Shift-invariant spaces on the real line

Abstract

We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in L p ( R ) L_p(\mathbb {R}) ( 1 ≤ p ≤ ∞ ) (1\le p\le \infty ) . 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 \mathbb {Z} . Moreover, we show that such a shift-invariant space provides L p L_p -approximation order k k if and only if it contains all polynomials of degree less than k k .