= Heat evolution of a triangular profile
{title2=$u(t,x)=\frac1{\pi}\int_{\mathbb R}e^{-t\omega^2+ix\omega}\frac{\sin^2\omega}{\omega^2}\,d\omega$}
On the real line, let $T(x)=\max(1-|x|/2,0)$. The <Fourier transform of a triangular function> gives $\widehat T(\omega)=2\sin^2\omega/\omega^2$, with value two at zero. The <Gaussian heat kernel> has transform $e^{-t\omega^2}$. Hence $u=q_t*T$ solves the <heat equation> with initial profile $T$, and the <inverse Fourier transform> gives the displayed integral. Differentiating for positive time is justified by Gaussian decay. The <uniform continuity> of $T$ and the vanishing Gaussian mass outside any fixed neighborhood of zero prove uniform recovery of $T$ as $t\downarrow0$. Convolution preserves total integral two, keeps the solution nonnegative, and smooths the corners for every positive time. Unlike the finite-interval problem in <heat flow from a triangular initial temperature>, this evolution has no endpoints through which heat can escape.
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