Donoho-Stark Uncertainty Principle

• The Donoho-Stark uncertainty principle is a framework that provides explicit lower bounds on the product of the support sizes of a function and its Fourier transform, central to harmonic analysis and sparse signal recovery.
• Extensions of the principle include its generalization to noncommutative settings and refinements using additive combinatorial invariants, yielding sharper and more structured inequalities.
• Applications span deterministic recovery in sparse signal processing, time-frequency analysis, and operator generalizations in both classical and quantum frameworks.

A Donoho-Stark-type uncertainty principle provides an explicit lower bound, usually in product form, on the simultaneous concentration of a function and its Fourier transform (or its generalization) in their respective "domains." Unlike classical Heisenberg-type bounds, Donoho-Stark inequalities are formulated in terms of the size (measure, cardinality, or algebraic dimension) of supports, potentially allowing for exact (hard) localization, and are central both in harmonic analysis and @@@@1@@@@. The canonical results first appeared in the work of D.L. Donoho and P.B. Stark [SIAM J. Appl. Math. 49 (1989)] and have since been generalized to noncommutative and combinatorial settings.

1. Fundamental Formulation and Core Examples

The archetypal Donoho-Stark uncertainty principle states that for a function $f$ and its Fourier transform $\widehat f$, the measures (or cardinalities) of their respective supports satisfy

$$|\mathrm{supp}(f)| \cdot |\mathrm{supp}(\widehat{f})| \geq N$$

when $f$ is defined on a finite abelian group $G$ ($|G| = N$), or

$$|T| \cdot |\Omega| \geq (2\pi)^d (1 - \epsilon_T - \epsilon_\Omega)^2$$

for $f$ on $\mathbb{R}^d$, where $T$ and $\Omega$ are sets in which $f$ and $\widehat f$ are $\epsilon_T$- and $\epsilon_\Omega$-concentrated, respectively (Poria, 2020, Fujikawa et al., 2015, Borello et al., 2020). In the case of exact (i.e., $\epsilon=0$) supports and the standard Fourier transform, the product bound is sharp and achieved for indicator functions of cosets or intervals.

2. Noncommutative and Generalized Frameworks

Donoho-Stark-type inequalities generalize beyond classical abelian groups. In subfactor theory, the Donoho-Stark principle holds in the $2$-box spaces $\mathscr{P}_{2,\pm}$ of a finite-index subfactor planar algebra $\mathscr{P}$, with the Ocneanu (one-click) Fourier transform $\mathcal F$: $$\boxed{ \mathcal S(x) \; \mathcal S(\mathcal F(x)) \geq \delta^2 }$$ Here, $\mathcal S(x)$ is the trace of the range projection of $x$, and $\delta = \sqrt{[\mathcal M:\mathcal N]}$ is the index of the subfactor. This result holds in all settings admitting a suitable planar algebraic formalism (e.g., Kac algebras, modular tensor categories), and minimizers are bi-shifts of biprojections, generalizing indicator functions (Jiang et al., 2014).

A broad noncommutative extension, the Noncommutative Donoho-Stark–Elad–Bruckstein–Ricaud–Torrésani principle, is proved for Hilbert $C^*$-modules with modular Parseval frames $\{\tau_n\}$ and $\{\omega_m\}$: $$\|\theta_\tau x\|_0 \; \|\theta_\omega x\|_0 \geq \frac{1}{\sup_{n,m} \| \langle \tau_n, \omega_m \rangle \|^2 }$$ where $\theta_\tau$ and $\theta_\omega$ are analysis operators and $\| \cdot \|_0$ counts nonzero coefficients (Krishna, 2024). This result subsumes both the classical Donoho-Stark and the Elad–Bruckstein mutual coherence formulation for pairs of bases.

3. Additive Energy, Structure, and Refined Inequalities

Recent advances sharpen the Donoho-Stark regime by introducing additive combinatorial invariants, notably additive energy $\Lambda_2(A)$: $$\Lambda_2(A) = |\{ (a_1, a_2, a_3, a_4) \in A^4 : a_1 + a_2 = a_3 + a_4 \} |$$ The improved uncertainty principle for $f: \mathbb{Z}_N^d \to \mathbb{C}$ reads

$N^d \leq |E| \cdot \Lambda_2(\Sigma)^{1/3}$

for $E = \mathrm{supp}(f)$, $\Sigma = \mathrm{supp}(\widehat f)$, with strict strengthening whenever supports are not cosets (i.e., $\Lambda_2(\Sigma) < |\Sigma|^3$). Further refinement introduces explicit correction terms quantifying the deviation from coset structure, leading to

$N^d \leq |E| \left( B - C(E,\Sigma) \right)^{1/3}$

where $B = \Lambda_2(\Sigma)$ and $C(E,\Sigma)$ incorporates the product and additive structure in both supports. These augmented bounds sharpen uniqueness guarantees in signal recovery beyond the classical product threshold (Aldahleh et al., 20 Apr 2025, Bortnovskyi et al., 30 Oct 2025).

