= Convolution and a finite slit array
Use <Fourier transform> convention $\widehat f(q)=\int f(x)e^{-iqx}dx$. The <convolution theorem> gives
$$
\boxed{\widehat{f*g}=\widehat f\,\widehat g},\qquad
\boxed{\widehat{fg}=\frac1{2\pi}\widehat f*\widehat g}.
$$
A finite array of very narrow slits at $x=jD$, $j=0,\ldots,N-1$, has transmission proportional to $\sum_j\delta(x-jD)$. It can be constructed by multiplying an infinite <Dirac comb> by a finite window. The infinite comb transforms to a reciprocal comb; the finite window convolves each reciprocal peak with a narrow sinc-shaped amplitude. Thus increasing the illuminated length narrows each <Fraunhofer diffraction> peak.
The exact finite sum makes the argument quantitative:
$$
\widehat t(q)\propto\sum_{j=0}^{N-1}e^{-iqjD}
=e^{-iq(N-1)D/2}\frac{\sin(NqD/2)}{\sin(qD/2)},\qquad q=\frac{2\pi}{\lambda}\sin\theta.
$$
Its normalized intensity is $[\sin(NqD/2)/(N\sin(qD/2))]^2$. Principal maxima occur at $qD=2\pi m$; their first zeros are displaced by $|\Delta(qD)|=2\pi/N$. Replacing the ideal narrow slit by a finite slit function convolves the array in real space, multiplying this field by the single-slit <Fourier transform>. Field amplitudes are transformed and combined before their squared modulus is taken.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2017/paper-2-b8-finite-grating.png]
{title=A finite narrow-slit array and its Fourier intensity, showing how the principal peaks narrow with slit count}
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