Rigidity of optimal bases for signal spaces (Q2012348)

Let \(\Omega \subset {\mathbb R}^N\) be a smooth bounded domain. Let \((e_j)_{j=1}^{\infty}\) be an orthonormal basis of \(L^2(\Omega)\) consisting of the eigenfunctions of the Laplace operator with homogeneous Dirichlet boundary conditions, where the related eigenvalues \(\lambda_j\) are ordered by \(0 < \lambda_1 \leq \lambda_2 \leq\,\ldots\). The authors show that \((e_j)_{j=1}^{\infty}\) is the only orthonormal basis of \(L^2(\Omega)\) that provides an optimal approximation of arbitrary \(f\in H_0^1(\Omega)\) in the sense \[ \| f - \sum_{j=1}^n (f,e_j)\,e_j\|_{L^2(\Omega)}^2 \leq \frac{1}{\lambda_{n+1}}\, \| \nabla f\|_{L^2(\Omega)}^2 \] for all \(n\in \mathbb N\). This nice result solves an open problem raised by \textit{Y. Aflalo} et al. [C.R., Math., Acad. Sci. Paris 354, No. 12, 1155--1167 (2016; Zbl 1361.94022)].