Three term recurrence relation modulo ideal and orthogonality of polynomials of several variables
This lengthy paper is bent on a thorough treatment of polynomials in several variables, orthogonal with respect to a positive measure supported on an algebraic set. To achieve their goal, the authors give a systematic treatment of the underlying concepts. The main difference to the study of orthogonal polynomials in one variable lies in the fact that there the equivalence between the determining measure of the linear functional and the well-known three term recurrence relation can be exploited. In the several variables case, there does not exist such a `simple' description; for a recent account on the theory (connected with the order in which the orthonormalization process is applied to `basis' functions), the reader is referred to [\textit{C. F. Dunkl, Y. Xu}, ``Orthogonal polynomials of several variables'', Encyclopedia of Mathematics and Its Applications. 81. (2001; Zbl 0964.33001)]. Apart from the problem of correctly defining a three term recurrence relation, there is also the possibility that the recurrence relation does not determine any orthogonality measure at all. A solution to these problems is found in the introduction of recurrence relations of metric type, satisfied modulo an ideal. The right type of ideals can be found using the methods and results due to \textit{J. Stochel} [Ark. Mat. 30, 133--48 (1992; Zbl 0819.47015)]. Implementing the program sketched in the introduction of the paper under review, the authors devote a substantial part of the paper to so-called quasi-orthogonality. The first four sections (page 13: Prerequisites; page 15: Monomial bases related to an ideal \(V\); page 18: \(V\)-bases; page 22: Rigid \(V\)-bases) contain basic ingredients, while in section 5 (page 24: Quasi-orthogonality: the real case) polynomials of several variables quasi-orthogonal with respect to a Hermitian linear functional are treated. The results of section 5 are looked into and discussed more deeply in the two sections thereafter (page 34: Quasi-orthogonality: degree versus rank; page 39: Quasi-orthogonality: the complex case). Then, in section 8 (page 42: Orthogonality), the case of full orthogonality with respect to a positive definite linear functional is investigated. An orthogonalizing functional is shown to be derived from a positive Borel measure on \textbf{R}\(^n\) only if the attached ideal is a set ideal, directly leading to algebraic sets and the Zariski topology (section 9, page 45: Algebraic sets as supports of orthogonalizing measures). The remaining three sections study the existence of orthogonalizing measures (page 48: `general approach'; page 56: `instances'; page 60: `matrix approach'). This is an excellent paper (actually a type of monograph: it could easily be extended to a companion volume to the Encyclopedia volume 81 mentioned above), that will be the main reference for a long time.
scientific article; zbMATH DE number 2174967
scientific article