Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier transform

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1 NTMSCI 6, No., ) 175 New Trends in Mathematical Sciences Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier transform Abdullah Abdullah and Afroz Afroz Department of Mathematics, Zakir Husain Delhi College, University of Delhi), New Delhi, India Department of Mathematics, School of sciences, Maulana Azad National Urdu University, Hyderabad, India Received: 11 February 017, Accepted: 1 April 017 Published online: May 018. Abstract: We investigate semi-orthogonal wavelet frames on positive half-line and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. Keywords: Frame multiresolution analysis, wavelet frame, semi-orthogonality, Walsh Fourier transform. 1 Introduction The concept of frames in a Hilbert space was originally introduced by Duffin and Schaeffer [7] in the context of non-harmonic Fourier series. In signal processing, this concept has become very useful in analyzing the completeness and stability of linear discrete signal representations. Frames did not seem to generate much interest until the ground-breaking work of Daubechies et al. [6]. They combined the theory of continuous wavelet transforms with the theory of frames to introduce wavelet affine) frames for R). Since then the theory of frames began to be more widely investigated, and now it is found to be useful in signal processing, image processing, harmonic analysis, sampling theory, data transmission with erasures, quantum computing and medicine. Recently, more applications of the theory of frames are found in diverse areas including optics, filter banks, signal detection and in the study of Bosev spaces and Banach spaces. We refer [5] for an introduction to frame theory and its applications. In recent years, wavelets have been generalized in many different settings, for example locally compact abelian group, abstract Hilbert spaces, locally compact Cantor dyadic group, Vilenkin group, local fields and positive half-line. In this paper our interest is in positive half-line. Farkov [8] has given general construction of compactly supported orthogonal p-wavelets in R + ). An algorithm for biorthogonal wavelets related to Walsh functions on positive half line was given in [9]. Dyadic wavelet frames on the positive half-line R+ were constructed by Shah and Debnath in [15] using Walsh Fourier transforms. They have established necessary and sufficient conditions for the system ψ j,k x) = j/ ψ j x k), j Z,k Z + to be a frame for R + ). Wavelet frame packets related to the Walsh polynomials were deeply investigated by Shah and Debnath in [16]. A constructive procedure for constructing tight wavelet frames on positive half-line using extension principles was recently considered by Shah in [17], in which he has pointed out a method for constructing affine frames in R + ). Moreover, the author has established sufficient conditions for a finite number of functions to form a tight affine frames for R + ). Abdullah [1] has given characterization of nonuniform wavelet sets on positive half-line and Characterization of wavelets on positive half line by means of two basic equations in the Fourier domain was established in []. Corresponding author abd.zhc.du@gmail.com; afrozjmi13@gmail.com

2 176 A. Abdullah and A. Afroz: Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier... In this paper, we extend the notion of wavelet frames to semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier transform. This paper is organized as follows: In Sec., we present a brief review of generalized Walsh functions and polynomials, the Walsh Fourier transform and FMRA in R + ). A characterization of semi-orthogonal wavelet frames on positive half-line is given in Section 3. Notations and preliminaries et p be a fixed natural number greater than 1. As usual, let R + =[0, ) and Z + =0,1,... Denote by [x] the integer part of x. For x R + and for any positive integer j, we set where x j, x j 0,1,..., p 1. x j =[p j x]mod p), x j =[p 1 j x]mod p), 1) Consider the addition defined onr + as follows with x y= ξ j p j 1 + ξ j p j ) j<0 j>0 ξ j = x j + y j mod p), j Z\0, 3) where ξ j 0,1,,..., p 1 and x j, y j are calculated by 1). Moreover, we write z=x y if z y=x. For x [0,1), let r 0 x) be given by [ ) 1, x 0, 1 p, r 0 x)= εp, j x [ jp 1, j+ 1)p 1), j = 1,,..., p 1, ) where ε p = exp πi p. The extension of the function r 0 to R + is defined by the equality r 0 x+1)=r 0 x), x R +. Then the generalized Walsh functionsω m x) m Z + are defined by ω 0 x)=1, ω m x)= k j=0 r0 p j x )) µ j, where m= k j=0 µ j p j, µ j 0,1,,..., p 1, µ k 0. For x, ω R +, let χx,ω)=exp πi p j=1 x j ω j + x j ω j ) ) 4), 5) where x j and ω j are calculated by 1). We observe that χ x, ) ) ) m x x p n 1 = χ p n 1, m = ω m p n 1 x [0, p n 1 ), m Z +.

