On Approximation by Ridge Functions

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作者
A. Kroó
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[1] Mathematical Institute of the Hungarian Academy of Sciences,
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Key words and phrases: Ridge functions, Homogeneous polynomials, Fundamentality, Rate of approximation. AMS Classification: 41A45, 41A63.;
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Ridge functions are defined as functions of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $f(\mbox{\footnotesize\bf a}\cdot \mbox{\footnotesize\bf x})$ \end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $f\colon\ {\Bbb R}\rightarrow {\Bbb R}$ \end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mbox{\footnotesize\bf x}\in {\Bbb R}^k$, and $\mbox{\footnotesize\bf a}$ \end{document} belongs to the given ``direction'' set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $A\subset {\Bbb R}^k$ \end{document} . In this paper we study the fundamentality of ridge functions for variable directions sets A and discuss the rate of approximation by ridge functions.
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页码:447 / 460
页数:13
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