Some numerical applications of generalized Bernstein Operators

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 can be suitable chosen in order to improve the rate of convergence to the function f, as the smoothness of f increases, overcoming the well-known low degree of approximation achieved by the classical Bernstein polynomials or by the 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.