×

Two variable orthogonal polynomials and structured matrices. (English) Zbl 1136.42305

Summary: We consider bivariate real valued polynomials orthogonal with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are established and used to give necessary and sufficient conditions for the existence of a positive linear functional. Links to the theory of matrix orthogonal polynomials are developed as well the consequences of a zero assumption on one of the coefficients in the the recurrence formulas.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30E05 Moment problems and interpolation problems in the complex plane
47A57 Linear operator methods in interpolation, moment and extension problems
PDFBibTeX XMLCite
Full Text: DOI