Three term recurrence relation modulo ideal and orthogonality of polynomials of several variables

Abstract

Orthogonality of polynomials in several variables with respect to a positive Borel measure supported on an algebraic set is the main theme of this paper. As a step towards this goal quasi-orthogonality with respect to a non-zero Hermitian linear functional is studied in detail; this occupies a substantial part of the paper. Therefore necessary and sufficient conditions for quasi-orthogonality in terms of the three term recurrence relation modulo a polynomial ideal are accompanied with a thorough discussion. All this enables us to consider orthogonality in full generality. Consequently, a class of simple objects missing so far, like spheres, is included. This makes it important to search for results on existence of measures representing orthogonality on algebraic sets; a general approach to this problem fills up the three final sections.

Holomorphic Hermite polynomials in two variables

“…The orthogonality relations (19) can be also shown using the exponential generating function (7). However such a proof, when performed like in [11], would depend on the evaluation…”

“…which would provide the crucial argument for changing the integration and summation when passing from (19) to the exponential generating functions (7). Taking into account its own interest this way of arguing is developed in Appendix, p. 14).…”

“…Because our ultimate goal is to consider and apply the Segal-Bargmann transform focusing exclusively on holomorphic polynomials (4) forces us to abandon the other option. A reputable recommendation for interested readers to learn more on polynomials in several variables is, besides the monograph [10], the influential survey article [28]; to see how subtle algebra (and algebraic geometry) is behind polynomials of several variables take a look at [7].…”

Subnormality and Operator Multidimensional Moment Problems

“…However, in some specific cases, this is possible. Following [12,40], we say that a complex polynomial p in 2d and let {a k,l } k,l∈Z d + ⊆ C be a finitely supported 2d-sequence such that…”

“…It is straightforward to formulate the vector counterpart of Proposition 6.5, replacing 'type A o ' by 'type A'. The latter notion seems to be weaker than the former one (see [12,40] for definitions and properties of types A and A o ).…”