On approximation of entire functions by exponential polynomials
Let \(K\) be a compact set in \(\mathbb{R}^n\) and \(B_K\) be a set of all continuous bounded functions on \(\mathbb{R}^n\), whose spectrum belongs to \(K\). The author investigates the question of approximation of the function \(f \in B_{K}\) on arbitrary compact sets \(S \subset \mathbb{R}^n\) by finite exponential sums \[ P_m(x) = \sum_\nu C^{(m)}_\nu e^{i\langle \alpha^{(m)} \nu, x\rangle},\;\nu \in \mathbb{Z},\;\alpha^{(m)} \in (0,\varepsilon), \] which additionally satisfy the condition \(\sup \{|P_m(x) |: x \in \mathbb{R}^n\} \leq \sup \{|f(x)|: x\in \mathbb{R}^n\}\). Some estimates of the rate of the approximation are given. The case of convex compact \(K\) is considered separately.
scientific article; zbMATH DE number 813158