Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions

作者：Hamm Keaton^*

来源：Journal of Approximation Theory, 2015, 189: 101-122.

DOI：10.1016/j.jat.2014.10.011

• Bandlimited functions
• Interpolation
• Scattered data
• Gaussian interpolation

A Riesz-basis sequence for L-2[n-, n-] is a strictly increasing sequence X := (xi))jez in R such that the set of functions (e ixj()) is a Riesz basis for L2[g, 7r1. Given such a sequence and a parameter j EZ 0 < h < 1, we consider interpolation of functions g E 141(R) at the set (hxj)jEz via translates of the Gaussian kernel. Existence is shown of an interpolant of the form I hX (g)() Eafe-(x -hxj)2, x E jEz which is continuous and square-integrable on R, and satisfies the interpolatory condition 112X (g)(hxj) = g(hxj), j E Z. Moreover, the use of the parameter h gives approximation rates of order hk. Namely, there is a constant independent of g such that (g) gIlL2(R) < Chk I g Iwkoa). Interpolation using translates 2 of certain functions other than the Gaussian, so-called regular interpolators, is also considered and shown to exhibit the same approximation rates.

• 出版日期2015-1