Periodic Non-uniqueness Sets for Shift-invariant Spaces and Parity-Based Obstructions to the Frame Property for Gabor Systems

Abstract: The goal of this note is twofold. First, we provide explicit examples of periodic (though not necessarily lattice) sets that give rise to Gabor systems failing to form frames. Our constructions depend only on the parity of the window function $g$. Second, for a wide range of finite-dimensional function spaces $V$ we show that $V$ contains a function $g$ such that a lattice of high density fails to generate a Gabor frame. In particular, we prove that the Gröchenig-Lyubarskii theorem is sharp in the finite-dimensional space of polynomials with Gaussian weight. More precisely, for every $N\in\mathbb{N}$ and every $α,β>0$ satisfying $αβ=\frac{1}{N+1}$, we give an explicit algorithm for finding an even or odd polynomial $p$ of degree at most $N$ such that $$\mathcal{G}(p(x)e^{{-πx^2},} α\mathbb{Z} \times β\mathbb{Z})$$ does not form a frame. The proofs are constructive, elementary, and based on linear algebra.

• The paper presents explicit constructions of periodic non-uniqueness sets that obstruct the Gabor frame property due to parity constraints on the window function.
• It proves sharp density thresholds for Gaussian-weighted polynomial generators by solving finite-dimensional linear systems, highlighting limits of classical bounds.
• The work extends to multi-generator spaces, showing that even high-density lattices fail as uniqueness sets when affected by parity-based obstructions.

Parity-Based Obstructions and Periodic Non-uniqueness Sets in Gabor Frame Theory

Introduction and Context

This paper ("Periodic Non-uniqueness Sets for Shift-invariant Spaces and Parity-Based Obstructions to the Frame Property for Gabor Systems" (2606.31450)) addresses fundamental questions in time-frequency analysis regarding the conditions under which Gabor systems constitute frames in $L^2(\mathbb{R})$. Of particular interest is the intricate dependence of the frame property on the parity of the generator function, the density of sampling sets, and their periodic structure, extending beyond classical lattice settings.

The primary contributions are twofold: (1) explicit construction of periodic (not necessarily lattice) sets yielding Gabor systems that fail to be frames due solely to parity constraints on the window function, and (2) demonstration, for a wide spectrum of finite-dimensional spaces (notably Hermite-type generators and polynomials with Gaussian weight), of situations where highly dense lattices do not produce frames, thus showing sharpness in known density bounds.

Gabor Systems, Shift-invariant Spaces, and Uniqueness Sets

The paper develops results leveraging the classical correspondence between Gabor systems and shift-invariant spaces. For a function $g$ with stable integer shifts in the Wiener amalgam space $W_0$, the shift-invariant space $V^\infty(g)$ consists of functions expressible as finite or infinite sums of integer translates of $g$. The non-uniqueness of certain periodic sets for $V^\infty(g)$ directly translates, via a duality lemma, to the failure of the associated Gabor system $\mathcal{G}(g, \Lambda \times \mathbb{Z})$ to achieve the frame property.

A key technical focus is explicit periodic set construction (denoted $\Lambda_k$, $\Gamma_k$, $\Theta_k$), parameterized by integer $g$0, whose density can approach or even exceed critical thresholds but which, due to parity-based obstructions, are proven not to be uniqueness sets for $g$1 and thus cannot yield Gabor frames.

Main Theoretical Results

Non-uniqueness Sets by Parity

The authors state and prove that for any odd generator $g$2 with stable integer shifts, none of the sets $g$3 (periodic, cardinality $g$4 per period) are uniqueness sets for $g$5; correspondingly, $g$6 is not a Gabor frame. For even generators, the obstructions are captured by sets $g$7 (odd $g$8 case) and $g$9 (even $W_0$0 case).

This establishes a wide class of parity-derived obstructions, generalizing previous lattice-based results ("Gabor frames with rational density" [MR3027914]), and focusing not just on the density but the structure and symmetry of the sampling set.

Sharp Density Thresholds for Frames

For generators of the form $W_0$1, where $W_0$2 is a polynomial of degree at most $W_0$3, the classical Gröchenig-Lyubarskii theorem prescribes a strict frame region $W_0$4. The paper proves sharpness: for every $W_0$5 and $W_0$6 with $W_0$7, there exists a polynomial $W_0$8 (even or odd according to $W_0$9) such that the Gabor system $V^\infty(g)$0 fails to be a frame. The construction is explicit and algorithmic, relying on finding appropriate coefficients by solving linear systems.

This result is highly relevant for the spectral theory of Gabor frames generated by Hermite functions and Gaussian-weighted polynomials, establishing that the known density lower bounds cannot be improved for these settings.

Multi-generator Spaces and High-density Lattices

For spaces generated by $V^\infty(g)$1 mutually independent even or odd generators, the paper constructs periodic sets (lattices $V^\infty(g)$2 and $V^\infty(g)$3) with density $V^\infty(g)$4 (odd case) or $V^\infty(g)$5 (even case), which are not uniqueness sets for $V^\infty(g)$6. Therefore, even with densities far exceeding the minimal sampling requirement ($V^\infty(g)$7), parity and linear independence can obstruct the frame property.

Moreover, it is shown that for each such high-density lattice, one can find a generator $V^\infty(g)$8 in the span of $V^\infty(g)$9 for which the corresponding Gabor system is not a frame.

Technical Approach and Rigorous Construction

The proofs are constructive and elementary, based on algebraic properties of shift-invariant spaces and the parity of generators. For both the single-generator and multi-generator cases, the authors develop and analyze families of periodic functions (with auxiliary constructs such as $g$0, $g$1), establish their linear independence, periodicity, and vanishing conditions, and show how non-trivial elements exist that vanish on large portions of the sampling sets. The explicit algorithm for sharp polynomial construction is given via solving finite-dimensional linear systems, and all steps are amenable to numerical computation.

Implications and Future Directions

Theoretical Implications

• The results illuminate the nuanced interplay between window function parity, sampling set structure, and density in determining the frame property for Gabor systems. The explicit identification of parity-based obstructions provides a new lens on frame set geometry, especially in prominent cases involving Hermite and B-spline generators.
• The sharpness of density theorems for Gaussian-weighted polynomial generators closes an important gap in Gabor frame theory for finite-dimensional settings, confirming that further expansion of the frame region is not possible in these cases.

Practical Relevance

• The explicit algorithms for constructing non-frame-generating polynomials may inform practical time-frequency applications, including signal processing scenarios where window parity and lattice density are engineering constraints.
• The extension to multi-generator spaces and explicit high-density periodic obstructions may impact sampling theory for wavelet-type and multicomponent signals.

Prospects for Further Research

The classification of non-uniqueness sets suggests several avenues for future research:

• Investigation of similar parity-based obstructions in irregular (non-periodic) settings and other function classes.
• Analysis of the frame property in higher dimensions or for more general window functions, possibly with additional symmetry constraints.
• Development of efficient numerical algorithms for detecting obstructions and non-uniqueness sets in practical signal analysis problems.

Conclusion

This paper rigorously establishes that parity constraints on the generator function can systematically obstruct the frame property for Gabor systems, regardless of sampling set density and periodicity. The constructive approach, sharp density results, and explicit algorithms extend the foundational landscape of Gabor analysis. The findings have concrete implications for both theoretical sampling theorems and applied time-frequency signal processing, suggesting that parity and structure, not merely density, dictate the efficacy of Gabor frame constructions.