Whittaker–Shannon interpolation formula

Remarks

The proof expands $\hat f$, restricted to $[-B, B]$, in a Fourier series of period $2B$. Writing $\hat f(\xi) = \sum_k c_k e^{-2\pi i k \xi/(2B)}$ on $[-B, B]$, Fourier inversion gives $c_k = \frac{1}{2B} f(\frac{k}{2B})$. Inverting term by term, the inverse transform of $e^{-2\pi i k \xi/(2B)} \mathbf{1}_{[-B, B]}(\xi)$ is $2B \operatorname{sinc}(2Bx - k)$, which gives the formula. The functions $\sqrt{2B}\, \operatorname{sinc}(2Bx - k)$, $k \in \Z$, form an orthonormal basis of the band-limited functions of bandwidth $B$, so the series converges in $L^2$, with $\sum_k \abs{f(k/(2B))}^2 = 2B \norm{f}_2^2$ by the Plancherel theorem. Uniform convergence follows from the bound $\sup \abs{g} \le \sqrt{2B} \norm{g}_2$ for band-limited $g$ of bandwidth $B$.

Example: for $f = \operatorname{sinc}$, $B = \frac12$, the samples are $f(k) = 0$ for $k \ne 0$ and $f(0) = 1$, and the formula reduces to $f(x) = \operatorname{sinc}(x)$.

The sample spacing $1/(2B)$ cannot be enlarged; if the band is wider than $[-B, B]$, different functions can have the same samples, which is aliasing.

Going deeper

For a student

This is the explicit form of the sampling theorem. It is also the expansion of $f$ in an orthonormal basis of the Paley–Wiener space, made of translates of one sinc function. In practice the slow decay of sinc makes truncation errors large, so oversampling and smoother interpolation kernels are used.