On uncertainty principles in the finite dimensional setting

The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known qualitative uncertainty principles into quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform. An application to trigonometric polynomials is also given.

💡 Research Summary

The paper studies quantitative uncertainty principles in finite‑dimensional complex Hilbert spaces. Let Φ={Φ_j} and Ψ={Ψ_j} be two orthonormal bases of ℂ^d and define the coherence M(Φ,Ψ)=max_{j,k}|⟨Φ_j,Ψ_k⟩|. For subsets S,Σ⊂{0,…,d−1} the authors introduce the notion of a strong annihilating pair: there exists a constant C(S,Σ) such that for every vector a∈ℂ^d, ‖a‖₂ ≤ C(S,Σ)·(‖a‖{ℓ²(Φ,S^c)}+‖a‖{ℓ²(Ψ,Σ^c)}). The main theorem (Theorem 2.3) shows that if |S|·|Σ| < M(Φ,Ψ)^{-2} then (S,Σ) is a strong annihilating pair with an explicit constant C(S,Σ)=1+1/√{1−M(Φ,Ψ)·|S|·|Σ|}. The proof uses the unitary change‑of‑basis operator U that maps Ψ to Φ, together with orthogonal projections P_S and P_Σ. By estimating the operator norm ‖P_Σ U P_S‖_{2→2} and showing it is strictly less than one under the stated condition, the authors obtain the desired inequality. The result refines earlier qualitative statements (e.g., Donoho‑Stark, Tao) by providing a concrete bound on C.

In the special case of unbiased bases (M=1/√d, such as the standard basis and the discrete Fourier basis), the condition reduces to |S|·|Σ|<d, reproducing the classical discrete uncertainty principle. Moreover, the paper proves that when M=1/√d and |Σ|≤d−√(240d), one can always find a set S with |S|≥(d−|Σ|)²/(240d) that forms a strong annihilating pair with Σ.

The second part of the work transfers these results to the discrete short‑time Fourier transform (STFT). For a window g∈ℓ²(d) with ‖g‖₂=1, the STFT is defined by V_g f(j,k)=d^{-1/2}∑{ℓ=0}^{d-1} f(ℓ) g(ℓ−j) e^{2π i kℓ/d}. Lemma 3.1 shows that the Fourier transform of the product of two STFTs is again an STFT, which allows the authors to apply the strong annihilating pair estimate to V_g. Theorem 3.2 (Theorem B) states that for any Σ⊂{0,…,d−1} with |Σ|<d, ‖f‖₂ ≤ 2√2·(1−|Σ|/d)^{-1/2}·\Big(∑{(j,k)∉Σ}|V_g f(j,k)|²\Big)^{1/2}. This inequality strengthens the earlier result of Krahmer, Pfander, and Rashkovic by giving an explicit dependence on the size of Σ. A probabilistic improvement is also presented (Corollary 3.4), showing that random choices of Σ lead to even better constants with high probability.

Finally, the authors discuss connections with the Uniform Uncertainty Principle (UUP) from compressed sensing. They show that the existence of a strong annihilating pair (S,Σ) implies a UUP for the pair (T,Σ^c) with restricted isometry constant δ_s = 1 − 1/C(Σ)·(1−|Σ|/d)^{1/2}, where C(Σ)=sup_{|S|=s} C(S,Σ). Conversely, a UUP yields a strong annihilating pair with an explicit constant. Thus the paper bridges the gap between classical uncertainty principles and modern compressed‑sensing theory.

Overall, the contribution consists of (1) a quantitative bound for the uncertainty principle in terms of coherence and support sizes, (2) an extension of this bound to the discrete STFT, and (3) a clear link to restricted isometry properties used in sparse recovery. The results are proved using elementary linear‑algebraic tools (unitary change‑of‑basis operators, projection estimates) and improve upon several known qualitative statements by providing concrete constants, thereby offering both theoretical insight and potential practical impact in signal processing and sparse reconstruction.