Folded sine approximation to a Dirac delta
= Folded sine approximation to a Dirac delta
{title2=$m\sin(m|t|)\to2\delta_0$}
As <distributions> on the line, $m\sin(m|t|)\to2\delta_0$. Pairing with a <test function> $A$ folds the integral to $\int_0^\infty m\sin(ms)[A(s)+A(-s)]\,ds$. One <integration by parts> yields $2A(0)$ plus a cosine integral tending to zero by the <Riemann-Lebesgue lemma>. Pointwise convergence is unnecessary.