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

Carsten Elsner. "On the approximation of real continuous functions by series of solutions of a single system of partial differential equations." Colloquium Mathematicae 104.1 (2006): 57-84. <http://eudml.org/doc/284024>.

@article{CarstenElsner2006,
abstract = {We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f: ℝ^\{s\} → ℝ$ can be approximated with arbitrary accuracy by an infinite sum $∑_\{r=1\}^\{∞\} H_\{r\}(x₁,...,x_\{s\}) ∈ C^\{∞\}(ℝ^\{s\})$ of analytic functions $H_\{r\}$, each solving the same system of universal partial differential equations, namely $P(x_\{σ\};H_r,∂H_\{r\}/∂x_\{σ\},...,∂⁵H_\{r\}/∂x⁵_\{σ\}⁵) = 0$ (σ = 1,..., s).},
author = {Carsten Elsner},
journal = {Colloquium Mathematicae},
keywords = {ordinary differential equations; approximation by solutions; universal differential equation},
language = {eng},
number = {1},
pages = {57-84},
title = {On the approximation of real continuous functions by series of solutions of a single system of partial differential equations},
url = {http://eudml.org/doc/284024},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Carsten Elsner
TI - On the approximation of real continuous functions by series of solutions of a single system of partial differential equations
JO - Colloquium Mathematicae
PY - 2006
VL - 104
IS - 1
SP - 57
EP - 84
AB - We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f: ℝ^{s} → ℝ$ can be approximated with arbitrary accuracy by an infinite sum $∑_{r=1}^{∞} H_{r}(x₁,...,x_{s}) ∈ C^{∞}(ℝ^{s})$ of analytic functions $H_{r}$, each solving the same system of universal partial differential equations, namely $P(x_{σ};H_r,∂H_{r}/∂x_{σ},...,∂⁵H_{r}/∂x⁵_{σ}⁵) = 0$ (σ = 1,..., s).
LA - eng
KW - ordinary differential equations; approximation by solutions; universal differential equation
UR - http://eudml.org/doc/284024
ER -