Unbounded Growth of One-Sided-Bounded Band-Limited Functions (Cohen's Theorem)
For the class of L2 functions band-limited to [-1,1] and bounded by 1 on the negative real axis, the supremum M(x) of |f(x)| for any x>0 is infinite. In other words, constraining a band-limited function on one side of the axis places no bound whatsoever on how large it can be on the other side (unlike two-sided boundedness). This refutes the natural conjecture that M(x) grows only exponentially. The counterexample multiplies the entire function cosh(a*sqrt(x)) - which is bounded on the negative axis but grows with the parameter a for x>0 - by a rapidly decaying band-limited function to place it in L2, then sends a to infinity.
Unbounded Growth of Band-Limited Functions Lloyd N. Trefethen ∗ January 3, 2025 Let f be a real or
A short analysis note (Trefethen) resolving a conjecture about how fast a band-limited function can grow. A function band-limited to wave numbers in [-1,1] is, by the Paley-Wiener theorem, the invers…