Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces |
| |
Authors: | Franz Luef |
| |
Institution: | a Department of Mathematics, University of California, Berkeley, CA 94720-3840, United States b Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria |
| |
Abstract: | In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space. |
| |
Keywords: | Gabor frames Noncommutative tori Twisted group _method=retrieve& _eid=1-s2 0-S0022123609002468& _mathId=si1 gif& _pii=S0022123609002468& _issn=00221236& _acct=C000054348& _version=1& _userid=3837164& md5=cb70a99e1c01144976299914c7f729a4')" style="cursor:pointer C&lowast" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">C&lowast -algebras Spectral invariance Standard Hilbert _method=retrieve& _eid=1-s2 0-S0022123609002468& _mathId=si2 gif& _pii=S0022123609002468& _issn=00221236& _acct=C000054348& _version=1& _userid=3837164& md5=8cf73ae49a64c6aeaeed22b8c399d578')" style="cursor:pointer C&lowast" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">C&lowast -module frames |
本文献已被 ScienceDirect 等数据库收录! |
|