Unbounded growth of band-limited functions

Abstract

It is shown that a band-limited function bounded by 1 for negative x can grow arbitrarily fast for positive x.

Figures (3)

• Figure 1: The sinc function is the basic example of a band-limited function, and the starting point of sampling theory.
• Figure 2: The blue curve is an analytic function $f(x)$. Numerical analytic continuation jjiam to the right of the gray interval gives the dashed red curve, which matches $f(x)$ for about one wavelength.
• Figure 3: The function $g(x)$ of $(\ref{['ga']})$ on the negative real axis for $a=10$ (above) and $100$ (below). Despite the increasingly rapid oscillations as $a\to\infty$, all such functions are uniformly band-limited after multiplication by the fixed envelope $\psi(x)$, an example of superoscillation. For $x>0$, they take huge values.