Generalized polynomials and associated semigroups

In this work we construct and study families of generalized orthogonal polynomials with hermitian matrix argument associated to a family of orthogonal polynomials on R. Different normalizations for these polynomials are considered and we obtain some classical formulas for orthogonal polynomials from the corresponding formulas for the one-dimensional polynomials. We also construct semigroups of operators associated to the generalized orthogonal polynomials and we give an expression of the infinitesimal generator of this semigroup and, in the classical cases, we prove that this semigroup is also Markov. For d-dimensional Jacobi expansions we study the notions of fractional integral (Riesz potentials), Bessel potentials and fractional derivatives. We present a novel decomposition of the L2 space associated with the d-dimensional Jacobi measure and obtain an analogous of Meyer's multiplier theorem in this setting. Sobolev Jacobi spaces are also studied.

Autorentext

Cristina Balderrama is a mathematician graduated from theUniversidad Central de Venezuela in 2002 and she got her PhD inMathematics from the Universidad Central de Venezuela in 2008 andfrom the Université d'Angers in 2009. She is currently working at the mathematics department of the Universidad Central de Venezuela.