= Riemann summation of a convergent trigonometric series
{c}
{title2=$\sum_r a_re^{irt}\operatorname{sinc}^2(rh/2)\to\sum_r a_re^{irt}$}
Suppose the symmetric partial sums converge at a fixed point. Group positive and negative frequencies together, let $S_n$ be the resulting partial sums and set $w(x)=(\sin x/x)^2$, with $w(0)=1$. <Summation by parts> expresses the weighted sum as $\sum_{n\ge0}S_n[w(nk)-w((n+1)k)]$, where $k=|h|/2$. The total variation of these weights is at most $\int_0^\infty|w'(x)|dx<\infty$, independently of $k$, and each fixed difference tends to zero. Subtract the limiting value of $S_n$ and split into a finite prefix and a uniformly small tail to prove convergence to that value. This is a regular summation method even though the sinc-squared weights need not decrease monotonically.
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