3 NTMSCI 6, No., ) / The Walsh Fourier transform of a function f 1 R + ) is defined by where χx,ω) is given by 5). If f R + ) and then f is defined as limit of J a f in R + ) as a. fω)= fx)χx,ω)dx, 6) R + a J a fω)= fx)χx, ω)dx a<0), 7) 0 The properties of Walsh Fourier transform are quite similar to the classical Fourier transform. It is known that systems χα,.) α=0 and χ.,α) α=0 are orthonormal bases in 0,1). et us denote by ω the fractional part of ω. For l Z +, we have χl,ω)=χl,ω). If x,y,ω R + and x y is p-adic irrational, then χx y,ω)= χx,ω)χy,ω), χx y,ω)= χx,ω)χy,ω). 8) For given Ψ =ψ 1,ψ,...,ψ R + ), define the wavelet system XΨ)=ψ l, j,k : 1 l, j Z,k Z + 9) where ψ l, j,k x) = p j/ ψ l p j x k). The wavelet system XΨ) is called a wavelet frame if there exist positive numbers A and B with 0<A B< such that A f k Z + f,ψl, j,k B f 10) for all f R + ). The largest A and the smallest B for which 10) holds are called frame bounds. A frame is a tight frame if A and B are chosen so that A=Band is a normalized tight frame if A=B=1. The collection XΨ) is said to be a semi-orthogonal wavelet frame if ψ l, j,k,ψ l, j,k =0 whenever j j and k,k Z +, 1 l. The characterization of wavelet frames on positive half-line has been studied in detail by Abdullah []. Theorem 1. SupposeΨ =ψ 1,ψ,...,ψ R + ). The affine system XΨ) is a tight frame with constant 1 for R + ), i.e., if and only if f = k Z + < f,ψ l, j,k > for all f R + ) ˆψ l p j ξ) = 1 for a. e. ξ R +, 11)

4 178 A. Abdullah and A. Afroz: Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier... and j=0 ˆψ l p j ξ) ˆψ l p j ξ s))=0 for a. e. ξ R + and for all \pz +. 1) In particular, Ψ is a set of basic wavelets of R + ) if and only if ψ l = 1 for l = 1,,..., and 11) and 1) hold. Definition 1. The collectionψ l. k) : 1 l,k Z + forms a wavelet frame for W 0 if and only if there exist positive numbers A and B such that where B=maxB l, A=minA l and S l =ξ : ˆψ l ξ k) 0. A k Z + ˆψ l ξ k) B a.e. ξ S l, 13) 3 Semi-orthogonal wavelet frames on positive half-line In this section, we characterize semi-orthogonal wavelet frames on positive half-line by virtue of the Walsh-Fourier transform. Our results generalize the characterization of wavelets on Euclidean spaces by means of two basic equations. For each l = 1,,...,, we define ˆψ l ξ)= ˆψ l ξ) ˆψ l ξ k) l Z + ) if ξ S l, 0 otherwise, where S l = ξ : ˆψ l ξ k) 0. Then, the system XΨ ) obtained by the combined action of dilation and translation of a finite number of functions Ψ =ψ 1,ψ,...,ψ R + ) is given by XΨ )=ψ l, j,k : 1 l, j Z,k Z+. 14) Theorem. et XΨ) and XΨ ) be as defined in 9) and 14), respectively. Then the following statements are equivalent: a) XΨ) is a semi-orthogonal wavelet frame with frame bounds A and B. b) For each l = 1,,...,, there exist positive constants A=minA l and B=minB l such that inequality 13) holds and and j=0 ˆψ l p j ξ) = 1 for a. e. ξ R +, 15) ˆψ l p j ξ) ˆψ l p j ξ s))=0 for a. e. ξ R + and for all \pz +. 16) c) There exist positive numbers A and B such that functions ψ 1,ψ,...,ψ satisfy conditions 13), 16) and k Z + ˆψ l ξ k) ˆψ l p j ξ k))=0 for a. e. ξ R + j 1, 17) A ˆψ l p j ξ ) B for a. e. ξ R +. 18)

