Approximation by simple partial fractions and the Hilbert transformстатья
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Дата последнего поиска статьи во внешних источниках: 18 июля 2013 г.
Аннотация:We study the problem of approximation of functions in Lp
by simple partial fractions on the real axis and semi-axis. A simple partial
fraction is a rational function of the form g(t) =
\sum_{k=1}^n \frac{1}{t-z_k}, where
z_1, . . . , z_n are complex numbers. We describe the set of functions that can
be approximated by simple partial fractions within any accuracy and the
set of functions that can be approximated by convex combinations of them
(the cone of simple partial fractions). We obtain estimates for the norms
of simple partial fractions and conditions for the convergence of function
series \sum_{k}\frac{1}{t-z_k}
in the space L_p. Our approach is based on the use of the
Hilbert transform and the methods of convex analysis.