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Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type

D. V. Gorbachev, V. I. Ivanov

Tula State University

Abstract: Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type, are established. They generalize quadrature formulae involving zeros of Bessel functions, which were first designed by Frappier and Olivier. Bessel quadratures correspond to the Fourier-Hankel integral transform. Some other examples, connected with the Jacobi integral transform, Fourier series in Jacobi orthogonal polynomials and the general Sturm-Liouville problem with regular weight are also given.
Bibliography: 39 titles.

Keywords: Gauss and Markov quadrature formulae, entire function of exponential type, Sturm-Liouville problem, Jacobi transform, Jacobi functions and polynomials.

UDC: 517.518.87

MSC: Primary 41A55; Secondary 30D15, 34B25

Received: 31.07.2014 and 14.11.2014

DOI: 10.4213/sm8413

 English version: Sbornik: Mathematics, 2015, 206:8, 1087–1122

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