Shannon's sampling theorem

The functions $$\{\text{sinc}(\cdot -k)\}_{k\in \mathbb Z}$$ form an orthonormal basis for the Paley-Wiener space (Th. 7.4.5 ). In other words, any function whose Fourier transform has compact support can be represented in the afore-listed basis. The coefficients for a function $$f(x)$$ in this basis expansion are simply the values $$f(k)$$ for $$k \in \mathbb Z$$. That is

$$f(x) = \sum_{k\in \mathbb Z} f(k) \text{sinc}(x-k)$$