On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Carsten Elsner ¹

¹ Institut für Mathematik Universität Hannover Welfengarten 1 D-30167 Hannover, Germany

Colloquium Mathematicum, Tome 104 (2006) no. 1, pp. 57-84

Résumé

We prove the existence of an effectively computable integer polynomial $P(x,t_0,\dots ,t_5) $ having the following property. Every continuous function $f: {\mathbb R}^s \to {\mathbb R} $ can be approximated with arbitrary accuracy by an infinite sum $$ \sum_{r=1}^{\infty} H_r(x_1,\dots ,x_s) \in C^{\infty}({\mathbb R}^s) $$ of analytic functions $H_r $, each solving the same system of universal partial differential equations, namely $$ P\bigg( x_{\sigma}; H_r , \frac{\partial H_r}{\partial x_{\sigma}} , \dots , \frac{{\partial}^5 H_r}{\partial x_{\sigma}^5} \bigg) = 0 \quad\ (\sigma =1, \dots ,s) . $$

Détail

DOI : 10.4064/cm104-1-4

Keywords: prove existence effectively computable integer polynomial dots having following property every continuous function mathbb mathbb approximated arbitrary accuracy infinite sum sum infty dots infty mathbb analytic functions each solving system universal partial differential equations namely bigg sigma frac partial partial sigma dots frac partial partial sigma bigg quad sigma dots