The Cauchy Transform * (Mathematical Surveys and Monographs 125) * By Joseph A. Cima, Alec L. Matheson and William T. Ross: * 272 pp., {pound}43.75 (US$75), ISBN 0-8218-3871-7 * (American Mathematical Society, Providence, RI, 2006).

In the border country between complex analysis, harmonic analysisand differential equations, there can be found many populartransforms, of which those of Fourier and Laplace are probablythe most commonly used. This book is concerned with anotherone, which is more rarely sighted. The Cauchy transform of a finite complex measure µ definedon the unit circle in the complex plane is the analytic functionKµ defined by

forz in the unit disc . Its name derives from the observation thatCauchy's integral formula, when applied to an analytic functionf, is equivalent to the statement that the Cauchy transformof the measure is just     

Spectral properties of banded Toeplitz matrices * By Albrecht Bottcher and Sergei M. Grudsky: * 411 pp., US$ 95.00, ISBN 0-89871-599-7 * (Society for Industrial and Applied Mathematics, Philadelphia, 2005)

Recall that an infinite Toeplitz matrix, , say, is one for which the values t_jk dependonly on j – k, so we may write t_jk = a_{j – k}; thusthe entries are constant on diagonals sloping down from topleft to bottom right:

Sucha matrix is said to be banded if there are only finite manynon-zero a_j, say . The propertiesof such a matrix, when defining a linear operator T on a sequencespace, are often closely related to the properties of the Laurentpolynomial

for zlying on the unit circle in the complex plane. For example,suppose that the matrix acts on the most familiar sequence space,the Hilbert space ²; then it has been known for many years thatthe operator norm of T is the maximum