Riemann summation of a convergent trigonometric series (${\sum }_r{a}_r{e}^{irt}{\mathrm{sinc}}^2(rh/2)\to {\sum }_r{a}_r{e}^{irt}$)

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 ${\sum }_{n\ge 0}{S}_n[w(nk)-w((n+1)k)]$, where $k=|h|/2$. The total variation of these weights is at most ${\int }_0^{\infty }|w^{\prime }(x)|dx<\infty$, 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.