Multivariate orthogonal polynomials and homogeneous Pad\'e approximants
It is well-known that the denominators of Pad\'e approximants can be considered as orthogonal polynomials with respect to a linear functional. This is usually shown by defining Pad\'e-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 Pad\'e 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.
Furthermore, in the univariate case the connection between orthogonal polynomials, Pad\'e approximants and Gaussian quadrature is well-known. In the past, several generalizations to the multivariate case have been suggested for all three concepts, however without reestablishing a fundamental and clear link. We will elaborate on definitions for multivariate Pad\'e and Pad\'e-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new $m$-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree $2m-1$, in any number of variables. A numerical application of the new integration rules illustrates the results.