= Solution
For fixed frequency $\omega$, the <Fourier transform of a derivative> gives $\widehat{u_{xx}}=-\omega^2\widehat u$. The transformed <initial value problem> is therefore
$$
\partial_t\widehat u(t,\omega)=-\omega^2\widehat u(t,\omega),\qquad
\widehat u(0,\omega)=\widehat T(\omega).
$$
Solving this scalar <ordinary differential equation> yields
$$
\boxed{\widehat u(t,\omega)=e^{-t\omega^2}\widehat T(\omega)
=2e^{-t\omega^2}\frac{\sin^2\omega}{\omega^2},}
$$
where the value at $\omega=0$ is interpreted by continuity and equals two. Thus diffusion suppresses high frequencies by the Gaussian multiplier while preserving the zero-frequency mass.
The differentiation and transform steps are justified for the <heat kernel> solution $u=q_t*T$: for positive time its spatial derivatives are integrable, since Gaussian derivatives are integrable and $T$ is bounded with compact support. Alternatively, the transformed calculation first suggests the formula, and the next part verifies the resulting solution directly.
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