Orthogonal polynomials in several variables for measures with mass points

Abstract

Let dν be a measure in R d obtained from adding a set of mass points to another measure dμ. Orthogonal polynomials in several variables associated with dν can be explicitly expressed in terms of orthogonal polynomials associated with dμ, so are the reproducing kernels associated with these polynomials. The explicit formulas that are obtained are further specialized in the case of Jacobi measure on the simplex, with mass points added on the vertices, which are then used to study the asymptotics kernel functions for dν.

On Sobolev orthogonal polynomials

“…For inner product that contains additional point evaluations of functions, as those discussed in Section 7, the Krall type construction of orthogonal polynomials can be extended to several variables, as shown in [29]. The same holds true if the point evaluations involve derivatives, see [30] for an example.…”

Multivariate Orthogonal Polynomials and Modified Moment Functionals

“…Our next result gives explicit formulas for the reproducing kernels associated with v, which we denote by Next theorem establishes a relation between the kernels of both families. Similar tools as those used in the proof of Theorem 2.5 in [11] can be applied to obtain this result. Theorem 3.4.…”

“…If v is quasi-definite and centrally symmetric, then relation (4.4) is given by 11) where N n = H n A t n−2,2Ĥ −1 n−2 , n 2.…”

“…In the multivariate case, the addition of Dirac masses to a multivariate measure was studied in [11] and [13]. Moreover, Uvarov modification for disk polynomials was analysed in [10].…”

“…In that case, both moment functionals are positive-definite, and both orthogonal polynomial systems exist. As a consequence, since it is possible to work with orthonormal polynomials, in [11] the first family of orthogonal polynomials is considered orthonormal, and formulas are simpler than in the general case considered here.…”