= Solution
Assume sufficient decay and <differentiability> to justify <integration by parts> and exchange of <integrals>; <Schwartz functions> are one convenient class. With the stated <Fourier transform> convention, integrate first in $x$:
$$
\begin{aligned}
\mathcal F[u_x]
&=\int_{\mathbb R}e^{-ik_yy}\left([u e^{-ik_xx}]_{-\infty}^{\infty}+ik_x\int_{\mathbb R}u e^{-ik_xx}\,dx\right)dy\\
&=\boxed{ik_x\widehat u}.
\end{aligned}
$$
The boundary contribution vanishes under the decay assumption. The <Fourier transform of a derivative> consequently turns constant-coefficient spatial <differential operators> into multipliers. Repeating it gives $\mathcal F[u_{xx}]=-k_x^2\widehat u$, so the <heat equation> becomes an independent temporal <ordinary differential equation> at each <wavevector>.
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