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Frames and bases: An introductory course

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During the last several years, frames have become increasingly popular; they have appeared in a large number of applications, and several concrete constructions of frames of various types have been presented. Most of these constructions were based on quite direct methods rather than the classical sufficient conditions for obtaining a frame. Consequently, there is a need for an updated book on frames, which moves the focus from the classical approach to a more constructive one.

Based on a streamlined presentation of the author's previous work, An Introduction to Frames and Riesz Bases , this new textbook fills a gap in the literature, developing frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field.

Key features and topics:

* Results presented in an accessible way for graduate students, pure and applied mathematicians as well as engineers.

* An introductory chapter provides basic results in finite-dimensional vector spaces, enabling readers with a basic knowledge of linear algebra to understand the idea behind frames without the technical complications in infinite-dimensional spaces.

* Extensive exercises for use in theoretical graduate courses on bases and frames, or applications-oriented courses focusing on either Gabor analysis or wavelets.

* Detailed description of frames with full proofs, an examination of the relationship between frames and Riesz bases, and a discussion of various ways to construct frames.

* Content split naturally into two parts: The first part describes the theory on an abstract level, whereas the second part deals with explicit constructions of frames with applications and connections to time-frequency analysis, Gabor analysis, and wavelets.

Frames and Bases: An Introductory Course will be an excellent textbook for graduate students as well as a good reference for researchers working in pure and applied mathematics, mathematical physics, and engineering. Practitioners working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find the book a useful self-study resource.

Author(s): Ole Christensen
Series: Applied and Numerical Harmonic Analysis
Edition: 1
Publisher: Birkhäuser Boston
Year: 2008

Language: English
Pages: 326

cover.jpg......Page 1
front-matter.pdf......Page 2
ANHA Series Preface......Page 7
Preface......Page 13
Frames in Finite-dimensional Inner Product Spaces......Page 18
Basic frames theory......Page 19
Frames in Cn......Page 29
The discrete Fourier transform......Page 34
Pseudo-inverses and the singular value decomposition......Page 37
Applications in signal transmission......Page 42
Exercises......Page 47
Normed vector spaces and sequences......Page 50
Operators on Banach spaces......Page 53
Hilbert spaces......Page 54
Operators on Hilbert spaces......Page 55
The pseudo-inverse operator......Page 57
A moment problem......Page 59
The spaces Lp(R), L2(R) , and 2(N)......Page 60
The Fourier transform and convolution......Page 63
Operators on L2(R)......Page 64
Exercises......Page 66
Bases......Page 67
Bessel sequences in Hilbert spaces......Page 68
General bases and orthonormal bases......Page 71
Riesz bases......Page 75
The Gram matrix......Page 80
Fourier series and trigonometric polynomials......Page 85
Wavelet bases......Page 88
Bases in Banach spaces......Page 94
Sampling and analog--digital conversion......Page 99
Exercises......Page 102
Bases in L2(0,1) and in general Hilbert spaces......Page 105
Gabor bases and the Balian--Low Theorem......Page 108
Bases and wavelets......Page 109
Frames in Hilbert Spaces......Page 112
Frames and their properties......Page 113
Frames and Riesz bases......Page 120
Frames and operators......Page 123
Characterization of frames......Page 127
Various independency conditions......Page 131
Perturbation of frames......Page 136
The dual frames......Page 141
Continuous frames......Page 144
Frames and signal processing......Page 145
Exercises......Page 148
B-splines......Page 153
The B-splines......Page 154
Symmetric B-splines......Page 160
Exercises......Page 162
Frames of Translates......Page 164
Frames of translates......Page 165
The canonical dual frame......Page 175
Compactly supported generators......Page 178
Frames of translates and oblique duals......Page 179
An application to sampling theory......Page 188
Exercises......Page 189
Frame-properties of shift-invariant systems......Page 191
Representations of the frame operator......Page 203
Exercises......Page 206
Gabor Frames in L2(R)......Page 207
Basic Gabor frame theory......Page 208
Tight Gabor frames......Page 222
The duals of a Gabor frame......Page 224
Explicit construction of dual frame pairs......Page 228
Popular Gabor conditions......Page 232
Representations of the Gabor frame operator and duality......Page 236
The Zak transform......Page 239
Time--frequency localization of Gabor expansions......Page 243
Continuous representations......Page 249
Exercises......Page 252
Translation and modulation on 2(Z)......Page 255
Gabor systems in 2(Z) through sampling......Page 256
Shift-invariant systems......Page 263
Exercises......Page 264
Wavelet Frames in L2(R)......Page 265
Dyadic wavelet frames......Page 266
The unitary extension principle......Page 272
The oblique extension principle......Page 288
Approximation orders......Page 297
Construction of pairs of dual wavelet frames......Page 298
The signal processing perspective......Page 302
A survey on general wavelet frames......Page 308
The continuous wavelet transform......Page 312
Exercises......Page 315
back-matter.pdf......Page 316
Index......Page 0