= Fourier decay from a Green-function derivative jump
{title2=$\int H(s)e^{-iqs}ds\sim-[H']e^{-iqy}/q^2$}
If $H$ is continuous, piecewise twice differentiable, and has derivative jump $[H']=H'(y+)-H'(y-)$ at $y$, then its distributional second <derivative> is $H''_{\mathrm{reg}}+[H']\delta_y$. If the regular part is integrable and the boundary terms vanish, two <integrations by parts> give the displayed <asymptotic expansion> by the <Riemann-Lebesgue lemma>. A jump $[H]$ in $H$ itself instead produces $[H]e^{-iqy}/(iq)$ at leading order. This distinguishes the different decay rates of the <streamfunction> and streamwise <velocity> in <linear inviscid damping in unbounded Couette flow>.
Back to article page
