Discrete Ingham Inequalities

• Discrete Ingham inequalities are quantitative uncertainty principles that establish strong lower bounds for discrete exponential sum models with well-separated nodes.
• Recent advancements improve the dimension dependence from O(√d) to O(log d) by employing higher-order differential operators and specialized test functions.
• These inequalities are critical in ensuring unique identifiability in multivariate Prony's method, thereby enhancing robust reconstruction in algebraic signal processing.

Discrete Ingham inequalities are quantitative uncertainty principles central to parameter reconstruction in exponential sum models, notably in the context of discrete or finite trigonometric systems. They rigorously relate the spacing of frequencies (moments) and the separation of underlying parameters (“nodes”), showing that if samples are sufficiently concentrated in frequency and the nodes are well separated in space, then a strong lower frame-type inequality applies to the associated exponential sums. Recent developments have dramatically improved the known dimension dependence of such inequalities, particularly as they pertain to multivariate parameter estimation and Prony-type methods (Kunis et al., 2017).

1. Formal Statement of Multivariate Discrete Ingham Inequalities

Consider $M$ unknown nodes $t_j\in [0,1)^d$ (“Dirac locations”) with minimal separation $$\Delta = \min_{j\ne\ell,\, m\in\mathbb{Z}^d} \|t_j - t_\ell + m\|_\infty > q$$ for $q>0$. Let $n>0$ denote the maximal frequency (“order of moments”) and fix an even integer $p=2r$. The relevant frequency set is the discrete $\ell^p$-ball $S(n,p) = \{k\in\mathbb{Z}^d\,:\, \|k\|_p \le n\}$. Then, for any complex amplitudes $f_1,\ldots,f_M$, the multivariate discrete Ingham inequality can be formulated as

$$\sum_{k\in S(n,p)} \left| \sum_{j=1}^M f_j\, e^{2\pi i k\cdot t_j} \right|^2 \geq c\, \sum_{j=1}^M |f_j|^2,$$

where $c>0$ depends only on $(d,p,n,q)$, and provided the product $nq \geq C_p\, d^{1/p}$, with a constant $C_p$ satisfying $C_p \le (2p+3)/(e\pi)$. Optimizing over even $p$ yields $C(d) = 2\min_{p\text{ even}} C_p\,d^{1/p}$, and the lower bound holds whenever $nq \ge C(d)$ (Kunis et al., 2017).

2. Improvements over Classical Bounds

The classical multivariate Ingham inequalities utilized compactly supported eigenfunctions of the Laplacian, establishing the threshold $nq > \frac{1}{2}\sqrt{d}$ for a quantitative uniqueness guarantee. This led to a dimension-dependent constant of order $O(\sqrt{d})$ [Ingham 1936; Koenig–Logan 2004]. Recent advancements have replaced the Laplacian by higher-order partial differential operators (sum of $p$-th derivatives), allowing for tuning $p \approx 2\lceil \log d \rceil$ to minimize the expression $d^{1/p}$. Consequently, the threshold constant improves to $C(d)=O(\log d)$, a logarithmic dependence on dimension rather than a square-root one. This substantial gain is achieved through the construction of a family of test functions $\psi_p$ with spatial support in $[-q,q]^d$ whose Fourier transforms achieve sign changes exactly on prescribed discrete $\ell^p$-balls (Kunis et al., 2017).

3. Construction and Proof Strategy

The proof leverages the following steps:

• Dual Minorant Construction: Begin with a univariate “bump” function $\phi(x) = (1-(2x/q)^2)^r$ for $|x|<q/2$, with $\phi=0$ otherwise. Form a $d$-fold tensor convolution $\Phi = \phi*\phi \otimes\cdots\otimes \phi*\phi$, supported on $[-q,q]^d$.
• Test Polynomial Definition: Define $$\psi(x) = \left( (2\pi n)^p - (-1)^r\sum_{s=1}^d \partial_{x_s}^p\right)\Phi(x)$$. By Fourier analysis,

$$\hat\psi(v) = \left((2\pi n)^p - \sum_{s=1}^d (2\pi v_s)^p\right)\prod_{\ell=1}^d \hat\phi(v_\ell)^2,$$

with $\operatorname{sign}\hat\psi(v)=+$ for $\|v\|_p\le n$, negative outside.

