Seminario NumLab di Analisi Numerica
In occasione della visita della Prof.ssa Annie Cuyt,
Research Director del FWO (Flemish Research Council),
nell'ambito dei Seminari Numlab di Analisi Numerica
Giovedì 3 maggio, ore 16.30, Aula 2BC45
terrà un seminario dal titolo
Radial orthogonality, Symbolic-numeric integration,
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.