= Fourier solution of the strip Dirichlet problem
{title2=$\widehat u(s,y)=\widehat g(s)\sinh(s(1-y))/\sinh s$}
For the <Laplace equation> on $0<y<1$, prescribe bottom data $g$ and zero top data. A horizontal <Fourier transform> reduces the equation to $\widehat u_{yy}-s^2\widehat u=0$, giving the displayed multiplier and its limit $1-y$ at $s=0$. Integrable transformed data give a bounded inverse-transform solution. The bounded solution is unique by the <maximum principle for harmonic functions>, using barriers $\varepsilon\cosh(\delta x)\cos(\delta(y-1/2))$ for $0<\delta<\pi$ on expanding rectangles. Without a growth condition, one can add $e^{\pi x}\sin(\pi y)$, so the two boundary traces alone do not determine an arbitrary harmonic function on an infinite strip.
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