A First Course on Wavelets (Google eBook)

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CRC Press, Sep 12, 1996 - Mathematics - 489 pages
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Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets.

The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets.

Some of the topics treated include:
  • Several bases generated by a single function via translations and dilations
  • Multiresolution analysis, compactly supported wavelets, and spline wavelets
  • Band-limited wavelets
  • Unconditionality of wavelet bases
  • Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functions

    The authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented.

    Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.
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    Contents

    Bases for LČR
    1
    11 Preliminaries
    2
    12 Orthonormal bases generated by a single function the BalianLow theorem
    7
    13 Smooth projections on LČR
    11
    14 Local sine and cosine bases and the construction of some wavelets
    20
    15 The unitary folding operators and the smooth projections
    31
    16 Notes and references
    40
    Multiresolution analysis and the construction of wavelets
    43
    61 Wavelets and sampling theorems
    256
    62 LittlewoodPaley theory
    260
    63 Necessary tools
    267
    64 The Lebesgue spaces LpR with 1 p oo
    279
    65 The Hardy space HlR
    287
    66 The Sobolev spaces LPSR 1 p oo s 123
    296
    67 The Lipschitz spaces AQ R 0 a 1 and the Zygmund class A R
    313
    68 Notes and references
    323

    21 Multiresolution analysis
    44
    22 Construction of wavelets from a multiresolution analysis
    52
    23 The construction of compactly supported wavelets
    68
    24 Better estimates for the smoothness of compactly supported wavelets
    93
    25 Notes and references
    98
    Bandlimited wavelets
    101
    32 Completeness
    105
    33 The LemariéMeyer wavelets revisited
    115
    34 Characterization of some bandlimited wavelets
    123
    35 Notes and references
    138
    Other constructions of wavelets
    141
    42 Spline wavelets on the real line
    152
    43 Orthonormal bases of piecewise linear continuous functions for L2T
    164
    44 Orthonormal bases of periodic splines
    176
    45 Periodization of wavelets defined on the real line
    186
    46 Notes and references
    193
    Representation of functions by wavelets
    201
    52 Unconditional bases for Banach spaces
    206
    53 Convergence of wavelet expansions in V
    217
    54 Pointwise convergence of wavelet expansions
    225
    55 Hč and BMO on R
    230
    56 Wavelets as unconditional bases for HR and V with 1 p oo
    237
    57 Notes and references
    247
    Characterizations of function spaces using wavelets
    255
    Characterizations in the theory of wavelets
    331
    71 The basic equations
    332
    72 Some applications of the basic equations
    348
    73 The characterization of MRA wavelets
    354
    74 A characterization of lowpass filters
    365
    75 A characterization of scaling functions
    381
    76 Nonexistence of smooth wavelets in HČR
    386
    77 Notes and references
    393
    Frames
    397
    81 The reconstruction formula for frames
    398
    82 The BalianLow theorem for frames
    403
    83 Frames from translations and dilations
    410
    84 Smooth frames forHČl
    418
    85 Notes and references
    419
    Discrete transforms and algorithms
    427
    92 The discrete cosine transform DCT and the fast cosine transform FCT
    432
    93 The discrete version of the local sine and cosine bases
    436
    94 Decomposition and reconstruction algorithms for wavelets
    442
    95 Wavelet packets
    449
    96 Notes and references
    464
    References
    467
    Author index
    479
    Index
    485
    Copyright

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