Volume 63, pp. 540-565, 2025.

A dilation quadrature formula for hypersingular and highly oscillatory integrals on the positive half-line

Maria Carmela De Bonis and Valeria Sagaria

Abstract

The aim of this paper is to introduce a new quadrature rule for approximating integrals with highly oscillatory and hypersingular integrands defined on the positive half-line. After the integration interval is split into the subintervals $[0,M]$ and $[M,+\infty)$, so that the part on $[M,+\infty)$ is negligible, the interval $[0,M]$ is suitably dilated and decomposed into a sum of integrals, where each of them is approximated by a Gaussian quadrature rule. We prove that the formula is convergent when the function $f$ is bounded on $\mathbb{R}^+$ together with a certain number of its derivatives. Numerical tests compare the performance of the proposed rule with other formulas available in the literature.

Full Text (PDF) [406 KB], BibTeX , DOI: 10.1553/etna_vol63s540

Key words

Hadamard integrals, Cauchy integrals, hypersingular integrals, highly oscillatory functions, dilation rule, Gaussian rule

AMS subject classifications

65D30, 65R10, 41A05, 42B20

Links to the cited ETNA articles

[13]	Vol. 50 (2018), pp. 129-143 Maria Carmela De Bonis and Donatella Occorsio: A product integration rule for hypersingular integrals on $(0,+\infty)$
[26]	Vol. 61 (2024), pp. 28-50 Domenico Mezzanotte and Donatella Occorsio: Simultaneous approximation of Hilbert and Hadamard transforms on bounded intervals