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arxiv: 1907.08812 · v1 · pith:O2Q5WZ5Mnew · submitted 2019-07-20 · 🧮 math.FA

Uncertainty Principles for Fourier Multipliers

Michael Northington V This is my paper

Pith reviewed 2026-05-24 18:44 UTC · model grok-4.3

classification 🧮 math.FA
keywords Fourier multipliersSobolev regularityBalian-Low uncertainty principleGabor systemsshift-invariant systemstorusHausdorff dimensionmatrix-valued multipliers
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The pith

A function w on the d-torus with a zero whose reciprocal is a (p,q)-multiplier must obey quantified Sobolev regularity bounds.

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

The paper quantifies how much Sobolev regularity a function w must have when it vanishes at some point on the d-dimensional torus and its reciprocal is a Fourier multiplier of type (p,q). It handles cases where the zero set has positive Hausdorff dimension, matrix-valued multipliers, and non-symmetric regularity. The work connects the multiplier property to approximation properties of Gabor systems and shift-invariant systems. This connection is then used to refine and extend versions of the Balian-Low uncertainty principle in those settings. A reader cares because the refined principle gives sharper limits on how localized these systems can be while still forming bases or frames.

Core claim

When w has a zero on the torus and u = 1/w is a (p,q)-multiplier, the admissible Sobolev regularity of w is quantified; the same multiplier condition implies approximation properties for Gabor and shift-invariant systems, which in turn refine the Balian-Low uncertainty principle for those systems. The results also cover zero sets of positive Hausdorff dimension, matrix-valued multipliers, and non-symmetric Sobolev norms.

What carries the argument

The link between the (p,q)-multiplier property of 1/w on the torus and the approximation properties of associated Gabor and shift-invariant systems, which transfers regularity statements into refined uncertainty bounds.

If this is right

  • The Balian-Low principle extends to Gabor systems generated by windows whose reciprocals are multipliers.
  • Shift-invariant systems inherit similar refined uncertainty statements under the same multiplier condition.
  • Matrix-valued multipliers produce corresponding regularity results for vector-valued functions.
  • Zero sets of positive Hausdorff dimension are admissible without losing the quantified regularity.
  • Non-symmetric Sobolev norms yield more flexible bounds than the symmetric case.

Where Pith is reading between the lines

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

  • The multiplier-to-approximation link may produce density restrictions on time-frequency lattices beyond the classical Balian-Low setting.
  • Explicit trigonometric-polynomial multipliers could be tested to obtain sharp constants in the regularity statements.
  • The torus results suggest analogous statements on other compact groups or manifolds where Fourier multipliers are defined.

Load-bearing premise

The fact that 1/w is a (p,q)-multiplier on the torus implies the approximation properties for Gabor and shift-invariant systems that are required to obtain the refined uncertainty principle.

What would settle it

An explicit function w on the torus that vanishes somewhere, has 1/w as a (p,q)-multiplier, yet fails to meet the claimed Sobolev regularity, or a concrete Gabor system whose window satisfies the multiplier condition but violates the refined Balian-Low bound.

read the original abstract

The admittable Sobolev regularity is quantified for a function, $w$, which has a zero in the $d$--dimensional torus and whose reciprocal $u=1/w$ is a $(p,q)$--multiplier. Several aspects of this problem are addressed, including zero--sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non--symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift--invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian--Low uncertainty principle in these settings.

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

2 major / 2 minor

Summary. The paper quantifies admissible Sobolev regularity for a function w on the d-dimensional torus that vanishes at some point, under the assumption that u = 1/w is a (p,q)-Fourier multiplier. It treats zero sets of positive Hausdorff dimension, matrix-valued multipliers, and non-symmetric Sobolev norms, then links the multiplier condition to approximation properties of Gabor and shift-invariant systems in order to refine and extend Balian-Low-type uncertainty principles.

Significance. If the central claims hold, the work supplies a concrete bridge between Fourier-multiplier theory and uncertainty principles, extending Balian-Low results to a broader class of windows and lattices via approximation-theoretic consequences of the multiplier condition. The treatment of positive-dimensional zero sets and matrix-valued multipliers adds technical generality not present in earlier versions.

major comments (2)
  1. [§4] §4 (connection between multiplier condition and approximation properties): the claim that u being a (p,q)-multiplier implies the requisite approximation properties for Gabor/SI systems (used to refine Balian-Low) is load-bearing for the uncertainty-principle statements, yet the derivation appears to require additional frame-bound or window-regularity hypotheses that are not stated in the main theorems.
  2. [§3] Theorem on Hausdorff-dimension zero sets (likely §3): the Sobolev-regularity bound is asserted to remain valid for zero sets of positive Hausdorff measure, but the proof sketch relies on a capacity estimate whose dependence on the multiplier norm of u is not made fully explicit, leaving open whether the constant deteriorates with the dimension of the zero set.
minor comments (2)
  1. Notation for the (p,q)-multiplier norm and the precise definition of admissible Sobolev regularity should be recalled at the beginning of each main section rather than only in the introduction.
  2. The abstract contains the typographical error 'admittable' for 'admissible'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating where revisions will be incorporated to improve clarity and precision.

read point-by-point responses

  1. Referee: [§4] §4 (connection between multiplier condition and approximation properties): the claim that u being a (p,q)-multiplier implies the requisite approximation properties for Gabor/SI systems (used to refine Balian-Low) is load-bearing for the uncertainty-principle statements, yet the derivation appears to require additional frame-bound or window-regularity hypotheses that are not stated in the main theorems.

    Authors: We agree that the main theorems refining the Balian-Low principle should explicitly state the frame-bound and window-regularity assumptions used in the derivation of the approximation properties in §4. These hypotheses are implicit in the context of the Gabor and shift-invariant systems considered but were not listed in the theorem statements. In the revised version we will add them to the statements of the relevant theorems (and to the statement of the connection result in §4) so that the load-bearing conditions are fully transparent. This is a clarification only and does not alter the results. revision: yes

  2. Referee: [§3] Theorem on Hausdorff-dimension zero sets (likely §3): the Sobolev-regularity bound is asserted to remain valid for zero sets of positive Hausdorff measure, but the proof sketch relies on a capacity estimate whose dependence on the multiplier norm of u is not made fully explicit, leaving open whether the constant deteriorates with the dimension of the zero set.

    Authors: The referee is correct that the dependence of the capacity estimate on the multiplier norm ||u||_{M_{p,q}} should be stated explicitly. In the proof the constant in the Sobolev bound is controlled by ||u||_{M_{p,q}} and the ambient dimension d; it does not deteriorate further with the Hausdorff dimension of the zero set. We will revise the proof to display this dependence explicitly and add a short remark confirming uniformity with respect to the dimension of the zero set. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation quantifies Sobolev regularity of w (with zeros) given that 1/w is a (p,q)-multiplier, then connects this via approximation properties to refined Balian-Low statements for Gabor and shift-invariant systems. No step reduces by construction to a fitted input, self-definition, or self-citation chain; the abstract explicitly invokes external multiplier theory as the foundation, and the reader's assessment confirms the absence of internal reduction. The central claims therefore remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be identified.

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