Some numerical applications of generalized Bernstein operators

Year 2021, Volume: 4 Issue: 2, 186 - 214, 01.06.2021

Donatella Occorsıo , Maria Grazia Russo , Woula Themıstoclakıs

https://doi.org/10.33205/cma.868272

Cited By: 7

Abstract

In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected.
This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of
[0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence
to the function f, according with the smoothness of f. This means that the sequence does not suffer of
the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial
approximation. The applications considered here deal with the numerical integration and the simultaneous approximation.
Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals
in one or two dimensions, and usefully employed in Nyström methods for solving Fredholm integral equations. Moreover,
the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated.
For all the applications, some numerical details are given in addition to the error estimates, and the proposed
approximation methods have been implemented providing numerical tests which confirm the theoretical estimates.
Some open problems are also introduced.

Keywords

Bernstein polynomials, approximation by polynomials, numerical integration on uniform gridss, Fredholm integral equations on uniform grids

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