Aleksandr Krivoshein

Multivariate Wavelet Frames

We proved that for any matrix dilation and for any positive integer

Multivariate sampling-type approximation

n

A directional uncertainty principle for periodic functions

, there exists a compactly supported tight wavelet frame with approximation order

Smoothness of Wavelets

n

Construction of Wavelet Frames Generated by MEP

. Explicit methods for construction of dual and tight wavelet frames with a given number of vanishing moments are suggested.

Frame-Like Wavelet Expansions

Approximation properties of the expansions ∑k∈ℤdLf(M−j⋅)(−k)φ(Mjx + k), where L is a linear differential operator and M is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in Lp-norm, 2 ≤ p ≤∞, are given in terms of the Fourier transform of f. The approximation order depends on the smoothness of f, the order of L, the order of Strang–Fix condition for φ and M. A wide class of φ including both band-limited and compactly supported functions is considered, but a special condition of compatibility φ with L is required. Such differential expansions may be useful for engineers.

Approximation by frame-like wavelet systems ✩

In this paper we introduce a notion of a directional uncertainty product for multivariate periodic functions and multivariate discrete signals. It measures a localization of a signal along a particular direction. We study properties of the uncertainty product and give an example of well localized multivariate periodic Parseval wavelet frames.

On construction of multivariate symmetric MRA-based wavelets

The regularity of multivariate wavelet frames with an arbitrary dilation is studied by the matrix approach. The formulas for the Holder exponents in spaces C and \(L_p\) are obtained in terms of the joint spectral radius of the corresponding transition matrices. Some results on higher order regularity, on the local regularity, and on the asymptotics of the moduli of continuity in various spaces are presented.

Differential and Falsified Sampling Expansions

Sufficient conditions for a dual wavelet system to be a dual wavelet frame are studied. Algorithmic methods for the construction of tight and dual compactly supported wavelet frames, providing an arbitrary approximation order and other important features, are discussed.

Directional Heisenberg uncertainty product

A notion of frame-like wavelet systems is introduced and analyzed. Those systems are not dual wavelet frames although preserve important properties of frames. The construction of such systems is based on the matrix extension principle (MEP), but it is simpler than the construction of wavelet frames.