Approximation of entire functions of exponential type by trigonometric sums

Let $\sigma>0$. For $1\le p\le \infty$, the Bernstein space $B^p_{\sigma}$ 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_{\sigma} by finite trigonometric sums \[ P_{\tau}(x)=\chi_{\tau}(x) \sum_{|k|\le \sigma\tau/\pi}c_{k,\tau} e^{i\frac{\pi}{\tau}k x } \] in L^p$norm on$R$as\$\tau\to\infty$,\ where\$\chi_{\tau}$denotes the indicator function of$[-\tau, \tau]$.