73,046 research outputs found
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 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 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 -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 be a set of isolated points in \RR^d. Define a linear functional
\CL on the space of real polynomials restricted on , \CL f = \sum_{x \in
V} f(x)\rho(x), where is a nonzero function on . 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
- …