In this paper, we study orthogonal polynomials with respect to the bilinear formwhere u is a quasi-definite (or regular) linear functional on the linear space IP of real polynomials, c is a real number, AT is a positive integer number, A is a symmetric N x N real matrix such that each of its principal submatrices are regular, and F(c) = (/(c), /'(c),...,/^J V~1 H C ))> G(c) = (g(c)ig f (c),...,g( N~1 \c)). For these non-standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated with u.
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.
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