Topic on Shift-Invariant Spaces with Extra Invariance

Abstract

We consider shift-invariant spaces with extra invariance, namely translation-invarianr or 1/nZ--invariant for some n > I. In general a shift invariant space does not possess .any invariance other than translation by integers. As additional invariance is desirable in signal processing and wavelet analysis, one may circumvent the slow spatial-decay property by seeking principle shift-invariant spaces invariant under the translation by 1/nZ for large integer n. In fact, such principle shift-invariant spaces with integrable generators do exist. The most surprising result is that there is a uncertainty principle of Balian-Low type under the additional invariance of principal shift-invariant space and the time-frequency localization of its generator. Similar results can also be obtained for finitely generated shift-invariant spaces. In some signal and image processing applications, it is assumed that signals live in some principal shift­ invariant spaces. As the observation data is usually corrupted due to various reasons, it is natural to look for principal shift-invariant spaces (even with additional invariance) that best approximate a given data set in the sense of the least squares. We will also construct principal shift-invariant spaces with additional invariance that best approximate a given data set.