Radial orthogonality, Symbolic-numeric integration, Padé approximation and Lebesgue constants

Giovedì 3 maggio 2012, ore 16.30, Aula 2BC45, la Prof.ssa Annie Cuyt (Research Director del FWO - Flemish Research Council, Antwerp, Belgium), nell'ambito dei Seminari Numlab di Analisi Numerica, terra' un seminario dal titolo "Radial orthogonality, Symbolic-numeric integration, Padé approximation and Lebesgue constants".

Moving from one to more dimensions with polynomial-based numerical techniques leaves room for a lot of different approaches and choices. We focus here on Padé approximation, orthogonal polynomials, integration rules and polynomial interpolation, four very related concepts.
In one variable an m-point Gaussian quadrature formula can be viewed as an [m-1/m] Padé approximant where the nodes and weights of the Gaussian quadrature formula are obtained from a sequence of orthogonal polynomials. Furthermore, in polynomial interpolation, the same m nodes now enjoy the advantage that they provide Lebesgue constants with small rate of growth.
We show that this close connection can be preserved in several variables when starting from spherical orthogonal polynomials. We obtain Gaussian cubature rules with symbolic nodes and numeric weights, exactly integrating parameterized families of polynomial functions. The spherical orthogonal polynomials are also related to the homogeneous Padé approximants introduced a few decades ago. And their zero curves provide sets of bivariate interpolation points on the disc with the smallest Lebesgue constants currently known.