Gabor Orthonormal Bases with Maximal Localization and Gabor Frame Operator on Local Fields

Abstract: We provide an explicit construction of a Gabor orthonormal bases for a local field $K$ that provides maximal localization in both time and frequency. Such a localization is not true in case of $\mathbb{R}$ due to the uncertainty principle. In particular, we construct examples of functions $f \in L<sup>2(K)$ such that the support of the ambiguity function of $f$ is of minimum measure. Moreover, we establish a quantitative uncertainty principle for local fields, which follows as a consequence of Lieb's inequalities for general locally compact abelian group. In addition, we develop fundamental operator representations for Gabor systems defined over local fields.

• The paper establishes that maximally localized Gabor orthonormal bases exist on local fields, bypassing the Balian–Low obstruction.
• It derives sharp uncertainty principles and explicit operator representations (Walnut, Wexler–Raz, Janssen) for non-Archimedean time–frequency analysis.
• The findings pave the way for advanced signal processing and p-adic quantum mechanics by leveraging compact open subgroup structures.

Gabor Orthonormal Bases with Maximal Localization and Gabor Frame Operator on Local Fields

Overview and Motivation

This paper addresses the explicit construction and operator-theoretic analysis of Gabor orthonormal bases on local fields—totally disconnected, locally compact fields, such as the $p$-adic numbers. The focus is on achieving maximal simultaneous localization in time and frequency, an objective unattainable in real or complex settings due to the classical uncertainty principle and the Balian–Low theorem. The authors establish sharp uncertainty principles for local fields, construct maximally localized orthonormal Gabor bases, and develop explicit operator representations (Walnut, Wexler–Raz, Janssen) for Gabor systems in this setting.

These results hold relevance for mathematical physics (notably $p$-adic quantum mechanics), time-frequency analysis over non-Archimedean structures, the study of pseudodifferential operators, and signal analysis in domains like genetics and geophysics exploiting $p$-adic models.

Background: Local Fields and Time-Frequency Analysis

Local fields—primarily $\mathbb{Q}_p$ and its finite extensions or analogues in positive characteristic—admit compact open subgroups, enabling fundamentally different harmonic-analytic phenomena compared to the real/complex case. In particular, they allow for functions and their Fourier transforms to be compactly supported and give rise to small uncertainty cells in the time-frequency plane $K\times\widehat K$. Standard constructions in classical Gabor theory—such as STFT, ambiguity functions, and lattice sampling—admit analogues, but with qualitative and quantitative distinctions stemming from non-Archimedean geometry.

Quantitative Uncertainty Principle and Localization

Unlike the Euclidean setting, local fields support nontrivial functions $f \in L^2(K)$ such that both $f$ and $\widehat{f}$ are compactly supported. The authors provide a sharp quantitative uncertainty principle for local fields: for any nonzero window $g$ and $f \in L^2(K)$, the measure of the support $p$0 of the ambiguity function (in this setup, $p$1) satisfies $p$2, with equality if and only if $p$3 is supported on a maximal compact open isotropic subgroup and is constant there. This result leverages the sharpness of Lieb-type inequalities for general locally compact abelian groups [lieb1990integral].

They explicitly construct such maximally localized functions, for example, the characteristic function of a compact open subgroup, showing both $p$4 and $p$5 possess compact supports with minimal measure product.

Construction of Maximally Localized Gabor ONBs

A fundamental distinction from the real line is the existence of Gabor orthonormal bases with windows having optimal time-frequency localization. The main result (Theorem~8) is that for $p$6, where $p$7 denotes the characteristic function of the compact open subgroup $p$8, the set $p$9 forms an orthonormal basis for $p$0, and the associated ambiguity function is restricted to minimal cells in phase space. This construction explicitly bypasses the Balian–Low obstruction present in the Euclidean case [nicola2024maximally]. Additionally, the associated STFT $p$1 belongs to $p$2, further underlining the strong localization.

