Introduction to Finite Frame Theory
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 expa
- PDF / 577,777 Bytes
- 53 Pages / 439.37 x 666.142 pts Page_size
- 107 Downloads / 280 Views
Introduction to Finite Frame Theory Peter G. Casazza, Gitta Kutyniok, and Friedrich Philipp
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. Keywords Applications of finite frames · Construction of frames · Dual frames · Frames · Frame operator · Grammian operator · Hilbert space theory · Operator theory · Reconstruction algorithms · Redundancy · Tight frames
1.1 Why Frames? The Fourier transform has been a major tool in analysis for over 100 years. However, it solely provides frequency information, and hides (in its phases) information concerning the moment of emission and duration of a signal. D. Gabor resolved this
P.G. Casazza Mathematics Department, University of Missouri, Columbia, MO 65211, USA e-mail: [email protected] G. Kutyniok () · F. Philipp Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany e-mail: [email protected] F. Philipp e-mail: [email protected] P.G. Casazza, G. Kutyniok (eds.), Finite Frames, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_1, © Springer Science+Business Media New York 2013
1
2
P.G. Casazza et al.
problem in 1946 [92] by introducing a fundamental new approach to signal decomposition. Gabor’s approach quickly became the paradigm for this area, because it provided resilience to additive noise, quantization, and transmission losses as well as an ability to capture important signal characteristics. Unbeknownst to Gabor, he had discovered the fundamental properties of a frame without any of the formalism. In 1952, Duffin and Schaeffer [79] were studying some deep problems in nonharmonic Fourier series for which they required a formal structure for working with highly overcomplete families of exponential functions in L2 [0, 1]. For this, they introduced the notion of a Hilbert space frame, in which Gabor’s approach is now a special case, falling into the area of time-frequency analysis [97]. Much later—in the late 1980s—the fundamental concept of frames was revived by Daubechies, Grossman and Mayer [76] (see also [75]), who showed its importance for data processing. T
Data Loading...
