Degree of approximation by superpositions of a sigmoidal function期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Degree of approximation by superpositions of a sigmoidal function

Authors:	Chen Debao

Affiliation:	(1) Department of Mathematics, University of Texas at Austin, 78712 Austin, TX, U.S.A.

Abstract:	In this paper we study the degree of approximation by superpositions of a sigmoidal function. We mainly consider the univariate case. If f is a continuous function, we prove that for any bounded sigmoidal function σ, $sigma ,d_{n,sigma } (f) leqslant \|\|sigma \|\|omega (f,frac{1}{{n + 1}})$ . For the Heaviside function H(x), we prove that $d_{n,H} (f) leqslant omega (f,frac{1}{{2(n + 1)}})$ . If f is a continuous function of bounded variation, we prove that $d_{n,sigma } (f) leqslant frac{{\|\|sigma \|\|}}{{(n + 1)}})V(f)$ and $d_{n,H} leqslant (f,frac{1}{{2(n + 1)}})V(f)$ . For he Heaviside function, the coefficient 1 and the approximation orders are the best possible. We compare these results with the classical Jackson and Bernstein theorems, and make some conjectures for further study.

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