Mark A. Pinsky

Review of *Wavelets through a Looking Glass: The World of the Spectrum* by Ola Bratteli and Palle Jorgensen, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2002.

This book does a superlative job of demonstrating the richness of the theory of wavelets, which began as an outgrowth of classical harmonic analysis. The authors have demonstrated further connections with spectral theory, ergodic theory, homotopy theory and the theory of probability---just to name a few of the well-established areas of mathematics which are shown to touch the theory of wavelets. At the same time the material is beautifully documented by means of 61 figures, numerous tables and other illustrations which are freely distributed throughout the book. An exhaustive list of 200 references to the most current literature ensures scholarly care and the most up-to-date account of the topics covered.

The authors succeed admirably in achieving the two-fold purpose of the book. On the one hand the goal is to give a modern (but "timeless") presentation of wavelet theory while on the other hand the goal is to present new results that have not previously been published. The latter include material on homotopy of resolutions, approximation theory and results on the spectrum of the associated transfer operators and subdivision operators. The first goal is well-served by the many well-documented exercises which appear at the end of each chapter. The pedagogy is further enhanced by several paragraphs of illuminating prose at the beginning of each chapter---to set the stage for the technical material to follow. Although not written as a conventional text, one would expect that an industrious graduate student could profit enormously from a serious exposure to this book.

With respect to the literature on wavelets, it is difficult to recall any other book that is so well documented both with graphical and numerical details as well as mathematical proof. This volume will remain a central reference work for many years to come.

to "About the Book