The approximation of Сauchy singular integrals and their limiting values at the endpoints of the curve of integration

D. G. Sanikidze

Computational Center, Academy of Sciences of the Georgian SSR

Abstract: We examine a specific approximating process for the singular integral
$$ S^*(f;x)\equiv\frac1\pi\int_{-1}^{+1}\frac{f(t)}{\sqrt{1-t^2}(t-x)}\,dt\quad(-1<x<1), $$
taken in the principal value sense. We study the influence of some local properties of the function $f$ on the convergence of the approximations. Next, assuming that $S^*(f;c)=\lim\limits_{x\to c}S^*(f;x)$, where $c$ is an arbitrary one of the endpoints $-1$ and $1$, we show that the conditions which guarantee the existence of the limiting values $S^*(f;c)$ ($c=\pm1$) and, moreover, the convergence of the process at an arbitrary point $x\in(-1,1)$ are not always sufficient for convergence of the approximations at the endpoints.

UDC: 517.5

Received: 20.02.1973

 English version: Mathematical Notes, 1974, 15:4, 313–318

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