A CHARACTERIZATION OF THE APPROXIMATION ORDER OF TRANSLATION INVARIANT SPACES OF FUNCTIONS

Let S be a space of functions on R with the following properties: (i) S is translation invariant, i.e., f is-a-element-of S implies f(. +/- 1) is-a-element-of S; (ii) dim S\[0, 1] < infinity; (iii) S is closed under uniform convergence on compact sets. In this paper we characterize the approximation order of S by proving the following: Theorem. S provides approximation of order k if and only if S contains a compactly supported function psi such that the Fourier transform psi-triple-overdot of psi satisfies psi-triple-overdot(0) = 1 and D-alpha-psi-triple-overdot(2-pi-j) = 0 for 0 less-than-or-equal-to alpha < k and j is-a-element-of Z\ {0}.