Approximation by rational functions, and an analogue of the M. Riesz theorem on conjugate functions for $L^p$-spaces with $p\in(0,1)$

In this paper the solution to some problems concerning rational approximation in the $L^p$-metric ($p\in(0,1)$) is given. The following is a typical problem: to describe the closure in the space $L^p[-1,1]$ of the linear hull of the Cauchy family $\{1/(x-a)\}_{a\in[-1,1]}.$ In the paper it is shown that this closure consists of all functions $f\in L^p[-1,1]$ for which there exists a functon $\tilde f$, analytic in $\mathbf C\setminus[-1,1]$, decreasing to zero at infinity, and such that $f(x)=\lim_{y\to0+}\tilde f(x+iy)=\lim_{y\to0+}\tilde f(x-iy)$ for almost all $x\in[-1,1]$.
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