Multivariate Hybrid Orthogonal Functions

Abstract

We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions with respect to a 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. Results about existence conditions, three term relations with matrix coefficients, a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a method to construct bivariate hybrid orthogonal functions from univariate orthogonal polynomials and univariate orthogonal functions is presented. Finally, we give a complete description of a sequence of hybrid orthogonal functions on the unit disk on $\mathbb{R}^2$, that includes, as particular case, the classical orthogonal polynomials on the disk.