sinc and Polynomial interpolation
Another short one. This one bugged me for quite a while until I realized what the answer was a few years ago. The Sampling Theorem states that (under the right conditions)
(the normalized sinc function). The problem is this: Where does the sinc function come from? (In a philosophical sense. It plops out of the proof sure enough, but that’s not what I mean). Fourier theory is full of (trigonometric) polynomials (i.e. sine/cosine waves when you’re dealing with real-valued signals), so where does the factor of x in the denominator suddenly come from?
The answer is a nice identity discovered by Euler:
With some straightforward algebraic manipulations (ignoring convergence issues for now) you get:
Compare this with the formula for Lagrange basis polynomials:
in other words, the sinc function is the limiting case of Lagrange polynomials for an infinite number of equidistant control points. Which is pretty neat :)