Approximation of entire functions of exponential type by trigonometric sums

TL;DR

This paper investigates how well functions in Bernstein spaces, which are bandlimited, can be approximated by finite trigonometric sums in the $L^p$ norm as the sum length increases.

Contribution

It provides a detailed analysis of the approximation of bandlimited functions by finite trigonometric sums in $L^p$ spaces, extending understanding of approximation in Bernstein spaces.

Findings

Convergence of trigonometric sums to bandlimited functions as sum length increases.

Quantitative estimates of approximation errors in $L^p$ norm.

Conditions under which approximation is optimal or near-optimal.

Abstract

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 $[-\tau, \tau]$.