Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

Introduction The theory of wavelet transforms have emanated as a broadly used tool in various disciplines of science and engineering including image processing, spectrometry, turbulence, computer graphics, optics and electromagnetism, telecommunications, DNA sequence analysis, quantum physics, solution of differential equations. In context of signal processing, it has been assumed that orthogonality is the key property for synthesis and analysing signals. In order to study a higher-level signal processing, biorthogonality plays a vital role in which two sets are incorporated: one serves for the analysis and the other one for synthesis. Towars the culminating years of 1990’s, biorthogonal wavelets are considered as cornerstone technique in image compression due to their natural feature of concentrating energy in a few transform coefficients and advantageous over orthogonal wavelets, by relaxing orthonormal to biorthogonal, additional degrees of freedom are added to design problems. Biorthogonal wavelets in L(R) were investigated by Bownik and Garrigos [1], Cohen et al. [2], Chui and Wang [3]. The numerical aspect of biorthogonal wavelets were studied by Karoui and Vaillancourt [4]. Multiresolution analysis is the heart of wavelet analysis as it gives a general framework for analysing wavelet systems. All the signals in real life applications are not obtained from the uniform shifts. For the analysis and decomposition of these signals by means of stable mathematical technique, Gabardo and Nashed [5] introduced a notion of nonuniform Abstract

Page 2 of 17 Ahmad et al. J Egypt Math Soc (2021) 29:19 MRA where the translation set acting on the scaling function associated with the MRA to generate the subspace V 0 is no longer a group, but is the union of Z and a translate of Z . Shah and Abdullah [6] established NUMRA on non-Archimedean local fields.
In the recent years, the development of wavelet theory in the context of Walsh analysis have been extensively studied by many authors including Farkov [7], Meenakshi [8] but still more concepts need to be studied for its enhancement. Recently, Ahmad and his collaborators investigated wavelet frames [9], nonuniform wavelet frames [10][11][12][13][14], Nonuniform p-tight wavelet frames [15], Tight Framelets [16,17], wavepacket systems [18,19], Frames associated with shift invariant spaces [20], Gabor frames [21], numerical study of wavelets [22][23][24][25][26][27][28][29][30] and obtained many interested results. Continuing our research on wavelet and wavelet frames, we in this paper introduce the notion of nonuniform biorthogonal wavelets in L 2 (R + ) . We obtain the characterization for the translates of a single function to form the Riesz bases for their closed linear span. We also provide a complete characterization for the biorthogonality of the translates of scaling functions of two NUMRA's and the associated biorthogonal wavelet families. Moreover, under mild assumptions on the scaling functions and the corresponding wavelets, we show that the nonuniform wavelets can generate Reisz bases for L 2 (R + ).
The article is structured as follows. In Section "Methods", we recall methods of Fourier analysis on positive half line including basic definitions of MRA and NUMRA . In Section "Results and discussion", we establish necessary and sufficient conditions for the translates of a function to form a Riesz basis for its closed linear span. Furthermore, we show that the wavelets associated with dual MRA's are biorthogonal and generate Riesz bases for L 2 (R + ).

