Topic on Shift-Invariant Spaces with Extra Invariance
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.