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2013 | OriginalPaper | Chapter

1. Introduction to Finite Frame Theory

Authors : Peter G. Casazza, Gitta Kutyniok, Friedrich Philipp

Published in: Finite Frames

Publisher: Birkhäuser Boston

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Abstract

To date, frames have established themselves as a standard notion in applied mathematics, computer science, and engineering as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. The reconstruction procedure is then based on one of the associated dual frames, which—in the case of a Parseval frame—can be chosen to be the frame itself. In this chapter, we provide a comprehensive review of the basics of finite frame theory upon which the subsequent chapters are based. After recalling some background information on Hilbert space theory and operator theory, we introduce the notion of a frame along with some crucial properties and construction procedures. Then we discuss algorithmic aspects such as basic reconstruction algorithms and present brief introductions to diverse applications and extensions of frames. The subsequent chapters of this book will then extend key topics in many intriguing directions.

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next chapter Constructing Finite Frames with a Given Spectrum
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Metadata
Title
Introduction to Finite Frame Theory
Authors
Peter G. Casazza
Gitta Kutyniok
Friedrich Philipp
Copyright Year
2013
Publisher
Birkhäuser Boston
Book
Finite Frames

Print ISBN: 978-0-8176-8372-6

Electronic ISBN: 978-0-8176-8373-3

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-0-8176-8373-3_1

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