  These notes are about ridge functions. Recent years have witnessed a flurry
of interest in these functions. Ridge functions appear in various fields and
under various guises. They appear in fields as diverse as partial differential
equations (where they are called plane waves), computerized tomography and
statistics. These functions are also the underpinnings of many central models
in neural networks.
  We are interested in ridge functions from the point of view of approximation
theory. The basic goal in approximation theory is to approximate complicated
objects by simpler objects. Among many classes of multivariate functions,
linear combinations of ridge functions are a class of simpler functions. These
notes study some problems of approximation of multivariate functions by linear
combinations of ridge functions. We present here various properties of these
functions. The questions we ask are as follows. When can a multivariate
function be expressed as a linear combination of ridge functions from a certain
class? When do such linear combinations represent each multivariate function?
If a precise representation is not possible, can one approximate arbitrarily
well? If well approximation fails, how can one compute/estimate the error of
approximation, know that a best approximation exists? How can one characterize
and construct best approximations? If a smooth function is a sum of arbitrarily
behaved ridge functions, can it be expressed as a sum of smooth ridge
functions? We also study properties of generalized ridge functions, which are
very much related to linear superpositions and Kolmogorov's famous
superposition theorem. These notes end with a few applications of ridge
functions to the problem of approximation by single and two hidden layer neural
networks with a restricted set of weights.
  We hope that these notes will be useful and interesting to both researchers
and students.
