Abstract:
We study the problem of approximation of functions in $L_p$
by simple partial fractions on the real axis and semi-axis.
A simple partial fraction is a rational function of the form
$g(t)=\sum_{k=1}^n\frac1{t-z_k}$, where $z_1,\dots,z_n$ 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
$\sum_{k=1}^\infty\frac1{t-z_k}$ in the space $L_p$.
Our approach is based on the use of the Hilbert transform
and the methods of convex analysis.
Keywords:approximation, simple partial fraction, convergence of function series, Hilbert transform, entire function, logarithmic derivative.
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