  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
$[-\tau, \tau]$.
