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$$L^ 1$$-theory of approximation by ridge functions. (English) Zbl 0745.62063
Summary: To approximate multivariate functions by finite sums of ridge functions a basic tool is projection pursuit regression. Its $$L^ 2$$-theory has been investigated by several authors. We discuss its $$L^ 1$$-theory. The main results are: Any integrable multivariate function can be $$L^ 1$$- approximated by finite linear combinations of exponential ridge functions with rational directions (at most denumerable many different directions); if the probability distribution $$F$$ is exponentially integrable, we may use polynomial ridge functions to approximate any integrable multivariate function.

##### MSC:
 62H99 Multivariate analysis 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)