Generalized translation operator and approximation in several variables

Generalized translation operators for orthogonal expansions with respect to families of weight functions on the unit ball and on the standard simplex are studied. They are used to define convolution structures and modulus of smoothness for these regions, which are in turn used to characterize the best approximation by polynomials in the weighted $L^p$ spaces. In one variable, this becomes the generalized translation operator for the Gegenbauer polynomial expansions.Comment: 22 pages, 7th International Symposium on Orthogonal Polynomials and Special Functions, Copenhagen, August 200

Almost everywhere convergence of orthogonal expansions of several variables

For weighted $L^1$ space on the unit sphere of \RR^{d+1}, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in $h$-harmonics. The result applies to various methods of summability, including the de la Vall\'ee Poussin means and the Ces\`aro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of \RR^d.Comment: 23 page

On discrete orthogonal polynomials of several variables

Let $V$ be a set of isolated points in \RR^d. Define a linear functional \CL on the space of real polynomials restricted on $V$, \CL f = \sum_{x \in V} f(x)\rho(x), where $\rho$ is a nonzero function on $V$. Polynomial subspaces that contain discrete orthogonal polynomials with respect to the bilinear form = \CL(f g) are identified. One result shows that the discrete orthogonal polynomials still satisfy a three-term relation and Favard's theorem holds in this general setting.Comment: 15 pages, 2 figure