Formal orthogonal polynomials for an arbitrary moment matrix and Lanczos type methods
Adhemar Bultheel, Marc Van Barel · Lirias (KU Leuven) · 1994
We give a framework for formal orthogonal polynomials with respect to an arbitrary moment matrix. When the moment matrix is Hankel, this simplies to the classical framework. The relation with Pade approximation and with Krylov subspace methods is given. 1 Formal block orthogonal polynomials We consider a linear functional dened on the space of polynomials in two variables, dened by the moments ij = (w i z j ), i; j 2 N. Let M = [ ij ] be the (innite) moment matrix, then for two polynomials p(w) = wp and q(z) = zq (w = [1; w; w 2 ; : : :] and z = [1; z; z 2 ; : : :], p; q 2 C 11 ), we dene a formal inner product by hp; qi = (p (w)q(z)) = p Mq. We call g(!; ) = (1=[(! w)( z)]) the generator for the matrix M , since at least formally: g(!; ) = X i;j ij ! i+1 j+1 = X i g [i] () ! i+1 with g [i] () = w i =( z) = P j ij = j+1 , the generator for row i. Any innite matrix M can be factored as B MA = D where D =...