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

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.