arXiv subjects
Publications and source records attributed to M. Skopina.
Wavelet systems on the generalized Vilenkin groups are considered. An algorithmic method for the construction of orthogonal wavelet bases is presented. These bases consist of compactly supported test functions (i.e. functions whose Fourier transform is also compactly supported).
Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support.
Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions $φ_j$ and a sequence of distributions/functions $\widetildeφ_j$. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for $φ_j$ and the compatibility conditions for $φ_j$ and $\widetildeφ_j$. These estimates are given in terms of the Fourier coefficients of approximated functions and provide analogs of some known non-periodic results. Under some additional assumptions error estimates are given in other terms in particular using the best approximation. A number of examples are provided.
Approximation properties of multivariate Kantorovich-Kotelnikov type operators generated by different band-limited functions are studied. In particular, a wide class of functions with discontinuous Fourier transform is considered. The $L_p$-rate of convergence for these operators is given in terms of the classical moduli of smoothness. Several examples of the Kantorovich-Kotelnikov operators generated by the ${\rm sinc}$-function and its linear combinations are provided.
Differential and falsified sampling expansions $\sum_{k\in \mathbb{Z}^d}c_kϕ(M^jx+k)$, where $M$ is a matrix dilation, are studied. In the case of differential expansions, $c_k=Lf(M^{-j}\cdot)(-k)$, where $L$ is an appropriate differential operator. For a large class of functions $ϕ$, the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of $ϕ$ from this class is required. In the present paper, we obtain similar results for a class of band-limited functions $ϕ$ with the discontinuous Fourier transform. In the case of falsified expansions, $c_k$ is the mathematical expectation of random integral average of a signal $f$ near the point $M^{-j}k$. To estimate the approximation order of the falsified sampling expansions we compare them with the differential expansions. Error estimations in $L_p$-norm are given in terms of the Fourier transform of $f$.
Approximation properties of the expansions $\sum_{k\in{\mathbb z}^d}c_kϕ(M^jx+k)$, where $M$ is a matrix dilation, $c_k$ is either the sampled value of a signal $f$ at $M^{-j}k$ or the integral average of $f$ near $M^{-j}k$ (falsified sampled value), are studied. Error estimations in $L_p$-norm, $2\le p\le\infty$, are given in terms of the Fourier transform of $f$. The approximation order depends on how smooth is $f$, on the order of Strang-Fix condition for $ϕ$ and on $M$. Some special properties of $ϕ$ are required. To estimate the approximation order of falsified sampling expansions we compare them with a differential expansions $\sum_{k\in\,{\mathbb z}^d} Lf(M^{-j}\cdot)(-k)ϕ(M^jx+k)$, where $L$ is an appropriate differential operator. Some concrete functions $ϕ$ applicable for implementations are constructed. In particular, compactly supported splines and band-limited functions can be taken as $ϕ$. Some of these functions provide expansions interpolating a signal at the points $M^{-j}k$.
Ordinary and partial differential equation for unknown functions defined on the Cantor dyadic group are studied. We consider two types of equations: related to the Gibbs derivatives and to the fractional modified Gibbs derivatives (or pseudo differential-operators). We find solutions in classes of distributions and study under what assumptions these solutions are regular functions with some "good" properties.
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 of view, the additive group of $p$-adic numbers looks very similar to the Vilenkin group where analogs of the Daubechies wavelets (and even band-limited ones) do exist. We note that all $p$-adic wavelet bases and frames appeared in the literature consist of Schwartz-Bruhat functions (i.e., band-limited and compactly supported ones).
The concept of $p$-adic quincunx Haar MRA was introduced and studied in~\cite{KS10}. In contrast to the real setting, infinitely many different wavelet bases are generated by a $p$-adic MRA. We give an explicit description for all wavelet functions corresponding to the quincunx Haar MRA. Each one generates an orthogonal basis, one of them was presented in~\cite{KS10}. A connection between quincunx Haar bases and two-dimensional separable Haar MRA is also found.
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 study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame.
We described a wide class of $p$-adic refinable equations generating $p$-adic multiresolution analysis. A method for the construction of $p$-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example, which gives a new 3-adic wavelet basis. Another realization leads to the $p$-adic Haar bases which were known before.
The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We discuss a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of $p$ characteristic functions of mutually disjoint discs of radius $p^{-1}$. This refinement equation generates a MRA. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are described. We also constructed multidimensional 2-adic Haar orthonormal bases for ${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our bases in applications.
We proved that for any matrix dilation and for any positive integer $n$, there exists a compactly supported tight wavelet frame with approximation order $n$. Explicit methods for construction of dual and tight wavelet frames with a given number of vanishing moments are suggested.
In this paper, the notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We use a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is the sum of $p$ characteristic functions of disjoint discs of radius $p^{-1}$. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is periodic with period 1, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are constructed. Since $p$-adic pseudo-differential operators are closely related to wavelet-type bases, our bases can be intensively used for applications.