Shift-invariant spaces on the real line

We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in L(p)(R) (1 less than or equal to p less than or equal to infinity). Based on a study of linear independence of the shifts of the;generators, we characterize such shift-invariant spaces in terms of the semi-convolutions of the generators with sequences on Z. Moreover, we show that such a shift;invariant space provides L(p)-approximation order k if and only if it contains all polynomials of degree less than k.