Vector-valued nonuniform multiresolution analysis related to Walsh function

In this paper, we introduce vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh functions. We obtain the necessary and sufficient condition for the existence of associated wavelets.


Introduction
The idea of multiresolution analysis has become one of the main tools for constructing wavelets. In his paper, Mallat [15] first formulated the remarkable idea of multiresolution analysis (MRA) that deals with a general formlism for construction of an orthogonal basis of wavelet bases. A multiresolution analysis consists of a sequence of embedded closed subspaces {V j : j ∈ Z} for approximating L 2 (R) functions such that ∩ j∈Z V j = {0} , ∪ j∈Z V j is dense in L 2 (R) and which satisfies f (x) ∈ V j if and only if f (2x) ∈ V j+1 . Furthermore, there exist an element φ ∈ V 0 such that the collection of integer translates of φ, {φ(x − n)} n∈Z is a complete orthonormal system for V 0 . Any compactly supported wavelet must come from a MRA. The function φ ∈ V 0 is called a scaling function.
The idea of wavelet and multiresolution analysis has been extended to many different setups. One can replace the dilation factor 2 by an integer N ≥ 2 and one needs to construct N − 1 wavelets to generate the whole space L 2 (R). In general, in higher dimensions, it can be replaced by a dilation matrix A, in which case the number of wavelets required is |detA| − 1. But in all these cases, the translation set is always a group. In the two papers [7], [8], Gabardo and Nashed considered a generalization of Mallat's [15] celebrated theory of MRA, in which 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. More precisely, this set is of the form {0, r/N } + 2Z, where N ≥ 1 is an integer, 1 ≤ r ≤ 2N − 1, r is an odd integer relatively prime to N . They call this a nonuniform multiresolution analysis (NUMRA) and is based on the theory of spectral pairs. The idea of dyadic multiresolution analysis has been studied on L 2 (R + ), where R + denotes the positive halfreal line by Protasov and Farkov. Farkov [5] has given a general construction of compactly supported orthogonal p-wavelets in L 2 (R + ). Farkov et al. [6] gave an algorithm for biorthogonal wavelets related to Walsh functions on positive half line. Protasov and Farkov constructed examples of refinable functions that generate MRA on halfline and Vilenkin groups with the help of Cohen conditions in [18]. F. A. Shah [19], studied the construction 207 of p-wavelet packets associated with the multiresolution analysis defined by Farkov [5], for L 2 (R + ). Recently, Meenakshi et al. [17] studied NUMRA on positive half line and they also derived the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.
In [24], Xia and Suter, introduced vector-valued multiresolution analysis and vector-valued wavelets for vector-valued signal spaces. Sun and Cheng, introduced vector-valued multiresolution analysis with dilation factor α ≥ 2 and orthogonal vector-valued wavelets in [23]. The properties of vector-valued wavelets packets are presented in [3]. In [1], Abdullah, introduced vector-valued multiresolution analysis on local field of positive characteristic and obtained the necessary and sufficient condition for the existence of associated wavelets.
In the present paper, we study vector-valued nonuniform multiresolution analysis on positive half-line and we obtain the necessary and sufficient condition for the existence of associated wavelets.The paper is organized as follows. In Sec. 2, we discuss some preliminary facts about NUMRA on positive half-line. We introduce the notion of VNUMRA on positive half-line and find a necessary and sufficient condition for the existence of associated wavelets in Sec. 3.
Let N be an integer, N ≥ 1 and an odd integer r with 1 ≤ r ≤ 2N − 1, r and N are relatively prime, we consider the translation set Λ + r,N as Let [x] denotes integer part of x. For x ∈ R + and for any positive integer j, we set Non-uniform multiresolution analysis in L 2 (R + ) defined by Meenakshi et. al. is as follows: Definition 2.1. A nonuniform multiresolution analysis on positive half-line is a sequence of closed subspaces V j ⊂ L 2 (R + ), j ∈ Z such that the following properties hold: The function φ whose existence is asserted in (4) is called a scaling function of the given NUMRA.
When, N > 1, the dilation factor of N ensures that N Λ + r,N ⊂ Z + ⊂ Λ + r,N . Condition (3) and (4) implies that Since V 0 ⊂ V 1 , the function φ ∈ V 1 and has the Fourier expansion On taking the Walsh-Fourier transform, we havê where λ j and ξ j are calculated by (2.1) for a positive integer N .
where m 1 ℓ and m 2 ℓ are locally L 2 functions. In this case {ψ 1 , ψ 2 , ...., ψ N −1 } is a set of basic wavelets associated with a scaling function φ. It is easy to show } is an orthonormal basis of L 2 (R + ).

