Conditions for Convergence of Biorthogonal Expansions of Functions on a Closed Interval

Abstract

d = 0, . . . , 2n−1, where either μ0 = 1 (Problem 1) or μ0 = cλ2n−1 if |λ| ≥ 1 and μ0 = c = const 6= 0 if |λ| 0 and choose an arbitrary numerical sequence {λk}k=1, λk ∈ S1 = {λ ∈ C : Reλ ≥ 0, | Imλ| ≤ γ0}, and an arbitrary system {uk(x)} of root functions of the operator L corresponding to the spectral parameters {λk} and satisfying the following conditions A: (1) {uk} is a closed minimal system in L 0(G), and the biorthogonal system {vk} satisfies ‖uk‖r0 ‖vk‖r′ 0 ≤ c1 for all k; (2) ∑ 0≤|λk|−λ≤1 1 ≤ c2 for all λ ≥ 0; (3) ∥∥∥m−1 uk ∥∥∥ r0 ≤ c3αλ ∥∥∥muk∥∥∥ r0 , m = 1, . . . ,mk, where αλ = |λk| for Problem 1 and αλ = 1 for Problem 2; (4) ‖uk‖∞ ≤ c4 ‖uk‖r0 for all k, where ci, i = 1, 2, 3, 4, are some constants independent of λk. Let f(x) ∈ L 0(G), and let σλ(x, f) = ∑ |λk|≤λ fkuk(x), λ > 0, fk ≡ (f, vk), be a partial sum of the biorthogonal expansion of f(x). By Sλ(x, f) we denote the corresponding partial sum of the trigonometric Fourier series of f(x) treated as an orthogonal expansion of f for the operator L0 determined by L0u = u′′, u ∈ D2, u(0) = u(1), and u′(0) = u′(1). We assume that