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arxiv: 2605.17970 · v1 · pith:55OSFMUKnew · submitted 2026-05-18 · 🧮 math.CA · math.FA

Gabor unconditional bases and frames in L^p(mathbb{R})

Nir Lev , Anton Tselishchev This is my paper

Pith reviewed 2026-05-20 00:38 UTC · model grok-4.3

classification 🧮 math.CA math.FA
keywords Gabor systemsunconditional Schauder framesL^p spacestime-frequency shiftsBalian-Low theorembases in Banach spaces
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The pith

For p greater than 2, Gabor systems form unconditional Schauder frames in L^p(R) exactly when the time-frequency set Lambda satisfies a specific characterization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks whether a window g in L^p(R) exists so that its time-frequency shifts over a discrete set Lambda form an unconditional basis or Schauder frame when p is not 2. This question matters because L^p spaces lack the inner-product structure of L^2, so standard Hilbert-space frame results do not carry over and new obstructions appear. For p greater than 2 the authors give a complete characterization of the sets Lambda that permit unconditional Schauder frames. For 1 less than p less than 2 they prove non-existence whenever Lambda meets a natural separation condition. They further establish a Balian-Low-type obstruction showing that g cannot combine mild continuity with decay.

Core claim

We completely resolve this question for p>2; in particular, we characterize the sets Lambda such that an unconditional Schauder frame of this form exists. We also prove a Balian-Low type result, showing that the window function g cannot enjoy mild continuity and decay conditions. For 1<p<2, we prove that a Gabor system cannot form an unconditional basis or unconditional Schauder frame in L^p(R) if the set Lambda satisfies a natural separation condition.

What carries the argument

The characterization of discrete sets Lambda in the time-frequency plane that admit an unconditional Gabor Schauder frame in L^p for p>2.

If this is right

  • Unconditional Schauder frames exist for certain Lambda when p>2.
  • No unconditional bases or frames exist for separated Lambda when 1<p<2.
  • The window g must fail at least one of mild continuity or decay.
  • Density and separation properties of Lambda control the existence of these frames in a manner absent from the p=2 case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same density conditions that guarantee frames for p>2 may fail to do so in other Banach function spaces.
  • Explicit constructions of g for admissible Lambda could be used to test numerical stability of expansions in L^p for p>2.
  • The separation obstruction for p<2 suggests that overcomplete systems might still be feasible if Lambda is allowed to cluster.

Load-bearing premise

The results assume the Gabor system arises from time-frequency shifts of a single g over a discrete set Lambda obeying separation or density conditions.

What would settle it

Exhibit a concrete discrete set Lambda that the characterization declares admissible for p=3, then show that no g in L^3(R) makes the corresponding Gabor system an unconditional Schauder frame.

read the original abstract

We consider the following problem: given a set $\Lambda \subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 \pi isx}\}$, $(t,s) \in \Lambda$, consisting of time-frequency shifts of $g$, forms an unconditional basis or unconditional Schauder frame in the space $L^p(\mathbb{R})$? We completely resolve this question for $p>2$; in particular, we characterize the sets $\Lambda$ such that an unconditional Schauder frame of this form exists. We also prove a Balian-Low type result, showing that the window function $g$ cannot enjoy mild continuity and decay conditions. For $1<p<2$, we prove that a Gabor system cannot form an unconditional basis or unconditional Schauder frame in $L^p(\mathbb{R})$ if the set $\Lambda$ satisfies a natural separation condition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript examines whether, for p ≠ 2 and a set Λ ⊂ ℝ × ℝ, there exists g ∈ L^p(ℝ) such that the Gabor system of time-frequency shifts {g(x-t) e^{2π i s x}}}_{(t,s)∈Λ} forms an unconditional basis or unconditional Schauder frame in L^p(ℝ). For p > 2 the authors claim a complete characterization of those Λ that admit an unconditional Schauder frame; they also prove a Balian-Low-type theorem showing that g cannot satisfy mild continuity and decay conditions. For 1 < p < 2 they prove non-existence whenever Λ obeys a natural separation condition.

Significance. If the stated characterization holds, the work supplies a definitive answer to a natural extension of the Gabor-frame problem from Hilbert space to the Banach-space setting L^p(ℝ), p > 2. The explicit distinction between the p > 2 and 1 < p < 2 regimes, together with the Balian-Low-type obstruction, clarifies how the geometry of L^p affects the existence of unconditional time-frequency frames. The argument appears to rely on standard tools of time-frequency analysis and Banach-space theory without hidden reductions.

minor comments (2)
  1. The precise statement of the separation condition used for the 1 < p < 2 non-existence result should be recalled verbatim in the introduction so that the reader can compare it immediately with the density hypotheses appearing in the p > 2 characterization.
  2. Notation for the unconditional Schauder-frame property (e.g., the constant C in the frame inequality) is introduced in §2 but used without re-statement in the proof of the Balian-Low theorem; a short reminder paragraph would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of our results on unconditional Gabor frames in L^p for p ≠ 2. We appreciate the recommendation for minor revision and will incorporate improvements to presentation and clarity.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper delivers a direct mathematical characterization of admissible sets Λ for which unconditional Gabor Schauder frames exist in L^p(R) when p>2, together with a Balian-Low-type non-existence result under mild continuity/decay assumptions and a separation-based non-existence proof for 1<p<2. All steps rest on explicit separation or density conditions stated in the setup, Banach-space frame properties, and decay estimates that are independently derived rather than fitted or self-defined. No load-bearing step reduces by construction to a prior result from the same authors, a renamed empirical pattern, or an ansatz smuggled via citation; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the paper relies on standard axioms of functional analysis for L^p spaces and the definition of unconditional bases and frames.

axioms (1)
  • standard math Standard properties of L^p spaces as Banach spaces for 1 < p < infinity
    Invoked implicitly when discussing bases and frames in L^p(R)

pith-pipeline@v0.9.0 · 5710 in / 1196 out tokens · 32224 ms · 2026-05-20T00:38:59.603242+00:00 · methodology

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Reference graph

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