Volume 61, pp. 28-50, 2024.

Simultaneous approximation of Hilbert and Hadamard transforms on bounded intervals

Domenico Mezzanotte and Donatella Occorsio


In this paper, we propose a compound scheme of different product integration rules for the simultaneous approximation of both Hilbert and Hadamard transforms of a given function $f$. The advantages of such a scheme are multiple: a saving in the number of function evaluations and the avoidance of the derivatives of the density function $f$ when approximating the Hadamard transform. Stability and convergence of the proposed method are proved in the space of locally continuous functions in $(-1,1)$ with possible algebraic singularities at the endpoints, equipped with weighted uniform norms. The theoretical estimates are confirmed by several numerical tests.

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Key words

hypersingular integrals, finite Hilbert transform, Hadamard finite part integrals, polynomial approximation, extended Lagrange interpolation, orthogonal polynomials

AMS subject classifications

65D32, 65R10, 41A10, 41A28, 44A15

Links to the cited ETNA articles

[33]Vol. 54 (2021), pp. 443-459 Donatella Occorsio and Maria Grazia Russo: A mixed collocation scheme for solving second kind Fredholm integral equations in [-1,1]

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