗ φ(ξ)|2|ξ| dξ=∫K|ψˆ(ξ)|2|φˆ(ξ)|2|ξ| dξ≤ ||φˆ||2L∞(K)∫K|ψˆ(ξ)|2|ξ| dξ.This completes the proof of the theorem ✷Definition 2.4. Continuous wavelet transform (CWT) on local fieldsFor ψ(x) ∈ L2(K) and a, b ∈ K, a 6= 0, we define the unitary linear operator:U ba : L2(K)→ L2(K),116 Ashish PathakbyU ba(ψ(x)) = ψa,b(x) =1|a| 12 ψ(x− ba), (2.2)ψ is called mother wavelet and ψa,b(x) are called daughter wavelets, where a is adilation parameter, b is a translation parameter.The Fourier transform of ψa,b(x) is given byˆψa,b(ξ) = |a|12 ψˆ(aξ)χb(ξ), (2.3)where ψˆ is the Fourier transform of ψ.The CWT on local fieldsKψ : L2(K)→ L2(K×K),of a function f ∈ L2(K) with respect to a mother wavelet ψ is defined byf 7→ Kψf(a, b) = (f, ψa,b)L2(K)=∫Kf(x)ψa,b(x)dx=∫Kf(x)1|a| 12 ψ(x− ba)dx. (2.4)3. Basic Properties of CWT on local fieldsBefore giving the fundamental properties of CWT, we list their basic properties.Theorem 3.1. Let ψ and ϕ be two wavelets and f, g be two functions belonging toL2(K), then(1) LinearityKψ(ηf + ϑg)(a, b) = ηKψ(f)(a, b) + ϑKψ(g)(a, b), (3.1)where η and ϑ are any two scalers.(ii)Shift property(Kψf(x− ς))(a, b) = (Kψf)(a, b− ς), (3.2)where ς is any scaler.(iii) Scaling property If σ 6= 0 is any scaler, then the CWT of the scaled functionfσ(x) =1σf( 1σ) is(Kψfσ)(a, b) = Kψ(f)(aσ,bσ). (3.3)(iv) Symmetry(Kψf)(a, b) = (Kf (ψ))(1a,−1b). (3.4)Continuous Wavelet Transform on Local Fields 117(iv) Parity(KPψPf)(a, b) = (Kψf)(a,−b), (3.5)where P is a parity operator define by Pf(x) = f(−x).Proof: The proof is the straight forward application of CWT. ✷Theorem 3.2. Show that the continuous wavelet transform can also expressed as(Kψf)(a, b) =(f ∗ 1√aψ(xa))(b), (3.6)where the ∗ is defined as(f ∗ g)(t) =∫Kf(x)g(t− x)dx. (3.7)Proof: From definition of CWT we have(Kψf)(a, b) =∫Kf(x)1a12ψ(x− ba)dx=(f ∗ 1√aψ(xa))(b). (3.8)✷Theorem 3.3. If f is homogeneous function of degree n, then(Kψf)(λa, λb) = λn|λ| 12 (Kψf)(a, b), (3.9)where the λ is scaler .Proof: From definition of CWT we have(Kψf)(λa, λb) =∫Kf(x)1|λa| 12 ψ(x− λbλa)dx=∫Kf(λx)1|a| 12 ψ(x− ba)|λ|dx= λn|λ| 12 (Kψf)(a, b). (3.10)✷118 Ashish Pathak4. Main Properties of the CWTThis section describes important properties of the CWT, such as the Plancherel, inversion formula and associated convolution fist, we establish the Planchereltheorem.Theorem 4.1. (CWT Plancherel) Let f, g ∈ L2(K). Then we have( (Kψf)(a, b), (Kψg)(a, b) )L2(K×K) = cψ(f, g)L2(K), (4.1)where cψ is given in (2.1).Proof: By using parseval formula for Fourier we can write the wavelet transformas(Kψf)(a, b) =∫Kf(x)1|a| 12 ψ(x− ba)dx= (f, ψa,b)= (fˆ , ψˆa,b)=∫Kfˆ(ξ)|a| 12 ψˆ(aξ)χb(ξ)dξ. (4.2)SimilarlyKψ(g)(a, b) =∫Kfˆ(ξ)|a| 12 ψˆ(aξ)χb(ξ)dξ. (4.3)Now, by using above (4.2) and (4.3) we get∫K∫KKψ(f)(a, b)Kψ(g)(a, b)dadb|a|2 =∫K∫K|a|dadb|a|2∫Kfˆ(ξ)ψˆ(aξ)χb(ξ)dξ×∫Kgˆ(υ)ψˆ(aυ)χb(υ)dυ=∫K∫Kdadb|a|∫Kfˆ(ξ)ψˆ(aξ)χb(ξ)dξ×∫Kgˆ(υ)ψˆ(aυ)χb(υ)dυ=∫K∫KF (fˆ(ξ)ψˆ(aξ))(b)F (gˆ(υ)ψˆ(aυ))(b)dadb|a|=∫K∫Kfˆ(ξ)ψˆ(aξ)gˆ(ξ)ψˆ(aξ)dξda|a|=∫Kfˆ(ξ)gˆ(ξ)(∫Kψˆ(aξ)ψˆ(aξ)da|a|)dξ=∫Kfˆ(ξ)gˆ(ξ)(∫K|ψˆ(aξ)|2|a| da)dξContinuous Wavelet Transform on Local Fields 119=∫Kfˆ(ξ)gˆ(ξ)(∫K|ψˆ(ω)|2|ω| dω)dξ= cψ(fˆ , gˆ)L2(K)= cψ(f, g)L2(K). (4.4)✷Theorem 