[...]... 52 55 59 64 69 72 78 83 86 4 Bases and their Limitations 4.1 Bases in L2 (0, 1) and in general Hilbert spaces 4.2 Gabor bases and the Balian–Low Theorem 4.3 Bases and wavelets 89 89 92 93 5 Frames in Hilbert Spaces 5.1 Frames and their properties 5.2 Frames and Riesz bases 5.3 Frames and operators 5.4 Characterization of frames 5.5 Various independency... role in Gabor theory and wavelet analysis; these subjects are not treated in classical analysis courses and are therefore described in detail Chapter 3 deals with the theory for bases in Hilbert spaces and Banach spaces The most important part of the chapter is formed by a detailed discussion of Bessel sequences and Riesz bases The chapter also contains sections on Fourier analysis and wavelet theory,... the theory for bases and frames The content can naturally be split into two parts: Chapters 1–5 describe the theory on an abstract level, and Chapters 7–11 deal with explicit constructions in L2 -spaces The link between these two parts is formed by Chapter 6, which introduces B-splines and their main properties Some years ago, I published the book An Introduction to Frames and Riesz Bases [10], which...xii Contents 3 Bases 3.1 Bessel sequences in Hilbert spaces 3.2 General bases and orthonormal bases 3.3 Riesz bases 3.4 The Gram matrix 3.5 Fourier series and trigonometric polynomials 3.6 Wavelet bases 3.7 Bases in Banach spaces 3.8 Sampling and analog–digital conversion 3.9 Exercises ... figures Finally, I would like to thank Richard Laugesen and Azita Mayeli for correcting parts of the material, as well as Henrik Stetkær and Kil Kwon for several suggestions concerning the presentation of the material I also thank the staff at Birkh¨user, especially Tom Grasso, for assistance a and support Ole Christensen Kgs Lyngby, Denmark November 2007 Frames and Bases 1 Frames in Finite-dimensional Inner... parts of the book and with a large part of the research literature concerning abstract frame theory • A theoretical graduate course on bases and frames could be based on Chapter 2, Chapter 3, and Chapter 5 It would be natural to continue with one or more chapters on concrete frame constructions in L2 (R) • For a course focusing on either Gabor analysis or wavelets, the detailed analysis of frames in Chapter... compact and that the function m φ:C m → R, φ{dk }m k=1 |dk | := k=1 is continuous There are some important differences between Theorem 1.1.5 and Theorem 1.1.10 In Theorem 1.1.5, we find the sequence minimizing the 2 -norm of the coefficients in the expansion of f explicitly; it is unique, and it depends linearly on f On the other hand, Theorem 1.1.10 only gives the existence of an 1 -minimizer, and it... 139 140 146 148 7 Frames of Translates 7.1 Frames of translates 7.2 The canonical dual frame 7.3 Compactly supported generators 7.4 Frames of translates and oblique duals 7.5 An application to sampling theory 7.6 Exercises 151 152 162 165 166 175 176 8 Shift-Invariant Systems 8.1 Frame-properties of shift-invariant systems 8.2 Representations... on the properties one can obtain from bases Hereby, the reader is provided with motivation for considering the generalizations of bases studied in the rest of the book Chapter 5 contains the core material about frames in general Hilbert spaces It gives a detailed description of frames with full proofs, relates frames and Riesz bases, and provides various ways of constructing frames Preface xvii Chapter... Gabor analysis are based on results derived in the chapter about shift-invariant systems In general, careful cross-references (and, if necessary, repetitions) between Chapters 7–11 are provided Depending on the level and specific interests of the students, a graduate course based on the book can proceed in various ways: • Readers with a limited background in functional analysis (and readers who just want . Exercises 86 4 Bases and their Limitations 89 4.1 Bases in L 2 (0, 1) and in general Hilbert spaces 89 4.2 Gabor bases and the Balian–Low Theorem 92 4.3 Basesandwavelets 93 5 Frames in Hilbert. Karen ANHA Series Preface The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic. 97 5.1 Framesandtheirproperties 98 5.2 FramesandRieszbases 105 5.3 Frames and operators . . 108 5.4 Characterization of frames 112 5.5 Various independency conditions 116 5.6 Perturbation of frames