Frames and numerical approximation – supplementary materials

This document contains supplementary materials for the paper Frames and numerical approximation by B. Adcock & D. Huybrechs [3]. SM1 Example 1. Fourier frames for complex geometries Consider the frame (3.1) over a domain Ω ⊆ (−1, 1)d. SM1.1 The kernel of G We first characterize the kernel of the Gram operator: Proposition SM1.1. Let G be the Gram operator (2.7) of the frame (3.1). Then Ker(G) = { {f̂n}n∈Zd : f ∈ L(−1, 1), f(x) = 0 a.e. x ∈ Ω } , where f̂n = ∫ (−1,1)d f(t)φn(t) dt are the Fourier coefficients of f ∈ L 2(−1, 1)d. Proof. Let f ∈ L2(−1, 1)d with f |Ω = 0. Let x = {f̂n}n∈Zd . Then Gx = {〈f, φn〉}n∈Zd = 0. Conversely, if x ∈ Ker(G) then x = {f̂n}n∈Zd for some f ∈ L2(−1, 1)d. Notice that 0 = x∗Gx = ∥∥∥∥∥ ∑ n∈Zd xnφn ∥∥∥∥∥ 2