4.2. (Inversion Formula) Let f ∈ L2(K). Then we havef(x) =1cψ∫K∫KKψ(f)(a, b)ψa,b(x)dadb|a|2 , (4.5)where cψ is given in (2.1).Proof: Let h(x) ∈ L2(K) be any function, then by using above theorem, we havecψ(f, g)L2(K) =∫K∫K(Kψf)(a, b)Kψ(h)(a, b)dadb|a|2=∫K∫K(Kψf)(a, b)∫Kh(x)ψa,b(x)dxdadb|a|2=∫K∫K∫K(Kψf)(a, b)ψa,b(x)h(x)dadbdx|a|2=(∫K∫K(Kψf)(a, b)ψa,b(x)dadb|a|2 , h(x)).Hence the result follows. ✷If f = h,||f ||2L2(K) =∫K∫K|(Kψf)(a, b)|2 dadb|a|2 . (4.6)Moreover the wavelet transform is isometry from L2(K) to L2(K ×K).4.1. Associated convolution for CWT on local fieldsUsing Pathak and Pathak techniques [5], we define the basic function D(x, y, z), translation τx and associated convolution # operator for CWT.The basic function D(x, y, z) for (2.4) is defined asKφ[D(x, y, z)](a, b) =∫KD(x, y, z)φa,b(t)dt= ψa,b(z) χa,b(y), (4.7)120 Ashish Pathakwhere ψ, φ and χ are three wavelets satisfying certain conditions (2.1).Now, by using(4.5) we get,D(x, y, z) = C−1φ∫K∫Kψa,b(z) χa,b(y) φa,b(x) |a|−2dadb. (4.8)The translation τx is defined as [5](τxh)(y) = h∗(x, y) =∫KD(x, y, z)h(z)dz= C−1φ∫K∫K∫Kψa,b(z) χa,b(y) φa,b(x)h(z)|a|−2dadbdz.The associated convolution is defined as(h#g)(x) =∫Kh∗(x, y)g(y)dy=∫K∫KD(x, y, z)h(z) g(y)dydz= C−1φ∫K∫K∫K∫Kψa,b(z) χa,b(y) φa,b(x)h(z)g(y) |a|−2 dadbdzdy,(4.9)by using the inversion formula we can write the above equation as(h#g)(x) = C−1φ∫K∫K(Kψh)(a, b)(Kχg)(a, b)φa,b(x)|a|−2dadb= K−1φ [(Kψh)(a, b)(Kχg)(a, b)] (x);so thatKφ[h#g](a, b) = (Kψh)(a, b)(Kχg)(a, b)(x). (4.10)AcknowledgmentsThe author is thankful to the referees for their valuable comments and thisresearch work is supported by U.G.C start-up grant No:F.30-12(2014)/(BSR).References1. Daubechies,I, Ten Lectures on Wavelets ,CBMS/NSF Ser. Appl. Math., vol. 61, SIAM, (1992).2. L. Debnath, Wavelet Transforms and Their Applications, Birkhauser, Boston, (2002).3. C.K. Chui, An Introduction to Wavelets, Academic Press, 1992.4. M. Holschneider, Wavelet analysis over Abelian groups, Appl. Comput. Harmon. Anal. 2,52-60, (1995).5. R. S. Pathak and Ashish Pathak . On convolution for wavelets transform , Internationaljournal of wavelets, multiresolution and information processing , 6(5): 739-747, (2008).Continuous Wavelet Transform on Local Fields 1216. Ashish Pathak, P Yadav, M M Dixit, On convolution for general novel fractional wavelettransform , Journal of Advanced Research in Scientific Computing, 7(1), 30-37, (2015).7. D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, Graduate Texts inMathematics , Springer-Verlag, New York, 1999.8. H. Jiang, D. Li and N. Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl., 294,523- 532 , (2004.)9. M.H. Taibleson, Fourier Analysis on Local Fields, Princeton Univ. Press, 1975.Ashish PathakDepartment of Mathematics & Statistics,School of Mathematical and Physical Sciences,Dr. Harisingh Gour University,Sagar-470003, India.ashishpathak@dhsgsu.ac.in , pathak_maths@yahoo.com

Continuous wavelet transform on local fields

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Continuous wavelet transform on local fields

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