Maria Skopina

p-Adic nonorthogonal wavelet bases

A method for constructing MRA-based p-adic wavelet systems that form Riesz bases in L2(ℚp) is developed. The method is implemented for an infinite family of MRAs.

p-adic orthogonal wavelet bases

We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L2(ℚp).

Quincunx multiresolution analysis for L2(ℚ22)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.

SYMMETRIC MULTIVARIATE WAVELETS

For arbitrary matrix dilation M whose determinant is odd or equal to ±2, we describe all symmetric interpolatory masks generating dual compactly supported wavelet systems with vanishing moments up to arbitrary order n. For each such mask, we give explicit formulas for a dual refinable mask and for wavelet masks such that the corresponding wavelet functions are real and symmetric/antisymmetric. We proved that an interpolatory mask whose center of symmetry is different from the origin cannot generate wavelets with vanishing moments of order n > 0. For matrix dilations M with |det M| = 2, we also give an explicit method for construction of masks (non-interpolatory) m0 symmetric with respect to a semi-integer point and providing vanishing moments up to arbitrary order n. It is proved that for some matrix dilations (in particular, for the quincunx matrix) such a mask does not have a dual mask. Some of the constructed masks were successfully applied for signal processes.

Multivariate Wavelet Frames

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

Tight wavelet frames

n

2-Adic wavelet bases

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

Multivariate periodic wavelets

n

Wavelet frames on Vilenkin groups and their approximation properties

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

Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

holds for all f ∈ L2(R). The general scheme [1] for construction of compactly supported tight wavelet frames based on a multiresolution analysis looks as follows. Let a multiresolution analysis in L2(R) be generated by a compactly supported scaling function φ ∈ L2(R) satisfying the re nable equation φ(x) = m0(M∗x)φ(M∗x), where m0 is a trigonometric polynomial (re nable mask). For any trigonometric polynomial mν , there exists a unique set of trigonometric polynomials μνk, k = 0, . . . , m− 1, (polyphase components of mν) such that