In this note a general way to develop a cardinal interpolant for l 2 -data on the integer lattice Z n is shown. Further, a parameter is introduced which allows one to recover the original Paley-Weiner function from which the data came.
Abstract. This paper studies the cardinal interpolation operators associated with the general multiquadrics, φα,c(x) = ( x 2 + c 2 ) α , x ∈ R d . These operators take the formwhere Lα,c is a fundamental function formed by integer translates of φα,c which satisfies the interpolatory condition Lα,c(kWe consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter c → ∞. In the univariate case, we consider the norm of the operator Iα,c acting on p spaces as well as prove decay rates for Lα,c using a detailed analysis of the derivatives of its Fourier transform, Lα,c.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.