Properties of Multivariate Homogeneous Orthogonal Polynomials

Abstract

It is well-known that the denominators of Padé approximants can be considered as orthogonal polynomials with respect to a linear functional. This is usually shown by defining Padé-type approximants from so-called generating polynomials and then improving the order of approximation by imposing orthogonality conditions on the generating polynomials. In the multivariate case, a similar construction is possible when dealing with the multivariate homogeneous Padé approximants introduced by the second author. Moreover it is shown here, that several well-known properties of the zeroes of classical univariate orthogonal polynomials, in the case of a definite linear functional, generalize to the multivariate homogeneous case. For the multivariate homogeneous orthogonal polynomials, the absence of common zeroes is translated to the absence of common factors.