Invariance Theorems in Approximation Theory and their Applications

Abstract

Let B be a closed linear subspace of a Banach space F and let \(\{T_s\}_{s\in G}\) be a group of continuous linear operators \(T_s:F\to F\), where G is a compact topological group. We prove that if \(f\in F\) is invariant under \(\{T_s\}_{s\in G}\), then under some conditions on f, F, B, and G, there exists an element \(g^*\in B\) of best approximation to f that has the same property. As applications, we compute the bivariate Bernstein constant for \(L_1\) polynomial approximation of \(|x|^{\lambda}\) and solve a Braess problem on the exponential order of decay of the error of polynomial approximation of \(|x-a|^{-{\lambda}}\). Other examples and applications are discussed as well.