For Vilenkin group only the existence of multiwavelets associated with multiresolution analysis (MRA) is known. In this paper, we have shown that by using wavelet sets we can also construct single wavelet in case of Vilenkin group which are not associated with MRA.We have given characterization of single and multi-wavelet sets on Vilenkin group. Further, we have studied generalized scaling sets, some of their properties and relation between wavelet sets and generalized scaling sets.
We consider a class of $(1,M)$-elementary step functions on the $p$-adic Vilenkin group. We prove that $(1,M)$-elementary step function generates a MRA on $p$-adic Vilenkin group iff it is generated by a rooted tree on the set of vertices $\{0,1,\dots p-1\}$ with 0 as a root. Bibliography: 14 titles.
We consider a class of $(N,M)$-elementary step functions on the $p$-adic Vilenkin group. We prove that $(N,M)$-elementary step function generates a MRA on $p$-adic Vilenkin group iff it is generated by a special $N$-valid rooted tree on the set of vertices $\{0,1,\dots p-1\}$ with the vector $(0,...,0)\in \mathbb Z^N$ as a root. Bibliography: 15 titles.
Vilenkin groups, introduced by F. Ya Vilenkin, form a class of locally compact abelian groups. The present paper consists of the characterization of Parseval frame multiwavelets associated to multiresolution analysis (MRA) in the Vilenkin group. Further, we introduce the pseudo-scaling function along with a class of generalized low pass filters and study their properties in Vilenkin group.
A variety of different orthogonal wavelet bases has been found in L_2(R) for the last three decades. It appeared that similar constructions also exist for functions defined on some other algebraic structures, such as the Cantor and Vilenkin groups and local fields of positive characteristic. In the present paper we show that the situation is quite different for the field of $p$-adic numbers. Namely, it is proved that any orthogonal wavelet basis consisting of band-limited (periodic) functions is a modification of Haar basis. This is a little bit unexpected because from the wavelet theory point…
In this paper, Meyer wavelets with an arbitrary integer scaling factor $N>2$ are defined using wavelets with multiple scaling factors $MN>2$. Expressions for frequency functions of wavelets and corresponding filters are obtained.
In this paper a simple method of construction of scaling function $ϕ(x)$ and orthogonal wavelets with the compact support for any natural coefficient of scaling $N\ge 2$ is given. Examples of construction of wavelets for coefficients of scaling N=2 and N=3 are produced.
In this paper it is shown, that B-splines are scaling functions for any natural N. For any natural N construction of Haar wavelets, Kotelnikov-Shannon wavelets, nonorthogonal wavelets based on B-splines is given. Examples of construction of filters of N-channel decomposition of signal and filters of reconstruction based on B-splines are given
A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\bR^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the asso…
Two scaling functions $φ_A$ and $φ_B$ for Parseval frame wavelets are algebraically isomorphic, $φ_A \simeq φ_B$, if they have matching solutions to their (reduced) isomorphic systems of equations. Let $A$ and $B$ be $d\times d$ and $s\times s$ \thematrix matrices with $d, s\geq 1$ respectively and let $φ_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. There always exists a scaling function $φ_B$ associated with matrix $B$ such that \begin{equation*} φ_B \simeq φ_A. \end{equation*} An example shows that the assumption on the finiteness of the solutions c…
In this article we establish theory of semi-orthogonal Parseval wavelets associated to generalized multiresolution analysis (GMRA) for the local field of positive characteristics (LFPC). By employing the properties of translation invariant spaces on the core space of GMRA we obtain a characterization of semi-orthogonal Parseval wavelets in terms of consistency equation for LFPC. As a consequence, we obtain a characterization of an orthonormal (multi)wavelet to be associated with an MRA in terms of multiplicity function as well as dimension function of a (multi)wavelet. Further, we provide char…
In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vallée Poussin type. These scaling functions and their corresponding wavelets are determined by their Fourier coefficients, which are sample values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the one-dimensional de la Vallée Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the two-scale relation of two spaces of …
Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\em{diction…
A generalization of Mallat's classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed (J. Funct. Anal. 158, 209-241, 1998). In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum $Λ=\left\{0, r/N\right\}+2\mathbb Z$, where $N\ge 1$ is an integer and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear …
Explicit formulae are given for a type of Battle-Lemarié scaling functions and related wavelets. Compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to Nikolskii-Besov spaces on R.
We use Lorentz polynomials to present the solutions explicitly of equations (6.1.7) of [I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992] and (4.9) of [I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988), no. 7, 909--996] sot that we give an efficient way to prove Daubechies' results on the existence of spline type orthogonal scaling functions and to evaluate Daubechies scaling functions.
The spectral properties of the Ruelle transfer operator which arises from a given polynomial wavelet filter are related to the convergence question for the cascade algorithm for approximation of the corresponding wavelet scaling function.
Using the Daubechies conditions of compact support, orthogonal, and regularity, we were able to derive bivariate scaling functions with which to reproduce linear functions (planes). We describe how to create all possible masks of refinement coefficients that satisfy those conditions.
Let $d\geq 1$ be a natural number and $A_0$ be a $d\times d$ expansive integral matrix with determinant $\pm 2.$ Then $A_0$ is integrally similar to an integral matrix $A$ with certain additional properties. A finite solution to the system of equations associated with the matrix $A$ will result in an iterated sequence $\{Ψ^k χ_{[0,1)^d}\}$ that converges to a function $φ_A$ in $L^2(\R^d)$-norm. With this (scaling) function $φ_A,$ we will construct the Parseval's wavelet function $ψ$ with compact support associated with matrix $A_0.$
Modulation spaces $M^s_{p,q}$ were introduced by Feichtinger \cite{Fei83} in 1983. By resorting to the wavelet basis, Bényi and Oh \cite{BeOh20} defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possible applications in PDE. By carefully investigating the scaling properties of modulation spaces and their connections with the wavelet basis, we will introduce a class of generalized modulation spaces, which contain both Feichtinger's and Bényi and Oh's modulation spaces. As their applications, we will give a local well-posedness a…
Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For invertible expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with t…
We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame.
We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in $\R^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial functions in $\R^3$. We provide a simple way to construct a radial scaling function and a radial wavelet from an even classical scaling function on $\R$. Furthermore, decomposition and reconstruction algorithms are formulated.
