Uncertainty principle
Benedicks's theorem

Amrein–Berthier and Benedicks's theorem intuitively says that the set of points where f is non-zero and the set of points where ƒ̂ is non-zero cannot both be small.

Specifically, it is impossible for a function f in L (R) and its Fourier transform ƒ̂ to both be supported on sets of finite Lebesgue measure. A more quantitative version is ‖ f ‖ L 2 (R d) ≤ C e C | S | | Σ | (‖ f ‖ L 2 (S c) + ‖ f ^ ‖ L 2 (Σ c)). {\displaystyle \|f\|_{L^{2}(\mathbf {R} ^{d})}\leq Ce^{C|S||\Sigma |}{\bigl (}\|f\|_{L^{2}(S^{c})}+\|{\hat {f}}\|_{L^{2}(\Sigma ^{c})}{\bigr)}~.}

One expects that the factor Ce may be replaced by Ce, which is only known if either S or Σ is convex.
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