Continuous Frames, Function Spaces, and the Discretization Problem

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Groechenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.Comment: 44 page

An inhomogeneous uncertainty principle for digital low-pass filters

ABSTRACT. This article introduces an inhomogeneous uncertainty principle for digital lowpass filters. The measure for uncertainty is a product of two factors evaluating the frequency selectivity in comparison with the ideal filter and the effective length of the filter in the digital domain, respectively. We derive a sharp lower bound for this product in the class of filters with so-called finite effective length and show the absence of minimizers. We find necessary and certain sufficient conditions to identify minimizing sequences. When the class of filters is restricted to a given maximal length, we show the existence of an uncertainty minimizer. The uncertainty product of such minimizing filters approaches the unrestricted infimum as the filter length increases. We examine the asymptotics and explicitly construct a sequence of finite-length filters with the same asymptotics as the sequence of finite-length minimizers. 1