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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.

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