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.

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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.