An analog of the Jackson-Nikol'skij theorem on the approximation by superpositions of sigmoidal functions
A bounded function \(\varphi: \mathbb{R} \to\mathbb{R}\) is said to be sigmoidal if \(\lim_{x \to-\infty} \varphi(x) =0\) and \(\lim_{x \to+ \infty} \varphi(x) =1\). Let \(f:[-1,1] \to\mathbb{R}\) be an arbitrary bounded function with modulus of continuity \(\omega\). Theorem. For every sigmoidal function \(\varphi\) and \(n\in \mathbb{N}\), and number \(q>1\) there are \(\{a_i,b_i,c_i\}^n_{i=1} \subset \mathbb{R}\) such that \[ \left|f(x) -\sum^n_{i=1} c_i\varphi (a_ix+b_i) \right|\leq C(q,\varphi) \omega \left({\bigl(1- |x|\bigr) \over n}^{1-1/q} +{1\over n^q} \right). \] Here \(C(q, \varphi)\) depends only on \(q\) and \(\sup_{\mathbb{R}} |\varphi |\).
scientific article; zbMATH DE number 1037780