 Fourier support

The Fourier support of a function or distribution is the support of a function of its Fourier transform, with distributional support used when necessary. For an ordinary transform, it is the closure of the set of frequencies where $\hat{f}$ is nonzero. Thus a bandlimited function has Fourier support in a bounded frequency region.

Under the convention $\hat{f}(\xi )=\int f(x)e^{-ix\cdot \xi }dx$, multiplication by $e^{ia\cdot x}$ translates Fourier support by $a$. Translation of $f$ in physical space instead multiplies its transform by a phase, leaving Fourier support unchanged. This distinguishes where a signal occurs from which frequencies it contains.

For example, the distribution $e^{iax}$ has Fourier support $\{a\}$. A nonzero Gaussian function has transform nonzero at every real frequency, so it has no compact Fourier support despite rapid frequency decay. Small coefficients and exact vanishing describe different properties.