= Solution
<Termwise differentiation of a Fourier series> multiplies a mode by its wavenumber, so
$$
\boxed{\widehat{(f_x)}_{m,n}=i\pi m\widehat f_{m,n},
\qquad
\widehat{(f_y)}_{m,n}=i\pi n\widehat f_{m,n}}.
$$
The <convolution theorem> turns pointwise multiplication into <discrete convolution>:
$$
h=fg
=\sum_{p,q}\sum_{r,s}\widehat f_{p,q}\widehat g_{r,s}
e^{i\pi(p+r)x+i\pi(q+s)y},
$$
so
$$
\boxed{\widehat h_{m,n}
=\sum_{p,q\in\mathbb Z}\widehat f_{p,q}\widehat g_{m-p,n-q}}.
$$
The assumed <real analytic function>[real analyticity] justifies these termwise operations through rapid decay of the <Fourier coefficients>.
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