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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.
Received: 20.02.1973
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
Linking options:
https://www.mathnet.ru/eng/mzm7376 https://www.mathnet.ru/eng/mzm/v15/i4/p533
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Abstract page: | 164 | Full-text PDF : | 76 | First page: | 1 |
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