= Solution
The <convolution> of two <aperture> functions in one dimension is
$$
(f*g)(x)=\int_{-\infty}^{\infty}f(u)g(x-u)\,du.
$$
With <Fourier transform> convention $\widehat f(q)=\int f(x)e^{-iqx}dx$, the <convolution theorem> states
$$
\boxed{\widehat{f*g}(q)=\widehat f(q)\widehat g(q)},\qquad
\boxed{\widehat{fg}(q)=\frac1{2\pi}(\widehat f*\widehat g)(q)}.
$$
The first identity follows by substituting the convolution integral, putting $v=x-u$, and separating the resulting double integral into $\int f(u)e^{-iqu}du$ times $\int g(v)e^{-iqv}dv$. For an absolutely integrable pair the interchange is justified by absolute convergence; finite optical <apertures> satisfy suitable integrability assumptions.
In <Fraunhofer diffraction>, the observed field is the <Fourier transform> of the \i[transmitted complex <aperture> field]. For two identical slits, let $g(x)$ be one slit and use a pair of <Dirac delta> distributions to locate the centres:
$$
t(x)=g*[\delta(x-D/2)+\delta(x+D/2)].
$$
Consequently $\widehat t(q)=2\widehat g(q)\cos(qD/2)$. For a rectangular slit, $\widehat g=a\operatorname{sinc}(qa/2)$, giving a <single-slit diffraction> envelope times the <double-slit interference> factor in intensity. A $\pi$ <phase> reversal replaces the plus between the deltas by a minus, giving $\sin(qD/2)$ instead, as in B7.
More generally, an $N$-slit array is a single-aperture function convolved with a finite comb. Its field contains the factor
$$
\sum_{j=0}^{N-1}e^{-iqjD}=e^{-iq(N-1)D/2}\frac{\sin(NqD/2)}{\sin(qD/2)}.
$$
This separates the narrow <diffraction grating> maxima from the individual-aperture envelope. The same reasoning holds in two transverse dimensions. Multiplying an existing <aperture> by an additional finite mask instead convolves their field transforms and can broaden features. These are operations on complex fields: intensity is the squared modulus of the final field and cannot generally be substituted into the field-level <convolution theorem>.
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