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Open AccessArticle

Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes

1
Department of Scientific Computing, Helmholtz Zentrum München German Research Center for Environmental Health, Ingolstädter Landstrasse 1, 85764 Neuherberg, Germany
2
Applied Numerical Analysis, Fakultät für Mathematik, Technische Universität München, Boltzmannstrasse 3 85748 Garching bei München. Research Center, Ingolstädter Landstrasse 1, 85764 Neuherberg, Germany
3
Department of Mathematics, Computer Science and Economics, University of Basilicata, viale dell’Ateneo Lucano 10, 85100 Potenza, Italy
4
C.N.R. National Research Council of Italy, IAC Institute for Applied Computing “Mauro Picone”, via P. Castellino, 111, 80131 Napoli, Italy
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 542; https://doi.org/10.3390/math8040542
Received: 6 March 2020 / Revised: 31 March 2020 / Accepted: 1 April 2020 / Published: 7 April 2020
(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)
In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider [ 1 , 1 ] and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results. View Full-Text
Keywords: Hilbert transform; Hadamard transform; hypersingular integral; Bernstein polynomials; Boolean sum; simultaneous approximation; equidistant nodes Hilbert transform; Hadamard transform; hypersingular integral; Bernstein polynomials; Boolean sum; simultaneous approximation; equidistant nodes
MDPI and ACS Style

Filbir, F.; Occorsio, D.; Themistoclakis, W. Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes. Mathematics 2020, 8, 542.

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