A class of functions for which the trapezoidal rule gives the exact value of integral over the infinite interval (Q1096832)

The author deals with a classical problem of finding a class of functions for which the trapezoidal rule gives the exact value of an improper integral over the infinite interval \((-\infty,+\infty):\) \[ \lim_{R\to +\infty,S\to +\infty}\int^{R}_{-S}f(x)dx=\lim_{M\to +\infty,N\to +\infty}h\quad \sum^{M}_{n=-N}f(nh). \] It is shown that for an entire function of the exponential type A, the trapezoidal rule with mesh size \(h<2\pi /A\) leads to the exact value. Three lemmas and two theorems summarise the main results.