The Donoho-Stark inequality states that if \(\widehat{x}\) denotes the Fourier transform of a vector \(x\) thought of as a function on the group \(\mathbb{Z}_N\) – the integers modulo \(N\) –, then \[ |{\text{supp}}\, x||{\text{supp}}\, \widehat{x}|\geq N \] where \({\text{supp}}\, x\) is the subset of \(\mathbb{Z}_N\) of nonzero coordinates of \(x\). The inequality generalizes readily to finite Abelian groups [see, e.g., A. Terras, Fourier analysis on finite groups and applications. Lond. Math. Soc. Student Texts. 43. Cambridge: Cambridge University Press (1999; Zbl 0928.43001)]. The extension to a general compact group \(G\) is nontrivial because in that case the group Fourier transform is operator-valued (\(\widehat{G}\) is the group of irreducible representations of \(G\)). The analogue of a time-localization operator is an operator \(P\) on \(L^2(G)\) that commutes with projection onto any measurable subset of \(G\), while the analogue of frequency localization or, more generally, a translation invariant operator, can be thought of as an operator \(R\) that commutes with left multiplication by elements of \(G\). Here the authors prove that if \(G\) is any compact group with Haar measure \(H\), and if \(P\) and \(R\) are as indicated, then \(\| PR\|\leq \|P\cdot \chi_G\|_2\|R\|_2\). Here, \(\|\cdot\|\) denotes the operator norm and \(\|\cdot\|_2\) denotes both the \(L^2(G)\)-norm and the Hilbert-Schmidt norm.
The group Fourier transform of \(f\in L^2(G)\) evaluated at the irreducible representation \(\rho\) is
\[ \widehat{f}(\rho) = \int f(x) \rho(x)^{\ast} \,d\mu(x), \]
where \(\mu\) is the Haar measure on \(G\). A band-limiting operator can be thought of as a projection onto a subset \(T\subset \widehat{G}\) by
\[ R_T f(x) = \sum_{\rho\in T} d_{\rho} {\text{tr}} (\widehat{f}\rho \, \rho(x)). \] Here \(d_\rho\) is the dimension of \(\rho\) that accounts for the Plancherel theorem. The following analogue of the Donoho-Stark inequality is given: If \(G\) is a compact group with Haar measure \(\mu\) and \(f\) a nonzero element of \(L^2(G)\) then \(\mu (\text{supp}\, f) \cdot \sum_{\rho\in\widehat{G}} d_{\rho} \text{rk} \widehat{f} (\rho) \geq 1\) where \(\text{rk} \widehat{f} (\rho)\) is the rank of the operator \(\widehat{f} (\rho)\).
This result is more precise than a prior inequality due to K. S. Chua and W. S. Ng [Expo. Math. 23, No. 2, 147–150 (2005; Zbl 1070.43001)] in which the size of the support of \(\widehat{f}\) was quantified by \(\sum_{\rho:\, \widehat{f}(\rho)\neq 0} d_\rho^2\).