The limiting behavior of certain sampling series and cardinal splines

Abstract

Many families of cardinal sampling kernels {Φα(x):α∈I} consist of members that decay rapidly as x→±∞ and enjoy the property that, as the parameter α tends to its limiting value, Φα(x) tends to the classical cardinal sine, sinc(x). We study the limiting behavior of the sampling series ∑ncnΦα(x−n) when the data sequence {cn} remains fixed. In the case when the sampling kernels are damped cardinal sines, appropriate conditions on the damping functions allow us to characterize the limiting behavior of the corresponding sampling series for a wide class of data sequences {cn}.These results lead to conclusions regarding the limiting behavior of classical piecewise polynomial cardinal splines when the fixed data sequence {cn} is of polynomial growth and the order tends to ∞. One application of this development shows that the classical spline summability method is regular. Another consequence, perhaps more interesting and unexpected, is a resolution of I. J. Schoenberg’s conjecture concerning the recovery, in terms of their samples {f(n)}, of members f(x) of the Bernstein class via the spline summability procedure.