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Recent years have witnessed a growth of interest in the special
functions called ridge functions. These functions appear in various
fields and under various guises. They appear in partial
differential equations (where they are called plane waves), in
computerized tomography, and in statistics. Ridge functions are
also the underpinnings of many central models in neural network
theory. In this book various approximation theoretic properties of
ridge functions are described. This book also describes properties
of generalized ridge functions, and their relation to linear
superpositions and Kolmogorov's famous superposition theorem. In
the final part of the book, a single and two hidden layer neural
networks are discussed. The results obtained in this part are based
on properties of ordinary and generalized ridge functions. Novel
aspects of the universal approximation property of feedforward
neural networks are revealed. This book will be of interest to
advanced graduate students and researchers working in functional
analysis, approximation theory, and the theory of real functions,
and will be of particular interest to those wishing to learn more
about neural network theory and applications and other areas where
ridge functions are used.
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