5 NTMSCI 6, No., ) / Proof. For each j Z, we define and W j = spanψ l, j,k : 1 l, k Z + W j = spanψ l, j,k : 1 l, k Z+. a) b): Suppose that the affine system XΨ) given by 9) is a semi-orthogonal wavelet frame for R + ) with bounds A and B, i.e. A f By the scaling property of W j spaces,we have k Z + f,ψl, j,k B f, for all f R + ). k Z + f,ψ l, j,k = k Z + f,ψ l,0,k, for all f W 0. Therefore, we have which is equivalent to 13). A f k Z + f,ψl,0,k B f, for all f W 0, Since the W j are orthogonal to each other, i.e., W j 1 W j, j 1 j, we have k Z + ˆψ l ξ k) = 1 for a.e. ξ S l. Therefore, the system XΨ ) is a tight frame with frame bound 1 for R + ). Hence 15) and 16) are satisfied by Theorem 1. b) c): For each l = 1,,...,, we use equation 13), we have Or equivalently, 1 B l ˆψ l ξ) ˆψ l ξ) 1 A l ˆψ l ξ). 19) 1 ˆψ l p j ξ) B l By taking maxb l = B, mina l = A and applying 15), we get A ˆψ l p j ξ) ˆψ l p j ξ) B a.e. ξ R + 1 ˆψ l p j ξ). A l which shows 18). Moreover, it follows from 15) and 16) that the system ψ l,0,k : k Z+ forms a tight frame for W 0 with frame bound 1. By Theorem 1,ψ l, j,k :,k Z+ is a normalized tight frame for R + ). Since each ψ l lies in W 0, it follows from the tightness of both systemsψ l,0,k : k Z+ andψ l, j,k : j Z,k Z+ that ψl = ψ l,ψl, j,k = ψ l,ψl,0,k. k Z + k Z +

6 180 A. Abdullah and A. Afroz: Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier... Therefore, ψl,ψ l, j,k =0 for j 0. Also, for each l = 1,,..., and j N, we have This shows that Therefore, we have 0= ψ l,ψ l, j,k = ˆψ l, ˆψ l, j,k = p j/ R + ˆψ l ξ) ˆψ l p j ξ)χk, p j ξ) = p R j/ ˆψ + l p j ξ) ˆψ l ξ)χk,ξ) = p j/ ˆψ l p j ξ) ˆψ l ξ)χk,ξ) r Z + r = p j/ ˆψ l p j ξ r)) ˆψ l ξ r) χk,ξ). r Z + r Z + ˆψ l ξ r) ˆψ l p j ξ r))=0 for a.e. ξ R +, j 1. ˆψ l ξ k) ˆψ l p j ξ k))= ˆψ l p j ξ k)) 1 1 ˆψ l ξ k) ˆψ l ξ k) ˆψ l p j ξ k))=0. k Z + k Z + k Z + k Z + Hence, condition 17) is satisfied. c) a): we use condition 13)which shows that the affine system XΨ) constitutes a frame for W j. Moreover, identity 17) shows that W 0 is orthogonal to W j for j 0. Therefore, by means of change of variables, we have ψ l, j,k,ψ l,n,m = ψ l,0,k p 1 j m,ψ l,n j,0, 1 l, k,m Z +. It is immediate from the above relation that W j W n for j n. Thus, we conclude that XΨ) forms a frame for W = spanψ l, j,k : 1 l, j Z,k Z +. Next, we claim that W = R + ). It suffices to show that the system XΨ ) given by 14) is a frame for R + ). We set S 0 = f S : sup ˆf R + 0 which is dense in R + ). et f be in S 0. Applying Parseval s formula, we obtain f,ψ l, j,k = k Z + ˆf, ˆψ l, j,k = k Z + = k Z + p j/ p j/ k Z + = p j/ = p j/ ˆfξ) ˆψ R + l p j ξ)χk, p j ξ) ˆfp j ξ) ˆψ R + l ξ)χk,ξ) l ξ s) Fjξ) l, 0) where Fjξ)= l l ξ s), l = 1,,...,.

