Wavelet and its Applications

Abstract

We developed the dual Gramian analysis for frames in abstract Hilbert spaces. We show the dual Gramian analysis is a powerful tool for the analysis of frames, e.g. to characterize a frame, and to estimate the frame bounds. With the introduction of adjoint systems, the duality principle plays a key role in this analysis. The duality principle also lies in the core of the analysis of Gabor systems, by which we unify several classical identities. The duality principle leads to a new and simple way to construct filter banks, or tight/dual wavelet frames from a prescribed MRA. The new method reduces the construction to a constant matrix completion problem rather than the usual methods to complete matrices with trigonometric polynomial entries. The constructed wavelets easily satisfy desired properties, e.g. small support, symmetric/anti-symmetric. Several multivariate tight and dual wavelet frames from given refinable functions have been constructed.