= Fourier support
{c}
{title2=$\operatorname{supp}\widehat f$}
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 $\widehat f$ is nonzero. Thus a <bandlimited function> has Fourier support in a bounded frequency region.
Under the convention $\widehat f(\xi)=\int f(x)e^{-ix\cdot\xi}dx$, 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.
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