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**Time-frequency and time-scale methods. Adaptive decompositions, uncertainty principles, and sampling.**
*(English)*
Zbl 1079.42027

Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser (ISBN 0-8176-4276-5/hbk). xxii, 388 p. (2005).

Mathematical methods play a significant role in the development of signal and image processing. Among these methods, harmonic analysis takes a prominent place. Foundations for these applications have been in place a long time ago, let me just mention here the Nobel Prize winner Dennis Gabor and his method of analyzing signals in which time and frequency play symmetrical parts. This gave start to the so-called time-frequency methods in mathematics and engineering. Also, the time-scale methods can be seen already in the Whitney decomposition and in the famous theory of Calderón and Zygmund.

But the real boom for time-frequency and time-scale methods started since the advent of the wavelet theory and the observation of the role that multiresolution algorithms play in information theory. In particular, studies devoted to wavelets and their generalizations and applications renewed the mathematicians’ interests in the Gabor systems. Moreover, it has been observed again and again that the two theories are in many ways parallel to each other.

Many texts have been written in the recent years on the two subjects. However, the book under review is written from a relatively novel perspective: its goal is to carefully and deeply study the similarities and differences between time-frequency and time-scale methods. Thus, one may view the present book as a continuation of the famous textbook of I. Daubechies [“Ten lectures on wavelets” (1992; Zbl 0776.42018)]. In my opinion, the two books nicely complement each other and both are a “must have” for everyone who is interested in a deeper understanding of the time-frequency and time-scale methods.

Chapter 1 introduces wavelets associated with multiresolution analysis (MRA) and presents several constructions of MRA wavelets with certain regularity properties. Chapter 2 is devoted to the study of these aspects of wavelets which are potentially important in their applications to partial differential equations, like, e.g., membership in Sobolev spaces or symmetry-related questions. In Chapter 3, several aspects of sampling are investigated, in particular uniqueness and reconstruction problems, as well as issues related to complexity of sampling algorithms. Chapter 4 gives several constructions of bases with good time-frequency localization and it introduces the Walsh plane as a model which leads to clear geometrical interpretations of best basis algorithms. Chapter 5 is devoted to various realizations of the notion of uncertainty principle in harmonic analysis. In Chapter 6, collections with prescribed time-frequency properties are used as bases for function spaces and for decopositions of operators acting on these spaces. Finally, Chapter 7 investigates the advances that time-scale and time-frequency analysis enabled in mathematical physics and in operator theory.

But the real boom for time-frequency and time-scale methods started since the advent of the wavelet theory and the observation of the role that multiresolution algorithms play in information theory. In particular, studies devoted to wavelets and their generalizations and applications renewed the mathematicians’ interests in the Gabor systems. Moreover, it has been observed again and again that the two theories are in many ways parallel to each other.

Many texts have been written in the recent years on the two subjects. However, the book under review is written from a relatively novel perspective: its goal is to carefully and deeply study the similarities and differences between time-frequency and time-scale methods. Thus, one may view the present book as a continuation of the famous textbook of I. Daubechies [“Ten lectures on wavelets” (1992; Zbl 0776.42018)]. In my opinion, the two books nicely complement each other and both are a “must have” for everyone who is interested in a deeper understanding of the time-frequency and time-scale methods.

Chapter 1 introduces wavelets associated with multiresolution analysis (MRA) and presents several constructions of MRA wavelets with certain regularity properties. Chapter 2 is devoted to the study of these aspects of wavelets which are potentially important in their applications to partial differential equations, like, e.g., membership in Sobolev spaces or symmetry-related questions. In Chapter 3, several aspects of sampling are investigated, in particular uniqueness and reconstruction problems, as well as issues related to complexity of sampling algorithms. Chapter 4 gives several constructions of bases with good time-frequency localization and it introduces the Walsh plane as a model which leads to clear geometrical interpretations of best basis algorithms. Chapter 5 is devoted to various realizations of the notion of uncertainty principle in harmonic analysis. In Chapter 6, collections with prescribed time-frequency properties are used as bases for function spaces and for decopositions of operators acting on these spaces. Finally, Chapter 7 investigates the advances that time-scale and time-frequency analysis enabled in mathematical physics and in operator theory.

Reviewer: Wojciech Czaja (Wien)

### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42Cxx | Nontrigonometric harmonic analysis |

42B05 | Fourier series and coefficients in several variables |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

42B25 | Maximal functions, Littlewood-Paley theory |

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

42C15 | General harmonic expansions, frames |

42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |

62D05 | Sampling theory, sample surveys |

65T60 | Numerical methods for wavelets |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

81Q15 | Perturbation theories for operators and differential equations in quantum theory |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |

81Qxx | General mathematical topics and methods in quantum theory |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

94A20 | Sampling theory in information and communication theory |