Vector-valued nonuniform multiresolution analysis on positive half-line
We denote the set of all vector-valued functions f(x) by L 2 (R + , C d ),i.e.
where T denotes the transpose and C d denotes the d-dimensional complex Euclidean space. For f ∈ L 2 (R + , C d ), ∥f∥ denotes the norm of the vector-valued function f and is defined as: For any two vector-valued functions f, g ∈ L 2 (R + , C d ), < f, g > denotes the integration of the matrix product f (x).g * (x) as: where * denotes the transpose and the complex conjugate and < f, g > is matrix valued not scalar valued but satisfied the inner product properties: Therefore, the operation < ., . > is called an inner product.
The vector-valued function Φ(x) whose existence is asserted in (5) is called a scaling function in L 2 (R + , C d ).
, satisfy the following refinement equation sequence that has only finite number of terms. Taking the Walsh-Fourier transform on both side of Equation (3.1), we havê where and where λ j and ξ j are calculated by (2.1).
where I d , denotes the identity matrix of order d × d On taking Walsh-Fourier transform on both sides of Eq. (3.8), we havẽ and where λ j and ξ j are given by (2.1) for a positive integer N .  12) where δ λ,λ ′ denotes the Kronecker's delta.
By Eq.(3.9), we have It proves the orthonormality of the system {Ψ k (x ⊖ λ) : λ ∈ Λ + r,N , k = 0, 1, 2, ..., N − 1}. Now, we have the following result on the existence of a vector-valued wavelet function N , k=0,1,2,...,N −1 is an orthonormal system in V 1 . Then this system is complete Proof. Since the system {Ψ k (x ⊖ λ)} λ∈Λ + r,N , k=0,1,2,...,N −1 is an orthonormal system in V 1 . By Lemma 3.4, we have We will now prove its completeness. where On the other hand This condition is equivalent to ∑ j∈Z +f Therefore, from Eqs. (3.3) and (3.17), we have As similar to the identity (3.13) in Lemma 3.4, we have Then, the identity (3.13) implies that for any ξ ∈ R + , the column vectors in the N d × d matrix H ′ (ξ) and the column vectors in the N d × d matrix G ′ k (ξ) are orthogonal for l = 0, 1, 2, ..., N − 1 and these vectors form orthogonal basis of N d-dimensional complex Eucledian space C N d .
The identity (3.18) implies that the column vectors in N d × d matrix P ′ k (ξ) and the column vectors Therefore, from Eqs. (3.9) and (3.17), we havê
If Ψ 0 , Ψ 1 , Ψ 2 , ..., Ψ N −1 ∈ V 1 are as in Theorem 3.5, one can obtain from them as orthonormal basis for L 2 (R + , C d ) by procedure for construction of wavelet from given MRA. It can be easily checked that for every j ∈ Z the collection N , k=0,1,2,...,N −1 , is a complete orthogonal system for V m+1 .

Conclusion
In this study, we introduced a new concept called vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh functions where associated subspace V 0 of L 2 (R + , C d ) has, an orthonormal basis, a collection of translates of vector-valued function Φ of the form {Φ(x ⊖ λ)} λ∈Λ + r,N . We Also obtained the necessary and sufficient condition for the existence of associated wavelets. Our results will be mostly used by that part of mathematical society who works in wavelet analysis and their applications.