## A First Course on Wavelets (Google eBook)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: 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

1 | |

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 |

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 |

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 |

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 |

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 |

467 | |

479 | |

485 | |

### Common terms and phrases

1-periodic 27r-periodic function algorithm apply associated assume atomic Balian-Low theorem Banach space band-limited basis for L2(R bell function belongs bounded Chapter characterization compactly supported compactly supported wavelets condition construct convergence Corollary cosine decomposition deduce definition dilations dyadic equality equations equivalent example exists a constant f 7r fact fcez formula Fourier series Fourier transform frame Franklin wavelet given graph Haar wavelet Hardy space Hence Hilbert space implies inequality integral interval ipj,k L2-norm La(R Lebesgue Lemarie-Meyer wavelets Lemma linear low-pass filter LP(R Moreover MSF wavelets multiresolution analysis norm Observe obtain operator orthogonal orthonormal basis orthonormal system Plancherel theorem polarities proof of Theorem properties Proposition 1.11 prove satisfies scaling function sequence Shannon wavelet smooth spline wavelets subspaces supp Suppose Theorem 4.1 translations trigonometric polynomial unconditional basis unitary wavelet ip wavelet packets write zero

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Page 2 - PRELIMINARIES We assume that the reader is familiar with the basic notions of formal languages and codes, cf . , eg, [7] or [1].

Page x - Fourier analysis. In the latter case the goal is to measure the local frequency content of a signal, while in the wavelet case one is comparing several magnifications of this signal, with distinct resolutions.