Approximation of bandlimited functions by finite exponential sums

Abstract

For a compact set K in ℝ n , let B 2 K be the set of all functions f ∈ L 2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f of f is supported by K. We investigate the question of approximation of f ∈ B 2 K by finite exponential sums $$ \sum\limits_{k \in {\mathbb{Z}^n} \cap \left( {{\tau \mathord{\left/{\vphantom {\tau \pi }} \right.} \pi }} \right)K} {{c_k}{{\text{e}}^{i\frac{\pi }{\tau }\left( {x,k} \right)}}} $$ in the space \( L^{\rm 2}(\tau\mathbb{T}^n),\ \mathbb{T}^n=[-1,1]^n\), as τ → ∞.