Publications de l’Institut Mathématique, Nouvelle Série Vol. 100[114], No. 1/1, pp. 141–162 (2016) 

BIVARIATE GENERALIZED BERNSTEIN OPERATORS AND THEIR APPLICATION TO FREDHOLM INTEGRAL EQUATIONSDonatella Occorsio, Maria Grazia RussoDepartment of Mathematics, Computer Science and Economics, University of Basilicata, Potenza, ItalyAbstract: We introduce and study the sequence of bivariate Generalized Bernstein operators ${\left\{{\mathbf{B}}_{m,s}\right\}}_{m,s}$, $m,s\in \mathbb{N}$, $$$$ B $$$$ where ${\mathbf{B}}_{m}$ is the bivariate Bernstein operator. These operators generalize the ones introduced and studied independently in the univariate case by Mastroianni and Occorsio [Rend. Accad. Sci. Fis. Mat. Napoli 44 (4) (1977), 151–169] and by Micchelli [J. Approx. Theory 8 (1973), 1–18] (see also Felbecker [Manuscripta Math. 29 (1979), 229–246]). As well as in the onedimesional case, for $m$ fixed the sequence ${\left\{{\mathbf{B}}_{m,s}\left(f\right)\right\}}_{s}$ can be successfully employed in order to approximate “very smooth” functions $f$ by reusing the same data points $f\left(\frac{i}{m},\frac{j}{m}\right)$, $i=0,1,\cdots ,m$, $j=0,1,\cdots ,m$, since the rate of convergence improves as $s$ increases. A stable and convergent cubature rule on the square ${[0,1]}^{2}$, based on the polynomials ${\mathbf{B}}_{m,s}\left(f\right)$ is constructed. Moreover, a Nyström method based on the above mentioned cubature rule is proposed for the numerical solution of twodimensional Fredholm integral equations on ${[0,1]}^{2}$. The method is numerically stable, convergent and the involved linear systems are well conditioned. Some algorithm details are given in order to compute the entries of the linear systems with a reduced time complexity. Moreover the procedure can be significantly simplified in the case of equations having centrosymmetric kernels. Finally, some numerical examples are provided in order to illustrate the accuracy of the cubature formula and the computational efficiency of the Nyström method. Keywords: iterated Bernstein polynomials; multivariate polynomial approximation; cubature formula; Nyström method Classification (MSC2000): 41A10;41A63;65D32;65R20 Full text of the article: (for faster download, first choose a mirror)
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