Ilya A. Krishtal

Some simple Haar-type wavelets in higher dimensions

AbstractAn orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψj,kℓ}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form

Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

\psi _{j,k}^\ell (x) = |\det a|^{ - j/2} \psi ^\ell (a^{ - j} x - k) \equiv (D_a jT_k \psi ^\ell )(x)

Invertibility of the Gabor frame operator on the Wiener amalgam space

that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L = d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if a = (1-11 1) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having ¦det b¦ = 1.

Matricial filters and crystallographic composite dilation wavelets

A useful sampling-reconstruction model should be stable with respect to different kind of small perturbations, regardless whether they result from jitter, measurement errors, or simply from a small change in the model assumptions. In this paper we prove this result for a large class of sampling models. We define different classes of perturbations and present a way of quantifying the robustness of a model with respect to them. We also use the theory of localized frames to study the dual frame method for recovering the original signal from its samples.

Multi-window Gabor frames in amalgam spaces

We consider the problem of spatiotemporal sampling in which an initial state f of an evolution process ft = Atf is to be recovered from a combined set of coarse samples from varying time levels {t1, . . . , tN}. This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time ti there are not enough samples to recover the function f or the state fti . Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which f is to be recovered from AT f for a single time T . In this paper, we consider signals that are modeled by `(Z) or a shift invariant space V ⊂ L(R).

Spectral analysis of a differential operator with an involution

Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal-decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QR factorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question of when and under what conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU, QR, Cholesky, and polar factorizations.

Finite Dimensional Dynamical Sampling: An Overview

We use a generalization of Wieners 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space W(L^~,@?^1)(R^d), the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the canonical dual belongs also to W(L^~,@?^1)(R^d).

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

In [7, 8] Guo, Labate, Lim, Weiss, and Wilson introduced the theory of MRA composite dilation wavelets. We continue their work by studying the filter properties of such wavelets and present several important examples.