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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 28-50, 2024/05/15
In this paper, we propose a compound scheme ofdifferent product integration rules for the simultaneous approximation of bothHilbert and Hadamard transforms of a given function $f$. The advantages ofsuch a scheme are multiple: a saving in the number of function evaluations andthe avoidance of the derivatives of the density function $f$ when approximatingthe Hadamard transform. Stability and convergence of the proposed method areproved in the space of locally continuous functions in $(-1,1)$ with possiblealgebraic singularities at the endpoints, equipped with weighted uniform norms.The theoretical estimates are confirmed by several numerical tests.
Keywords: hypersingular integrals, finite Hilbert transform, Hadamard finite part integrals, polynomial approximation, extended Lagrange interpolation, orthogonal polynomials