74,058 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
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 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 -functional defined using the power of the spherical -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 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
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