Band-limited function

Remarks

If $f$ is band-limited with bandwidth $B$, then $\hat f$ lies in $L^2[-B, B] \subset L^1[-B, B]$, and $f$ agrees almost everywhere with the continuous function $x \mapsto \int_{-B}^{B} \hat f(\xi) e^{2\pi i x \xi}\, d\xi$; one always uses this continuous representative.

Example: $f(x) = \sin(\pi x)/(\pi x)$, with $f(0) = 1$, lies in $L^2(\R)$, and $\int_{-1/2}^{1/2} e^{2\pi i x \xi}\, d\xi = f(x)$. So $\hat f$ is the indicator of $[-\frac12, \frac12]$, and $f$ is band-limited with bandwidth $\frac12$.

Non-example: $f(x) = e^{-\pi x^2}$ has $\hat f(\xi) = e^{-\pi \xi^2}$, which is positive everywhere, so $f$ is not band-limited.

By the Paley–Wiener theorem, the band-limited functions with bandwidth $B$ are exactly the restrictions to $\R$ of certain entire functions. These are the entire functions of exponential type at most $2\pi B$ that are square-integrable on $\R$. In particular a nonzero band-limited function cannot vanish on an interval. The same definition is used for tempered distributions, with the condition that the support of the distribution $\hat f$ lies in $[-B, B]$. In this wider sense $\sin(2\pi B x)$ is band-limited, though not in $L^2$.