Cardinal interpolation with general multiquadrics: convergence rates

Abstract

This article pertains to interpolation of Sobolev functions at shrinking lattices hZ d from Lp shift-invariant spaces associated with cardinal functions related to general multiquadrics, φα,c(x) := (|x| 2 + c 2 ) α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, Lp error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h k ) for functions with derivatives of order up to k in Lp, 1 < p < ∞). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.

Stability and Robustness of RBF Interpolation

“…We note also that for the cases α = ±1/2, such considerations were already made by Buhmann [5] and Riemenschneider and Sivakumar [20]. Additionally, L p approximation rates of the same order as in Theorem 4.1 for interpolation at hZ can be found in [12].…”

“…which is supported on a closed interval. This theorem and the estimate in (12) imply that if one wishes to approximate a compactly supported f by its interpolant I N −1 X f , it suffices to consider the more simple uniform interpolant I N −1 Z f up to the penalty of a possibly larger constant C. The usefulness of this will be discussed further in the next section.…”

“…Some study of summability methods using cardinal functions formed from translates of a single radial basis function (RBF) -one which satisfies φ(x) = φ(|x|)-has been made [4,5,6,11,12,13,16,21,25]. Cardinal functions are those which satisfy the interpolatory condition L(k) = δ 0,k , k ∈ Z, of which sinc is the prototypical example.…”

Regular families of kernels for nonlinear approximation

“…the general multiquadrics may be found in [22,23] for a broad range of exponents α, whereas the particular cases of α = ±1/2, 1, −k + 1/2, k ∈ N were considered previously [4,10,12].…”

“…Note that this is different from T ♯ h f due to the fact that we use the samples of f at the lattice hZ d in the approximant rather than the values of f L τ (h) as defined previously. This object has been studied in various instances before [10,23,25], and is actually an interpolant of f . By definition of the cardinal functions, it is easy to see that I ♯ h f (hk) = f (hk), k ∈ Z d .…”