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

A new approach to the reconstruction of images from Radon projections

A new approach is proposed for reconstruction of images from Radon projections. Based on Fourier expansions in orthogonal polynomials of two and three variables, instead of Fourier transforms, the approach provides a new algorithm for the computed tomography. The convergence of the algorithm is established under mild assumptions.Comment: 28 pages, accepted by Adv. in Applied Mat

Weighted Approximation of functions on the unit sphere

The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of \RR^{d+1}, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a $K$-functional defined using the power of the spherical $h$-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of \RR^d.Comment: 25 page

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