title: Banach-Gelfand triples and applications in time-frequency analysis
Creators: name: Bannert, Veit-Severin
Publisher: name: :none
Year: 2010

description: Ziel der vorliegenden Arbeit ist es, Banach-Gelfand Triple im Kontext der Zeitfrequenzanalyse vorzustellen. Banach-Gelfand triple verbinden die Eigenschaften von Hilberträumen mit denen von allgemeinen Banachräumen von Distributionen. Ein Banach-Gelfand Triple ist ein Triple von Räumen, bestehend aus einem Hilbertraum H in den ein (kleinerer) Banachraum B eingebettet liegt. H selbst wiederum ist enthalten in dem Banachraum B′, dem Dualraum von B. Wir werden die Rolle aufzeigen, die Banach-Gelfand Triple als grundlegendes Konzept in der Zeit- Frequenz Analyse spielen. Kapitel 2 gibt eine kurze Einführung in das Themengebiet. Fouriertransformation, Modulations- und Verschiebungsoperator, die Kurzzeitfouriertransformation sowie Gaussfunktionen werden vorgestellt und deren, fu ̈r diese Arbeit wichtigen, Eigenschaften bewiesen. In Kapitel 2.2 werden Gabor Frames eingeführt und diskutiert. Sie erlauben eine Diskretisierung der Kurzzeitfouriertransformation. Kapitel 3 leitet den Hauptteil dieser Arbeit ein. Banach-Gelfand Triple werden allgemein definiert und wichtige Eigenschaften bewiesen. Kapitel 4 widmet sich im Anschluss daran dem Gelfand Triple (S0, L2, S0′), das eine Schlü sselrolle in der hier vorgestellten Form der Zeit-Frequenz Analyse spielt. In Kapitel 5 werden schließlich unterschiedliche Klassen der Operatordarstellung besprochen. Es werden Methoden vorgestellt, um Operatoren mit Funktionen auf der Zeit-Frequenz Ebene zu identifizieren.
description: A recurrent theme in mathematics is extension. The rational numbers, for example, were extended to the real numbers which in turn were extended to the complex numbers when it became clear that the respective number sets were too small to solve particular problems in. In this work we highlight a related theme in the context of time-frequency analysis. The natural domain of the Fourier transform, which is the fundamental opera- tion in time-frequency analysis, is L1. This is somewhat limited but luckily one can extend its domain to L2 by taking limits, so we are able to work in a Hilbert space, which is a very nice place to be, amongst others because many concepts of linear algebra can be carried over to this setting. But Hilbert spaces too, are limited. As will be pointed out in the beginning of Section 3, it is easy to pose questions which lead outside the Hilbert space setting and call for distribution spaces. In this work we will introduce Banach-Gelfand triples, a concept which encom- passes the “niceness” of Hilbert spaces and the generality of Banach spaces of distributions. A Banach-Gelfand triple is a triple of spaces, consisting of a Hilbert space H which contains a smaller Banach space B and is itself contained in the dual space B′. Our aim is to emphasize the role of Banach-Gelfand triples as a background to modern time-frequency analysis. Thus the subsequent section (Section 2) will give a short introduction to the key concepts of time-frequency analysis. The Fourier transform, translation and modulation operators, the short-time Fourier transform and Gaussian windows will be introduced and important properties will be proved. In Section 2.2 we will introduce Gabor frames which enable us to discretize the short-time Fourier transform. Section 3 and the subsequent sections encompass the main part of this work. We will introduce the general concept of Banach-Gelfand triples and subsequently in Section 4 its specific incarnation (S0,L2,S0′), which plays a key role in time- frequency analysis as presented here, consequently its discussion will make up a major part of this paper. Section 5 is dedicated to various classes of operator representation. Different methods to identify operators with functions on the time- frequency plane are introduced there.