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Limits of interpolatory processes

Authors:	W. R. Madych

Affiliation:	Department of Mathematics, U-9, University of Connecticut, Storrs, Connecticut 06269-3009

Abstract:	Given distinct real numbers and a positive approximation of the identity $phi_{epsilon}$ , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials $P_{epsilon}(x)= sum c_{epsilon , j} e^{-i nu_j x}$ which solve the interpolation problem $begin{displaymath}int P_{epsilon}(x) e^{i nu_k x} phi_{epsilon}(x)dx=f_{epsilon,k}, quad k=1, ldots, N,end{displaymath}$ with prescribed data $f_{epsilon,1}, dots, f_{epsilon,N}$ . More specifically, we are interested in the behavior of $P_{epsilon}(x)$ when the data is of the form $f_{epsilon, k}=int f(x) e^{i nu_k x} phi_{epsilon}(x)dx$ for some prescribed function . One of our results asserts that if is sufficiently nice and $phi_{epsilon}$ has sufficiently well-behaved moments, then $P_{epsilon}$ converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.

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