Spline-type Spaces in Gabor AnalysisHans G. Feichtinger given at esi05 (21.05.06) id: 306 length: min status: type: seminar talk LINK-Presentation: https://nuhagphp.univie.ac.at/dateien/feibomm00abs.pdf ABSTRACT: TeX: \documentstyle{article} % \documentstyle[12pt]{article} \begin{document} Abstract by Hans G. Feichtinger (version of August 28th, 2000) % \vspace{1cm} \begin{center} { Spline-type Spaces in Gabor Analysis} \end{center} % \begin{center} { Hans G. Feichtinger, NuHAG, Vienna} \end{center} {\it Gabor Analysis} is concerned with the representation of functions (resp.\ tempered distributions) $f$ on ${\bf R}^d$ as double series, whose terms are obtained from a single building block (the so-called Gabor atom $g$) by applying time-frequency shifts from some lattice $ \Lambda \subset {\bf R}^{2d}$. {\it Good} Gabor expansions make use of a building block $g$ which has both good decay and smoothness properties. As a matter of fact, however, they have to be {\it redundant} (in particular non-orthogonal), as a consequence of the Balian-Low principle. Equivalently one can describe Gabor analysis as the task of stable recovery of a function $f$ from a sampled (over $\Lambda$) STFT (i.e.\ the {\it short-time Fourier transform}) $S_g(f)$, for some known $g$. Since the short-time Fourier transform is an isometry from the Hilbert space $L^2({\bf R} ^d)$ into $L^2({\bf R}^{2d})$ both questions can be treated as problems concerning certain function spaces (containing the functions of the form $S_g(f)$, sharing some smoothness) over the TF-plane. % ($=$ time-frequency plane ${\bf R}^{2d}$. As a matter of fact these subspaces of $L^2({\bf R}^{2d})$ resemble very much so-called {\it spline type spaces} (also called principal invariant Banach spaces elsewhere), which are obtained from a given function $G$ using shifts along some lattice $ \Lambda \subset {\bf R}^{2d}$). Although this analogy cannot be used directly % (as will be explained), due to the phase factors that prevent time-frequency shifts from constituting a commutative group of operators there are nevertheless % various special cases where various recently established results in Gabor analysis (formulated in a group theoretical manner) which imply that at least for important special cases the key tasks of Gabor analysis can be recast into equivalent problems for spline-type spaces. Therefore refined ``standard results concerning spline-type spaces'' over locally compact Abelian groups have remarkable consequences for Gabor analysis. Among others it will be discussed in which sense the Zak transform finds a natural interpretation in the context of lca.\ groups. Indeed, this viewpoint allows to clarify under which conditions on a lattices $\Lambda$ within the abstract time-frequency plane $G \times \hat G$ a corresponding {\it abstract Zak transform} can be found in order to perform Gabor analysis of signals efficiently. \end{document} |
