Concerning sets of polynomials orthogonal simultaneously on several circles

On Discrete Orthogonal Polynomials of Several Variables

Let V be a set of points in R. Define a linear functional L on the space of polynomials, Lf = ∑ x∈V f(x)ρ(x), where ρ is a nonzero function on V . The structure of discrete orthogonal polynomials of several variables with respect to the bilinear form 〈f, g〉 = L(fg) is studied. For a given V , the subspace of polynomials that will generate orthogonal polynomials on V is identified. One result sh...

Zeros of Polynomials Orthogonal on Several Intervals

Let a1 < a2 < . . . < a2l, Ej = [a2j−1, a2j ], put E = ⋃l j=1 Ej and H(x) = ∏2l j=1(x − aj). Furthermore let pn(x) = x + . . . be the polynomial of degree n orthogonal on E with respect to a weight function of the form w/ √ −H with square root singularities at the boundary points of E and w ∈ C(E). We study and answer the following questions: how many zeros has pn in the interval Ej , j ∈ {1, ....

On Orthogonal Polynomials in Several Variables

We report on the recent development on the general theory of orthogonal polynomials in several variables, in which results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation.

Monomial orthogonal polynomials of several variables

A monomial orthogonal polynomial of several variables is of the form x−Qα(x) for a multiindex α ∈ N 0 and it has the least L2 norm among all polynomials of the form xα − P (x), where P and Qα are polynomials of degree less than the total degree of xα. We study monomial orthogonal polynomials with respect to the weight function ∏d+1 i=1 |xi| 2κi on the unit sphere Sd as well as for the related w...

Lecture Notes on Orthogonal Polynomials of Several Variable