Interpolation Through Approximation in a Bernstein Space

Abstract

Let B_σ be the Bernstein space of entire functions of exponential type at most σ bounded on the real axis. Consider a sequence Λ = {z_n}_n∈ℤ, z_n = x_n + iy_n, such that x_n+1 − x_n ≥ l > 0 and |y_n| ≤ L, n ∈ ℤ. Using approximation by functions from B_σ, we prove that for any bounded sequence A = {a_n}_n∈ℤ, |a_n| ≤ M, n ∈ ℤ, there exists a function f ∈ B_σ with σ ≤ σ₀(l,L) such that f|_Λ = A.