A mathematical introduction to wavelets.

*(English)*Zbl 0865.42026
London Mathematical Society Student Texts. 37. Cambridge: Cambridge University Press. xii, 261 p. (1997).

The book presents the essential mathematical core of the theory of wavelets. It is concentrated on orthonormal wavelets, and is especially intended for use by pure mathematicians. The book is suitable as a basis for wavelet courses for senior undergraduates in mathematics. The reader should be familiar with basics of real analysis, functions, function series and Lebesgue integration. Additionally, elementary knowledge of the Fourier transform and Hilbert spaces are needed. These facts are summarized in the Appendix. For the second part of the book, one needs the concept and basic properties of Banach spaces. Detailed constructions of most important wavelets are given and the usefulness of wavelet bases in decomposing functions is presented. All this is done in the framework of function spaces.

In the first part of the book one variable orthonormal wavelets are studied. As an introduction into the field, the Haar wavelet and the StrĂ¶mberg wavelet are discussed. Then the concept of multiresolution analysis and scaling functions is presented. It is shown, how associated wavelets can be constructed. The general theory is applied to different concrete cases. Meyer’s wavelets and spline wavelets are constructed and their smoothness and decay properties are discussed. In Chapter 3, examples of wavelets are considered, which are not associated with any multiresolution analysis. Further, a general approach to construction of compactly supported and smooth wavelets is presented. Chapter 5 is devoted to multivariable generalizations. The tensor product technique is briefly discussed, and the concept of multiresolution analysis is generalized to \(\mathbb{R}^d\). In this case, a finite wavelet set is needed instead of only one wavelet.

The second part of the book deals with wavelet expansions. In Chapter 6, a self-contained presentation of the basic theory of \(L_p\) spaces, \(H_1\) and BMO is given. Some necessary interpolation theorems are proved. Further, unconditional convergence of series of Banach spaces is introduced and the concept of unconditional basis is discussed. In Chapter 8, it is shown that wavelets can provide good unconditional bases in \(L_p(\mathbb{R}^d)\) and in \(H^1(\mathbb{R}^d)\). Finally, the moduli of continuity and Besov norm as well as their connection with wavelets are considered.

Each chapter ends with historical comments and exercises.

In the first part of the book one variable orthonormal wavelets are studied. As an introduction into the field, the Haar wavelet and the StrĂ¶mberg wavelet are discussed. Then the concept of multiresolution analysis and scaling functions is presented. It is shown, how associated wavelets can be constructed. The general theory is applied to different concrete cases. Meyer’s wavelets and spline wavelets are constructed and their smoothness and decay properties are discussed. In Chapter 3, examples of wavelets are considered, which are not associated with any multiresolution analysis. Further, a general approach to construction of compactly supported and smooth wavelets is presented. Chapter 5 is devoted to multivariable generalizations. The tensor product technique is briefly discussed, and the concept of multiresolution analysis is generalized to \(\mathbb{R}^d\). In this case, a finite wavelet set is needed instead of only one wavelet.

The second part of the book deals with wavelet expansions. In Chapter 6, a self-contained presentation of the basic theory of \(L_p\) spaces, \(H_1\) and BMO is given. Some necessary interpolation theorems are proved. Further, unconditional convergence of series of Banach spaces is introduced and the concept of unconditional basis is discussed. In Chapter 8, it is shown that wavelets can provide good unconditional bases in \(L_p(\mathbb{R}^d)\) and in \(H^1(\mathbb{R}^d)\). Finally, the moduli of continuity and Besov norm as well as their connection with wavelets are considered.

Each chapter ends with historical comments and exercises.

Reviewer: G.Plonka (Rostock)

##### MSC:

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

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

41A15 | Spline approximation |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |