Intelligent Systems Reference Library 19. Berlin: Springer (ISBN 978-3-642-21430-1/hbk; 978-3-642-21431-8/ebook). v, 107 p. EUR 99.95/net; SFR 143.50; £ 90.00; $ 129.00 (2011).

It is well known that feed forward neural networks are universal approximators. Continuous functions defined on compact sets can be, for instance, approximated to any degree of accuracy by a feed forward network. In implementations, specific activation functions are the sigmoidal and the hyperbolic tangent functions. The present work is devoted to the study of convergence rates and upper bounds of approximation errors. The architecture type of the network is the MLP with only one hidden layer. The author presents his own discoveries in the period 1997-2011 and the book is essentially related to one other article, namely [

*Z. Chen* and

*F. Cao*, “The approximation operators with sigmoidal functions”, Comput. Math. Appl. 58, No. 4, 758–765 (2009;

Zbl 1189.41014)]. Throughout all chapters of the book the same method, the same construction is used. The interpolation neural network operators are defined by means of a superposition of sigmoidal or hyperbolic tangent functions. The convergence rates are expressed via the modulus of continuity of the involved function or its higher order derivative and are given by Jackson type inequalities. Upper bounds to error quantities are then discussed. The first chapter, Univariate sigmoidal neural network quantitative approximation, is devoted to univariate approximations of real and complex valued continuous functions by means of interpolation operators based on sigmoidal activation function. In the second chapter, for the same approximation, interpolation operators based on superposition of hyperbolic tangent functions are considered. In the third chapter , Multivariate sigmoidal neural network quantitative approximation, results from previous chapters are extended to multivariate approximation of real and complex valued continuous multivariate functions, by means of multivariate quasi-interpolation sigmoidal neural networks. For the same approximation problem, in the forth chapter, multivariate quasi-interpolation hyperbolic tangent neural network operators are used. The book has 107 pages and references are added to each chapter. The formal presentation by the Springer Verlag is excellent.