Lasha Ephremidze

A New Method of Matrix Spectral Factorization

A new algorithm of matrix spectral factorization is proposed which can be applied to compute an approximate spectral factor of any positive definite matrix function which satisfies the Paley-Wiener condition.

An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem

A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.

A new efficient matrix spectral factorization algorithm

An absolutely new method of matrix spectral factorization is proposed which leads to the most simple computational algorithm. A demo version of the software implementation is located at www.ncst.org.ge/MSF-algorithm.

On the uniqueness of maximal functions

The uniqueness theorem for the one-sided maximal operator has been proved.

A remark on a polynomial matrix factorization theorem

Abstract. The Wiener–Hopf factorization of trigonometric polynomial matrix functions is considered and it is proved that the factors are matrices of trigonometric polynomials as well. The natural bounds for partial indices and for orders of the factors are obtained.

Numerical comparison of different algorithms for construction of wavelet matrices

Wavelets have found beneficial applicability in various aspects of wireless communication systems design, including channel modeling, transceiver design, data representation, data compression, source and channel coding, interference mitigation, signal denoising and energy efficient networking. Factorization of compact wavelet matrices into primitive ones has been known for more than 20 years. This method makes it possible to generate wavelet matrix coefficients and also to specify them by their first row. Recently, a new parametrization of compact wavelet matrices of the same order and degree has been introduced by the last author. This method also enables us to fulfill the above mentioned tasks of matrix constructions. In the present paper, we briefly describe the corresponding algorithms based on two different methods, and numerically compare their performance.

An approximation of Daubechies wavelet matrices by perfect reconstruction filter banks with rational coefficients

It is described how the coefficients of Daubechies wavelet matrices can be approximated by rational numbers in such a way that the perfect reconstruction property of the filter bank be preserved exactly.

On a relationship between the integrabilities of various maximal functions

It is shown that the right-sided, left-sided, and symmetric maximal functions of any measurable function can be integrable only simultaneously. The analogous statement is proved for the ergodic maximal functions.

On a generalization of Smirnov’s theorem with some applications

Abstract We present a certain generalization of Smirnov’s theorem on functions from the Hardy spaces H p {H_{p}} . We provide some applications of the proposed generalization. Namely, we give an equivalent characterization of outer analytic rectangular matrix functions, and give a simple proof of the uniqueness of spectral factorization of rank deficient matrices.

The Riesz "rising sun" lemma for arbitrary Borel measures with some applications

The Riesz “rising sun” lemma is proved for arbitrary locally finite Borel measures on the real line. The result is applied to study an attainability problem of the exact constant in a weak (1,1) type inequality for the corresponding Hardy-Littlewood maximal operator.