Mixed orthogonality on the unit ball

Abstract

We consider multivariate functions satisfying mixed orthogonality conditions with respect to a given moment functional. This kind of orthogonality means that the product of functions of different parity order is computed by means of the moment functional, and the product of elements of the same parity order is computed by a modification of the original moment functional. Three term relations and a Favard type theorem for this kind of mixed orthogonal functions are proved. In addition, a method to construct bivariate mixed orthogonal functions from univariate orthogonal polynomials and univariate mixed orthogonal functions is presented. Finally, we give a complete description of a sequence of mixed orthogonal functions on the unit disk on R2, that includes, as a particular case, the classical orthogonal polynomials on the disk.