• Poisson Summation and Separation: Exploiting Poisson’s summation formula, cross terms between well-separated nodes vanish due to the support of $\psi$ and the separation $\Delta$. Only the diagonal (self-interaction) terms contribute, and a lower bound emerges from $\psi(0)$. The constant in the inequality is controlled by maximizing $\psi(0)$ over allowable parameters.

This approach replaces the traditional Laplacian eigenfunction minorants with those from higher-derivative operators, directly controlling the sign structure of the Fourier transforms on high-dimensional frequency sets (Kunis et al., 2017).

4. Implications for Multivariate Prony's Method

The discrete Ingham inequalities are instrumental in establishing identifiability in multivariate Prony methods. For samples $f(k) = \sum_j f_j\, z_j^k$ taken on $k$ with $\|k\|_\infty \le n$, the associated multilevel Toeplitz matrix $$T_n = [f(k-\ell)]_{\|k\|_\infty,\,\|\ell\|_\infty \le n}$$ admits a Vandermonde factorization $T_n = A_n^* D A_n$ with $A_n = [z_j^k]$ and $D=\mathrm{diag}(f_j)$. The Ingham inequality ensures that $A_n$ has full column rank $M$ once $n q \geq C(d)$, so that $\ker T_n = \ker A_n$ defines a zero-dimensional variety.

The flat extension principle specifies that if $$\operatorname{rank} A_n = \operatorname{rank} A_{n+1} = M$$, then all larger degree Vandermonde matrices maintain rank $M$, localizing the common zero set to precisely the nodes $\{z_1,\ldots,z_M\}$. Recovery of the parameter set is then realized by root-finding or solving the generalized eigenproblem for this system—crucial for robust reconstruction in algebraic signal processing (Kunis et al., 2017).

5. Central Formulas, Constants, and Lemmas

Formula/Result	Statement / Definition	Context
Dual Minorant $\phi(x)$	$$\displaystyle \phi(x) = \max\left(0, 1-(2x/q)^2 \right)^r$$	Bump for test function construction
Test polynomial $\psi(x)$	$$\displaystyle \psi(x) = ((2\pi n)^p - \sum_{s=1}^d \partial_s^p ) ( \phi*\phi )^{\otimes d}$$	Structure for sign-localized Fourier transform
Threshold constant $C_p$	$$\displaystyle C_p = \left(\frac{\Gamma(p/2+1)\Gamma(p+3/2)}{ \pi^p \Gamma((p+3)/2)}\right)^{1/p} \le \frac{2p+3}{e\pi}$$	Controls $nq$ threshold in the inequality
Vandermonde factorization	$T_n = A_n^* D A_n$; $A_n=[z_j^k]_{\|k\|_\infty\le n}$	Prony system linear algebra

With proper optimization, choosing $p = 2\,\lceil \log d\rceil$, this yields $C(d) = O(\log d)$ for the critical constant.

6. Extensions, Variants, and Open Questions

Several avenues remain under active investigation:

• Achieving dimension–free constants is currently out of reach, as lower-bound constructions indicate $C(d)\to\infty$ is unavoidable. Whether $O(\log d)$ is optimal remains open.
• Generalizations to other $\ell^q$ norms in frequency domains and to non-tensorized test functions are of significant interest.
• Weighted Ingham inequalities and settings using nonuniform frequency sets beyond rectangular grids are pertinent for applications with irregular sampling.
• Stability and condition number estimates for Vandermonde systems beyond full rank, especially under noise, are an ongoing challenge.
• Tighter analysis of the flat extension principle—max-degree versus total-degree (Gröbner-style) matrix settings—could yield even sharper trade-offs between sample size and node separation.
• Connections to Beurling–Selberg extremal minorants/majorants and tensorization techniques in the context of discrete trigonometric approximation theory are also relevant (Kunis et al., 2017).

7. Historical and Cross-Disciplinary Context

The discrete Ingham inequality and its refinements build on foundational work by Ingham (“A contribution to Fourier series,” 1936), and later multivariate extensions by Koenig–Logan (2004) and Potts–Theis–Tasche (2013). Their analytical structures are closely related to extremal function theory and algebraic signal processing. Flat extension ideas draw upon commutative algebra (Hilbert functions, Gröbner bases), with variants studied in both the max-degree and total-degree regimes (e.g., CuFi00, LaMo09). These interconnections illuminate new directions in superresolution, sparse spectral estimation, and computational algebraic geometry (Kunis et al., 2017).