Rigidity of optimal bases for signal spaces

Abstract We discuss optimal L 2 -approximations of functions controlled in the H 1 -norm. We prove that the basis of eigenfunctions of the Laplace operator with Dirichlet boundary condition is the only orthonormal basis ( b i ) of L 2 that provides an optimal approximation in the sense of ‖ f − ∑ i = 1 n ( f , b i ) b i ‖ L 2 2 ≤ ‖ ∇ f ‖ L 2 2 λ n + 1 ∀ f ∈ H 0 1 ( Ω ) , ∀ n ≥ 1 . This solves an open problem raised by Y. Aflalo, H. Brezis, A. Bruckstein, R. Kimmel, and N. Sochen (Best bases for signal spaces, C. R. Acad. Sci. Paris, Ser. I 354 (12) (2016) 1155–1167).