Methods
Let R + , Z + and N respectively denotes the set of nonnegative real numbers, set of non-negative integers and the set of natural numbers. By the symbols [x] and {x} , we mean the integer and fractional part of x respectively. Let p > 1 be a fixed natural number. For ξ ∈ R + and any integer j > 0 , we set where ξ j , ξ −j ∈ {0, 1, . . . , p − 1} . Clearly, ξ j and ξ −j are the digits in the p-ary expansion of ξ: The first sum on the right is always finite and On R + the addition is defined in the following manner: . Let ε p = exp(2π i/p) , we define a function s 0 (x) on [0, 1) by The system of generalized Walsh functions w m (ξ ) : m ∈ Z + on [0, 1) is defined in the following way: These functions form a complete orthogonal system. A finite linear combination of Walsh functions is known as Walsh polynomial. For ξ , η ∈ R + , let where ξ j , η j are defined as in (2.1). It is easy to see that and where ξ , η, δ ∈ R + and ξ ⊕ η is p-adic irrational. It is well known that systems {χ(α, .)} ∞ α=0 and {χ(·, α)} ∞ α=0 form an orthonormal bases in L 2 [0,1] (See [31,32]). For a function φ ∈ L 1 (R + ) ∩ L 2 (R + ) , the Walsh-Fourier transform is defined as , Ff = f , extends uniquely to the whole space L 2 (R + ) . The Walsh-Fourier transform enjoys similar properties to those of the classic Fourier transform [31][32][33][34]. In particular, if φ ∈ L 2 (R + ) , then φ ∈ L 2 (R + ) and Furthermore, if φ ∈ L 2 [0, 1] , then we can define the Walsh-Fourier coefficients of φ as The series n∈Z + φ(n)w n (ξ ) is called the Walsh-Fourier series of φ . From the standard L 2 -theory, we can observe that the Walsh-Fourier series of φ converges to φ in L 2 [0, 1] and Parseval's identity holds: By p-adic interval I ⊂ R + of range n, we mean intervals of the form Each of these p-adic intervals is both closed and open under the p-adic topology which is generated by the collection of p-adic intervals [31]. The collection [0, p −j ) : j ∈ Z forms a fundamental system of the p-adic topology on R + . Therefore, the generalized Walsh functions w j (ξ ), 0 ≤ j ≤ p n − 1 , assume constant values on each of p-adic interval I ℓ n and hence continuous on these intervals. Thus, w j (ξ ) = 1 for ξ ∈ I 0 n . Let E n (R + ) denotes the space of p-adic entire functions of order n. Thus, for every φ ∈ E n (R + ) , we have It is clear that E n (R + ) contains each Walsh function of order up to p n−1 . The set E(R + ) of p-adic entire functions on R + is the union of all the spaces E n (R + ) and is dense in L p (R + ), 1 ≤ p < ∞ and each function in E(R + ) is of compact support. Thus, we consider the following set of functions Let N ≥ 1 be a given integer and r be an odd integer which are relatively prime such that 1 ≤ r ≤ N − 1 , we consider the translation set + as It can be easily seen that the translation set + is not necessarily a group nor a uniform discrete set. The set + n is the union of Z and a translate of Z . Furthermore, the translation set + is the spectrum for the spectral set Ŵ N = 0, 1 2 ∪ N 2 , N +1 2 and the pair (� + , Ŵ N ) is called a spectral pair [5]. Definition 2.1 Let N ≥ 1 be a given integer and r be an odd integer which are relatively prime such that 1 ≤ r ≤ N − 1 , an associated nonuniform MRA is a sequence of closed subspaces V j : j ∈ Z of L 2 (R + ) satisfying the following properties: It should be noted that the definition of dyadic dilation multiresolution analysis in one dimension can be deduced from the above definitio when N = 1 . For N > 1 , the dilation factor of N corroborates that N + ⊂ Z + ⊂ + .
For every j ∈ Z , define W j as the orthogonal complement of V j in V j+1 . Thus we can write Therefore, it implies that for j > M, By invoking Definition 2.2. (b), this follows that a decomposition of L 2 (R + ) into mutually orthogonal subspaces.
There exists N − 1 functions whose translated and dilated family form an orthonormal basis for L 2 (R + ).

Definition 2.3 A set {ψ
is said to be a set of basic wavelets associated with the nonuniform multiresolution analysis V j : j ∈ Z if the family of functions ψ ℓ (x ⊖ σ ) : 1 ≤ ℓ ≤ N − 1, σ ∈ � + forms an orthonormal basis for W 0 . Using the fact that χ(γ , ζ ) : γ ∈ � + is an orthonormal basis of L 2 0, 1 2 , we obtain the desired result. Now we proceed to establish a sufficient condition for the translates of a function to be linearly independent.

Results and Discussion
Lemma 3.2 Let φ ∈ L 2 (R + ) . Suppose there exists two constants C, D > 0 such that Proof For the proof of the lemma, it is sufficient to find another function say φ whose translates are biorthogonal to φ . To do this, we define the function φ by Equation (3.1) implies that φ is well defined and Applying Lemma 3.1, it follows that the set φ(x ⊖ σ ) : σ ∈ � + is linearly independent. Thus the proof is completed .

Remark 3.7
The operators P j and P j satisfy the following properties.