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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1974, Volume 15, Issue 4, Pages 533–542 (Mi mz7376)

The approximation of Ñauchy 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.

Full text: PDF file (652 kB)

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

Bibliographic databases:

UDC: 517.5

Citation: D. G. Sanikidze, “The approximation of Ñauchy singular integrals and their limiting values at the endpoints of the curve of integration”, Mat. Zametki, 15:4 (1974), 533–542; Math. Notes, 15:4 (1974), 313–318

Citation in format AMSBIB
\Bibitem{San74} \by D.~G.~Sanikidze \paper The approximation of Ñauchy singular integrals and their limiting values at the endpoints of the curve of integration \jour Mat. Zametki \yr 1974 \vol 15 \issue 4 \pages 533--542 \mathnet{http://mi.mathnet.ru/mz7376} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=361642} \zmath{https://zbmath.org/?q=an:0296.44010} \transl \jour Math. Notes \yr 1974 \vol 15 \issue 4 \pages 313--318 \crossref{https://doi.org/10.1007/BF01095120}