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    Introduction to Frames and Riesz Bases

    ISBN-10: 0817642951
    ISBN-13: 9780817642952
    Edition: 2003
    Author(s): Ole Christensen
    Description: The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of  More...
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    List Price: $79.99
    Copyright Year: 2003
    Publisher: Birkhauser Boston
    Publication Date: 12/13/2002
    Binding: Hardcover
    Pages: 440
    Size: 6.25" wide x 9.25" long x 1.25" tall
    Weight: 1.782

    The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature. Key features: * Basic results presented in an accessible way for both pure and applied mathematicians * Extensive exercises make the work suitable as a textbook for use in graduate courses * Full proofs included in introductory chapters; only basic knowledge of functional analysis required * Explicit constructions of frames with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series * Selected research topics presented with recommendations for more advanced topics and further reading * Open problems to stimulate further research An Introduction to Frames and Riesz Bases will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference.

    Dept of Mathematics, Denmark Technical University, Lyngby, Denmark

    Frames in Finite-dimensional Inner Product Spaces
    Some basic facts about frames
    Frame bounds and frame algorithms
    Frames in C[superscript n]
    The discrete Fourier transform
    Pseudo-inverses and the singular value decomposition
    Finite-dimensional function spaces
    Infinite-dimensional Vector Spaces and Sequences
    Banach spaces and Hilbert spaces
    L[superscript 2] (R) and l[superscript 2] (N)
    The Fourier transform
    Operators on L[superscript 2] (R)
    Bases in Banach spaces
    Bessel sequences in Hilbert spaces
    Bases and biorthogonal systems in H
    Orthonormal bases
    The Gram matrix
    Riesz bases
    Fourier series and Gabor bases
    Wavelet bases
    Bases and their Limitations
    Gabor systems and the Balian-Low Theorem
    Bases and wavelets
    General shortcomings
    Frames in Hilbert Spaces
    Frames and their properties
    Frame sequences
    Frames and operators
    Frames and bases
    Characterization of frames
    The dual frames
    Tight frames
    Continuous frames
    Frames and signal processing
    Frames versus Riesz Bases
    Conditions for a frame being a Riesz basis
    Riesz frames and near-Riesz bases
    Frames containing a Riesz basis
    A frame which does not contain a basis
    A moment problem
    Frames of Translates
    Sequences in R[superscript d]
    Frames of translates
    Frames of integer-translates
    Irregular frames of translates
    The sampling problem
    Frames of exponentials
    Gabor Frames in L[superscript 2] (R)
    Continuous representations
    Gabor frames
    Necessary conditions
    Sufficient conditions
    The Wiener space W
    Special functions
    General shift-invariant systems
    Selected Topics on Gabor Frames
    Popular Gabor conditions
    Representations of the Gabor frame operator and duality
    The duals of a Gabor frame
    The Zak transform
    Tight Gabor frames
    The lattice parameters
    Irregular Gabor systems
    Applications of Gabor frames
    Wilson bases
    Gabor Frames in l[superscript 2] (Z)
    Translation and modulation on l[superscript 2] (Z)
    Discrete Gabor systems through sampling
    Gabor frames in C[superscript L]
    Shift-invariant systems
    Frames in l[superscript 2] (Z) and filter banks
    General Wavelet Frames
    The continuous wavelet transform
    Sufficient and necessary conditions
    Irregular wavelet frames
    Oversampling of wavelet frames
    Dyadic Wavelet Frames
    Wavelet frames and their duals
    Tight wavelet frames
    Wavelet frame sets
    Frames and multiresolution analysis
    Frame Multiresolution Analysis
    Frame multiresolution analysis
    Sufficient conditions
    Relaxing the conditions
    Construction of frames
    Frames with two generators
    Some limitations
    Wavelet Frames via Extension Principles
    The general setup
    The unitary extension principle
    Applications to B-splines I
    The oblique extension principle
    Fewer generators
    Applications to B-splines II
    Approximation orders
    Construction of pairs of dual wavelet frames
    Applications to B-splines III
    Perturbation of Frames
    A Paley-Wiener Theorem for frames
    Compact perturbation
    Perturbation of frame sequences
    Perturbation of Gabor frames
    Perturbation of wavelet frames
    Perturbation of the Haar wavelet
    Approximation of the Inverse Frame Operator
    The first approach
    A general method
    Applications to Gabor frames
    Integer oversampled Gabor frames
    The finite section method
    Expansions in Banach Spaces
    Representations of locally compact groups
    Feichtinger-Grochenig theory
    Banach frames
    Gabor systems and wavelets in L[superscript p] (R) and related spaces
    Appendix A
    Normed vector spaces and inner product spaces
    Linear algebra
    Some special normed vector spaces
    Operators on Banach spaces
    Operators on Hilbert spaces
    The pseudo-inverse
    Some special functions
    List of symbols

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