IJPAM: Volume 29, No. 4 (2006)

OPTIMAL $sk$-SPLINE APPROXIMATION AND
RECONSTRUCTION ON THE TORUS AND SPHERE

Alexander Kushpel$^1$, Chris Grandison$^2$, Dzung Minh Ha$^3$
$^{1,2,3}$Department of Mathematics
Ryerson University
350 Victoria Street, Toronto, Ontario, M5B 2K3, CANADA
$^1$e-mail: akushpel@ryerson.ca
$^2$e-mail: cgrandis@ryerson.ca
$^3$e-mail: haminh@ryerson.ca


Abstract.On the circle, $\torus^{1}$, the subspace of polynomial splines with equidistant nodes is optimal on $W^{r}_{1}(\torus^{1})$ in $L_{1}(\torus^{1})$ in the sense of different $n$-widths. Using rectangular grids on the torus, $\torus^{d}$, it is not possible to obtain the best possible order of convergence on Sobolev classes, Kushpel [#!ku034!#,#!ku6!#], and optimal methods of reconstruction of functions from these sets are connected with the theory of uniform distribution of sequences, Kushpel [#!kushpel6!#,#!ku07!#]. The fundamental number theoretic concepts in this area were advanced in Korobov [#!korobov!#] and Kuipers [#!kuipers!#]. A lot of effort has been spent trying to find analogs of equidistant distributions on the sphere, Glasser [#!gl!#], Grabner [#!gr!#] and Saff [#!saf!#]. An important phenomenon we present in this article lies in the fact that in the case of the sphere it is sufficient to use an equiangular grid to get an optimal order of convergence on Sobolev classes. We construct subspaces of splines on the torus and sphere and show that these subspaces are optimal on standard Sobolev classes in sense of respective $n$-widths.

Received: May 17, 2006

AMS Subject Classification: 41A15, 41A65, 41A10, 41A25

Key Words and Phrases: interpolation, approximation, reconstruction

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 29
Issue: 4