Quadrature processes and new applications

G. V. Milovanović

This survey on quadrature processes and their applications is an extended version of my public lecture given in the Serbian Academy of Sciences and Arts under the same title and it represents a continuation of my ``2007-lecture'' under the title ``Quadrature processes - development and new directions'' $[$Bull. Cl. Sci. Math. Nat. Sci. Math. {\bf33} {\rm(2008), 11-41])}. Beside the basic concept on the constructive theory of orthogonal polynomials on $\RR$, an account on Gaussian quadratures for nonclassical weights on the real semiaxis and their applications in summation of slowly convergent series, generalized Birkhoff-Young quadratures, nonstandard quadratures of Gaussian type $($interval quadratures and quadratures based on operator values$)$, as well as on some applications in solving Fredholm integral equations of the second kind in one and two dimensions are presented. The presented results have been obtained in our recent papers.