Seminars

Thurday April 26th  - 10:30am - Byron Beige (Y506)


Annie Cuyt

Mathematics & Computer Science
Universiteit Antwerpen (CMI)



"Radial orthogonality, Symbolic-numeric integration, Pad\'e 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 Pad\'e 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 $[m-1/m]$ Pad\'e 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 Pad\'e 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.