4. Time-Frequency, Cohen Class, and Operator Generalizations

In time-frequency analysis, Donoho-Stark-type uncertainty principles govern quadratic time-frequency representations such as the short-time Fourier transform (STFT), Wigner and Born-Jordan distributions, and more general Cohen-class operators: $$\int_U |V_g f(x,\omega)|^p\,dx\,d\omega \ge (1-\varepsilon) \|f\|_2^p \|g\|_2^p \implies |U| \ge 1-\varepsilon$$ for $V_g f$ the STFT (Poria, 2020, Albanese et al., 2024). For localization operators and Cohen-class transforms, explicit lower bounds relate the weighted measure of concentration domains to structure constants of the representation kernels and the concentration parameters.

Extensions encompass wavelet transforms (including Clifford and noncommutative settings (Arfaoui, 2022)), windowed special-function transforms (e.g., Opdam–Cherednik (Mondal et al., 2021)), and transforms on nonabelian groups such as the Strichartz Fourier transform on the Heisenberg group, where the principle takes the form

$$|V| \cdot |W| \geq (2\pi)^{-n} M^n (1 - \sqrt{\epsilon_V^2 + \epsilon_W^2})^2$$

with $V$ a spatial domain, $W$ a spectral domain with parameter $M$ (Dabra et al., 10 Nov 2025).

5. Characterization of Extremizers

The extremal cases in Donoho-Stark inequalities, achieving equality, are characterized by highly structured functions:

• For abelian groups or $\mathbb{Z}_N$, indicator functions of cosets (up to modulation and translation) and their transforms: for $f = c \chi_H$, $\widehat f$ supported on dual coset.
• In noncommutative or subfactor settings: bi-shifts of biprojections are exactly the extremizers, uniquely determined by the range projections of $x$ and $\mathcal F(x)$ (Jiang et al., 2014).
• For STFT on cyclic groups, extremal pairs correspond to cosets of subgroups in phase space (Nicola, 2022).

6. Signal Recovery, Applications, and Extensions

Donoho-Stark principles underpin deterministic recovery guarantees in sparse signal processing and compressed sensing. Exact recovery (via $\ell^1$ or $\ell^2$ minimization) from partial Fourier data is possible when the product of support sizes is less than the specified threshold, with refined guarantees when accounting for additive structure (Iosevich et al., 2023, Aldahleh et al., 20 Apr 2025). The philosophy extends naturally to coding theory (minimum distance bounds), convex geometry (via Blaschke–Santaló duality (Gosson, 11 May 2025)), and sampling theorems (Shannon–Nyquist as a Donoho–Stark instance (Fujikawa et al., 2015)).

Table: Representative Settings and Key Donoho-Stark Inequalities

Setting	Inequality	Reference
Finite abelian group $G$	$$|\mathrm{supp}(f)|\cdot|\mathrm{supp}(\widehat f)| \geq |G|$$	(Feng et al., 2018)
Subfactor planar algebra $\mathscr{P}$	$$\mathcal S(x)\mathcal S(\mathcal{F}(x)) \geq \delta^2$$	(Jiang et al., 2014)
Finite field (cyclic code)	$|\mathrm{supp}(f)|\cdot|\mathrm{supp}(F)| \geq n$	(Borello et al., 2020)
Noncommutative Hilbert $C^*$-module	$$\|\theta_\tau x\|_0 \|\theta_\omega x\|_0 \geq \frac{1}{\sup_{n,m} \|\langle \tau_n, \omega_m \rangle\|^2}$$	(Krishna, 2024)
Additive energy refined (finite group)	$N^d \leq |E| \Lambda_2(\Sigma)^{1/3}$	(Aldahleh et al., 20 Apr 2025)
Time-frequency STFT	$|\mathrm{ess\ supp}(V_g f)| \geq 1 - \epsilon$	(Poria et al., 2022)

7. Ongoing Developments and Open Problems

Active research continues to seek optimal constants, sharper bounds leveraging further algebraic/combinatorial structure (e.g., Salem sets, restriction theory (Iosevich et al., 2023)), and improvements under geometric or group-theoretic constraints. Open questions involve the characterization of extremizers in nonabelian and noncommutative frameworks, further connections to geometric functional analysis (e.g., Mahler volume bounds (Gosson, 11 May 2025)), and the design of efficient recovery algorithms exploiting refined uncertainty principles.

Collectively, the Donoho-Stark family of uncertainty principles forms a central axis in modern harmonic analysis, connecting the geometry of supports, spectral sparsity, and optimal recovery in both classical and quantum settings (Feng et al., 2018, Boggiatto et al., 2015, Krishna, 2024, Aldahleh et al., 20 Apr 2025, Bortnovskyi et al., 30 Oct 2025, Iosevich et al., 2023, Jiang et al., 2014, Borello et al., 2020, Fujikawa et al., 2015, Dabra et al., 10 Nov 2025, Gosson, 11 May 2025).