## A product integration rule for hypersingular integrals on $(0,+\infty)$

Maria Carmela De Bonis and Donatella Occorsio

### Abstract

In the present paper we propose a product integration rule for hypersingular integrals on the positive semi-axis. The rule is based on an approximation of the density function $f$ by a suitable truncated Lagrange polynomial. We discuss theoretical aspects by proving stability and convergence of the procedure for density functions $f$ belonging to weighted uniform spaces. Moreover, we give some computational details for the effective construction of the rule coefficients. For the sake of completeness, we present some numerical tests that support the theoretical estimates and some comparisons with other numerical methods.

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### Key words

Hadamard finite part integrals, approximation by polynomials, orthogonal polynomials, product integration rules

### AMS subject classifications

65D32, 65R20, 41A10

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