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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)
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