Fundamental Operator Representations

Walnut Representation

The paper generalizes Walnut's representation of the Gabor frame operator to the local field context. The frame operator for a Gabor system over $p$3 is shown to admit an integral representation in terms of convolutions with a kernel $p$4, which can be made explicit for windows that are locally constant and compactly supported. For suitable choices, such as balls in $p$5, the Walnut kernel computation collapses to an elementary formula, leading to an explicit Parseval frame characterization for spaces of functions with Fourier support in certain compact open subgroups.

Wexler–Raz Biorthogonality

The authors establish Wexler–Raz type biorthogonality relations for Gabor dual frames in $p$6, with normalization and dual lattice structure naturally dictated by the theory of compact open subgroups and their annihilators in local fields. These conditions are characterized by discrete orthogonality of cross-ambiguity functions sampled on the relevant lattices, and the standard density factor reduces to $p$7 given the Haar measure normalization.

Janssen Representation

An explicit Janssen representation is derived for the Gabor frame operator, expressing it as a potentially infinite sum of time-frequency shifts weighted by the sampled ambiguity function. Summability is ensured for window functions in the Feichtinger algebra $p$8; this space is well-adapted to the local field setting, inheriting desirable invariance and embedding properties from the Euclidean theory [Grochenig01]. The representation is unconditional in the strong operator topology and provides a clear structural perspective on frame operators in the non-Archimedean context.

Numerical Strength and Theoretical Claims

• The sharp uncertainty bound $p$9 is attained sharply, in contrast to all Euclidean settings.
• The construction of an orthonormal Gabor basis with window concentration on minimal uncertainty cells cannot be replicated in $\mathbb{Q}_p$0 due to Balian–Low type obstructions.
• The formulation of operator representations (Walnut, Wexler–Raz, Janssen) reproduces the analytic backbone of time–frequency theory in a genuinely non-Archimedean context, with explicit dictionary between compact open subgroups and the standard time–frequency lattice parameters.

Implications and Prospects

The mathematical implications are twofold: (1) the existence of maximally localized Gabor orthonormal bases suggests the potential for efficient and robust analysis/synthesis systems in $\mathbb{Q}_p$1-adic and related settings, relevant to signal processing and mathematical physics in non-Archimedean frameworks; (2) operator-theoretic tools—previously restricted to Euclidean analysis—are now systematically available for analysis of frames, duality, and spectral properties of time–frequency systems over local fields.

Practically, this advances the development of $\mathbb{Q}_p$2-adic models in areas such as quantum mechanics, where coherent states with minimal uncertainty are central, and opens avenues for extensions to non-Archimedean pseudodifferential operators [grochenig2007pseudodifferential] and $\mathbb{Q}_p$3-adic stochastic processes [bikulov1997p] or neural networks [albeverio1999p].

From a theoretical standpoint, the results support further generalizations to locally compact abelian groups with compact open subgroups, laying foundations for time–frequency analysis in much broader algebraic topological contexts.

Future Directions

Potential future developments include:

• Extension to multi-window and irregular Gabor systems over local fields.
• Analysis of boundedness and spectral properties of pseudodifferential operators derived from Gabor theory in these settings.
• Exploration of uncertainty, entropy, and localization phenomena in $\mathbb{Q}_p$4-adic quantum systems using maximally localized bases.
• Generalization to non-abelian settings or higher-rank local fields.

Conclusion

This work provides a comprehensive analytic foundation for Gabor frames and orthonormal bases over local fields. It demonstrates that, unlike the Euclidean case, local fields support maximally localized Gabor orthonormal bases, and classical operator representations carry over with explicit, computable structure. The results enable both the theoretical exploration of harmonic analysis and applied development of algorithms and models in non-Archimedean frameworks, highlighting a robust and flexible extension of time–frequency methods beyond the Archimedean paradigm.

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References:

• "Integral bounds for radar ambiguity functions and Wigner distributions" [lieb1990integral]
• "Maximally localized Gabor orthonormal bases on locally compact Abelian groups" [nicola2024maximally]
• "Foundations of Time–Frequency Analysis" [Grochenig01]
• "Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class" [grochenig2007pseudodifferential]
• "p-Adic Brownian motion" [bikulov1997p]
• "p-Adic dynamical systems and neural networks" [albeverio1999p]