Approximation of entire functions of exponential type by exponential sums*

Abstract

Let K be a compact set in \( {{\mathbb R}^n} \). For \( 1 \leqslant p \leqslant \infty \), the Bernstein space \( B_K^p \) is the Banach space of all functions \( f \in {L^p}\left( {{{\mathbb R}^n}} \right) \)such that their Fourier transform in a distributional sense is supported on K. If \( f \in B_K^p \), then f is continuous on \( {{\mathbb R}^n} \) and has an extension onto the complex space \( {{\mathbb C}^n} \) to an entire function of exponential type K. We study the approximation of functions in \( B_K^p \) by finite τ -periodic exponential sums of the form $$ \sum\limits_m {{c_m}{e^{2\pi {\text{i}}\left( {x,m} \right)/\tau }}} $$ in the \( {L^p}\left( {\tau {{\left[ { - 1/2,1/2} \right]}^n}} \right) \)-norm as τ → ∞ when K is a polytope in \( {{\mathbb R}^n} \).