We introduce and study the sequence of bivariate Generalized Bernstein
operators {Bm,s}m,s, m, s ? N, Bm,s=I?(I?Bm)s, Bi m = Bm(Bi?1 m),
where Bm 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
one-dimesional case, for m fixed the sequence {Bm,s(f)}s can be successfully
employed in order to approximate ?very smooth? functions f by reusing the
same data points f (i/m,j/m), i=0,1,...,m, j=0,1,...,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
Bm,s(f) is constructed. Moreover, a Nystrom method based on the above
mentioned cubature rule is proposed for the numerical solution of
two-dimensional 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 Nystrom method.
A Nyström method for Fredholm integral equations on the real line
