Bandlimited function

A bandlimited function has its Fourier transform supported in a bounded frequency region. For $f\in L^1(\mathbb{R})$ with support of $\hat{f}$ contained in $[-B,B]$, the transform is bounded and hence integrable. Fourier inversion supplies the bounded continuous representativeThe same formula for complex $x=z$ defines an entire function: derivatives can pass under the finite-interval integral, uniformly on compact subsets. It obeys $|f(z)|\le (2\pi {)}^{-1}\Vert \hat{f}{\Vert }_1{e}^{B|\mathrm{Im}z|}$.

For $a\ne 0$, the exact scaling law isThus $f(ax)$ has bandwidth $|a|B$, while the stretched function $f(x/a)$ has bandwidth $B/|a|$. This fixes the otherwise ambiguous phrase that time and bandwidth scale inversely. Products add frequency-support intervals through the convolution theorem.

Band limitation differs from compact spatial support. A nonzero entire representative cannot vanish on an open real interval by the identity theorem. Consequently a nonzero integrable bandlimited function cannot also have compact support. This matters when interpreting a sharply started signal or a wave-packet front.