arXiv subjects

V. I. Danchenko

Publications and source records attributed to V. I. Danchenko.

3 recordsLinked to original sources

arXiv2017

Extremal and approximation properties of simple partial fractions

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPF) as suggested by Eu.P. Dolzhenko. Many solved and unsolved extremal problems related to SPF are traced back to works of G. Boole, A.J. Macintyre, W.H.J. Fuchs, J.M. Marstrand, E.A. Gorin, A.A. Gonchar, Eu.P. Dolzhenko. At present, many authors systematically develop methods for approximation and interpolation by SPF and several their modifications. Simultaneously, related problems, being of independent interest, arise for SPF: inequalities of different metrics, estimation of derivatives, separation of singularities, etc. We systematize some of these problems which are known to us in Introduction of this survey. In the main part, we formulate principal results and outline methods to prove them if possible.

math.CA↗

arXiv2016

Extraction of harmonics from trigonometric polynomials by amplitude and phase operators

Extraction of harmonics of a given order from real trigonometric polynomials (signals) is one of the main problems in harmonic analysis. It has many applications in physics, radio and electrical engineering, in particular, in filtration of harmonic signals of different nature. There exist many methods (mainly, approximative) for solution of this problem. The most common ones are spectral methods based on Fourier transform and other resonance principles. In this paper we propose a new method for extracting harmonics by amplitude and phase transformation of trigonometric polynomials. These transformations use the two simplest operations -- multiplication by a real constant and phase shift -- to obtain polynomials similar to the initial ones. A harmonic is extracted by an amplitude and phase operator that simply overlays (sums up) a finite number of such similar polynomials. The overlay method enables us to obtain precise analytical formulas for calculating harmonics of a given order.

math.CA↗

arXiv2015

Sharp quadrature formulas and Nikol'skii type inequalities for rational functions

Sharp quadrature formulas for integrals of complex rational functions on circles, real axis and its segments are obtained. We also find sharp quadrature formulas for calculation of $L_2$-norms of rational functions on such sets. Basing on quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive sharp inequalities for different metrics (Nikol'skii type inequalities).

math.CA↗