Balian-Low

Theorem 1 (Balian-Low)

Suppose $g\mathrm{\in }{L}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}$ and $g_{m\mathrm{,}n}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}{e}^{\mathrm{2}\mathit{}\pi \mathit{}i\mathit{}m\mathit{}x}\mathit{}g\mathit{}\mathrm{(}x\mathrm{-}n\mathrm{)}$, where $m\mathrm{,}n\mathrm{\in }\mathrm{Z}$.If $\mathrm{\{}{g}_{m\mathrm{,}n}\mathrm{:}m\mathrm{,}n\mathrm{\in }\mathrm{Z}\mathrm{\}}$ is an orthonormal basis for $L^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}$, then either

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^2{|g(x)|}^2𝑑x=\mathrm{\infty }\mathit{\text{ or }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\xi }^2{|\widehat{g}(\xi )|}^2𝑑\xi =\mathrm{\infty }.$$