Approximation of entire functions of exponential type by trigonometric sums

Abstract

Let $\sigma >0$. For $1\le p\le \infty$, the Bernstein space $B_{\sigma }^p$ is a Banach space of all $f\in L^p(R)$ such that $f$ is bandlimited to $\sigma$; that is, the distributional Fourier transform of $f$ is supported in $[-\sigma ,\sigma ]$. We study the approximation of\ $f\in B^p_σ by finite trigonometric sums \[ P_τ(x)=χ_τ(x) \sum_{|k|\le στ/π}c_{k,τ} e^{i\fracπτk x } \] in $L^p$normon$R$as$τ\to\infty$,where$χ_τ$denotestheindicatorfunctionof$[-τ, τ]$.

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