FOURIER-ANALYSIS OF THE APPROXIMATION POWER OF PRINCIPAL SHIFT-INVARIANT SPACES

The approximation order provided by a directed set {S(h)}h>0 of spaces, each spanned by the hZ(d)-translates of one function, is analyzed. The ''near optimal'' approximants of [R2] from each s, to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions. The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s(h)} is obtained from s1 by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.