Ukrainian Mathematical Bulletin

Volume 4, Issue 1, 2007 pp. 1-19.

Approximation in the Mean by Matrix-values Polynomials on Rectifiable Curves

Authors: Lutz Klotz, Sergei M. Zagorodnyuk
Author institution: Lutz Klotz, Fachbereich Mathematik Universit"at, D-7010 Leipzig, Deutschland Sergei Mikhailovich Zagorodnyuk, Department of Mechanics and Mathematics, Karazin National Kharkov University, pl. Svobody 4, Kharkov, 61077, Ukraine zagorodnyuk@univer.kharkov.ua,

Summary: In this paper, we study problems on approximation with matrix-valued polynomials in the spaces $L^p(M),\ p\in(0,\infty),$ on Jordan curves. For full rank measures $M$, we obtain necessary and sufficient conditions for the closure, $L^p_0(M)$, of the set of matrix-valued polynomials to be dense in $L^p(M)$. We show that the equality $L^p_0(M)=L^p(M)$ does not depend on the singular part of the measure $M$. In the case where $L^p_0(M)\not=L^p(M)$, we solve the problem of describing the space $L^p_0(M)$ for full rank measures. We introduce a notion of a linearly regular space $L^p(M)$ and obtain sufficient conditions for $L^p(M)$ to be linearly regular. For a nonclosed Jordan curve, which could be not rectifiable, we show that the equality $L^p_0(M)=L^p(M)$ always holds.

Contents Full-Text PDF