Derivatives, Differences, Multiple Fourier Kernels
WISCONSIN UNIV-MADISON MADISON
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Identities and inequalities for Fourier kernels and for difference operators are related to a geometric series identity. The resulting machinery is applied to obtain, in the approximation theory for ordinary or partial derivatives of any order, necessary and sufficient conditions in place of classical sufficient conditions. Alternative formulations are given in terms of Tauberian Theorems, and in terms of Schwartz distributions. The results are achieved by making use, as in L. C. Youngs papers on Stochastic integrals and the like, of pairs of estimate functions in place of the classical higher moduli of continuity.