7 NTMSCI 6, No., ) / Moreover, we have Fj l ξ) = = l ξ s) ˆfp j ξ s)) ˆψ l ξ s) s Z + s Z + ˆfp j ξ s)) ˆψ l ξ s) = ˆfp j ξ) ˆψ R + l ξ) ˆfp j ξ) ) 1 ) 1 R + R + ˆψ l ξ). Since both f and ˆf are compactly supported and each ψl Fj lξ) ). Also, for any j Z, we have lies in R + ), it follows from the above inequality that and By applying the Plancherel formula, we get Hence equation0) becomes where Fjξ) l ˆfp j ξ s)) ˆψ l ξ s), l = 1,,...,. ˆfp j ξ) ˆψ R + l ξ)χk,ξ) = k Z + f,ψ l, j,k = k Z + p j/ = p j/ = = Fj l ξ)χk,ξ) = F l jξ)χk,ξ). Fj l ξ). l ξ s) l ξ s) l ξ s) p R j/ ˆfp j ξ) ˆψ + l ξ) l ξ s) R + ˆfξ) ˆψ l p j ξ) + R f), 1) R f)= ˆfξ) ˆψ R + l p j ξ) ˆfξ p j s) ˆψ l p j ξ s). For given, there is a unique pairk,m) with k Z + and m Z + \pz +, such that s= p k m. Since R f) is absolutely convergent, we can estimate R f) by rearranging the series, changing the order of summation and integration by the evi

8 18 A. Abdullah and A. Afroz: Semi-orthogonal wavelet frames on positive half-line using the Walsh Fourier... lemma as follows R f)= ˆfξ) ˆψ R + l p j ξ) = ˆfξ) R + k Z + = ˆfξ) R + = ˆfξ) R + = ˆfξ p j s) ˆψ l p j ξ s) m Z + \pz + ˆψ l p j ξ) ˆfξ p j p k m) ˆψ l p j ξ p k m) k Z + m Z + \pz + m Z + \pz + Using the above estimate of R f) in 1), we obtain f,ψ l, j,k = ˆfξ) k Z + R + + From 16), 19) and ), we have ˆfξ p j m) m Z + \pz + ˆψ l p j+k ξ) ˆfξ p j m) ˆψ l p j+k ξ p k m) k Z + ˆψ l p j+k ξ) ˆψ l pk p j ξ m)) R + ˆfξ) ˆfξ p j m) k Z + ˆψ l p j+k ξ) ˆψ l pk p j ξ m)). m Z + \pz + A/B f ˆψ l p j ξ) R + ˆfξ) ˆfξ p j m) k Z + f,ψ l, j,k B/A f. k Z + ˆψ l p j+k ξ) ˆψ l pk p j ξ m)). ) This means that the systemψ l, j,k : 1 l, j Z,k Z+ is a frame for R + ). Hence, we get the desired result. Competing interests The authors declare that they have no competing interests. Authors contributions All authors have contributed to all parts of the article. All authors read and approved the final manuscript. References [1] Abdullah, On the characterization of non-uniform wavelet sets on positive half line, J. Info. Comp. Sc., 10 1) 015), [] Abdullah, Characterization of p-wavelets on positive half line using the Walsh-Fourier transform, Int. J. Anal. Appl., 10 ) 016), [3] M. Bownik, On characterizations of multiwavelets in R n ), Proc. Amer. Math. Soc., ), [4] A. Calogero, A characterization of wavelets on general lattices, J. Geom. Anal., ), [5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 015. [6] I. Daubechies, A. Grossmann, Y. Meyer, Painless non-orthogonal expansions, J. Math. Phys., 7 5) 1986),

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