Approximation from Shift-invariant Spaces by Integral Operators∗

We investigate approximation from shift-invariant spaces by using certain integral operators and discuss various applications of this approximation scheme. We assume that our integral operators commute with shift operators and that their kernel functions decay at a polynomial rate. We prove that the approximation order provided by such an integral operator is m if and only if the integral operator reproduces polynomials of degree up to m − 1, where m is a positive integer. Using this result, we characterize the approximation order provided by a finitely generated shiftinvariant space whose generators decay in a polynomial rate and have stable shifts. We also review some already well-studied approximation schemes such as projection, cardinal interpolation, and quasi-interpolation by considering them as special cases of integral operators.