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A stable Nyström interpolant for some Mellin convolution equations

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Abstract

We consider the numerical solution of second kind integral equations of the form

$$u(y) - \int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx = f(y), 0 \le y \le 1,} $$

for some given kernelk(t). These equations, usually indicated as of Mellin type, arise in a variety of applications.

In particular, we examine a Nyström interpolant based on the following product quadrature rule:

$$\int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx \approx \sum\limits_{i = 0}^n {w_{ni} (y)u(x_{mi} ).} } $$

This rule is obtained by interpolatingu(x) by the Lagrange polynomial associated with the set of Gauss-Radau nodes {x ni}.

Under certain assumptions on the kernelk(t), we are able to prove the stability of our interpolant and derive convergence estimates.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, I-10129, Torino, Italy

    Giovanni Monegato

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  1. Giovanni Monegato
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Work performed under the auspices of the Ministero dell'Università e della Ricerca Scientifica e Tecnologica of Italy.

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Monegato, G. A stable Nyström interpolant for some Mellin convolution equations. Numer Algor 11, 271–283 (1996). https://doi.org/10.1007/BF02142502

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