Symmetrization Formulas and Norm Estimates of Projections in Multivariate Polynomial Approximation

Abstract

Suppose we have a bounded linear operator on a Banach space X. If we know in addition a compact group of operators acting on X, this fact can sometimes be exploited to get a lower bound for the operator norm ∥A∥ of A. For example let $\overline T$ be the one-dimensional torus group which can be identified with the interval [-π,π] and denote by $X = C(\overline T )$ the Banach space of all complex valued 2π-periodic continuous functions on ℝ with the Čebyšev norm ∥ ∥. $\overline T$ is realizable as a group of isometries on $C(\overline T )$ if we associate with each $\lambda \in \overline T$ the translation operator T_λ, which is defined by ${T_\lambda }f(x) = f(x + \lambda ),x,\lambda \in \mathbb{R}$. Then, we can define the symmetrization A of A relative to ${({T_\lambda })_{\lambda \in \overline T }}$ as the operator ${A_s} = 1/2\pi \int\limits_\pi ^\pi {{T_\lambda }} A{T_{ - \lambda }}d\lambda$ the integration of the (operator valued, continuous) integrand causing no difficulties. Clearly $\left\| {{A_S}} \right\| \leqslant \left\| A \right\|$. Now, in concrete situations, A_S can often be computed explicitly, hence also ∥A_S∥, which furnishes the desired lower bound of ∥A∥. This technique has been employed by D.L. Berman [1] to prove the minimal property of the Fourier projections and has been extended to compact groups by Rudin [14], Lambert [12], Daughavet [4], and Dreseler-Schempp [8].