arxiv: 2605.07673 · v1 · submitted 2026-05-08 · 🧮 math.CA · math.AP· math.CV

Recognition: 2 theorem links

· Lean Theorem

Hermite expansions of functions from the weighted Hardy class

Manish Chaurasia, Ramesh Manna, Satyajyoti Achar

Pith reviewed 2026-05-11 02:01 UTC · model grok-4.3

classification 🧮 math.CA math.APmath.CV

keywords Hermite expansionsweighted Hardy spacesPilipović spacesharmonic oscillatorSchrödinger equationuncertainty principleLaguerre expansionsshort-time Fourier transform

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The pith

Functions whose value and Fourier transform both decay like Gaussians admit decaying Hermite coefficients that control the time evolution under the harmonic oscillator.

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

The paper examines spaces of functions where both the function and its Fourier transform exhibit Gaussian decay together with exponential growth controlled by weight functions. For logarithmic weights these spaces coincide with Pilipović spaces, which permits explicit decay bounds on the coefficients of the Hermite expansion. Combining those bounds with the known asymptotics of Hermite functions produces decay rates for solutions of the Schrödinger equation with the harmonic-oscillator potential. The same framework yields exponential decay of the Hermite projection operators for a wider class of weights and supplies a partial sharpening of the subcritical Hardy uncertainty principle.

Core claim

Functions in the weighted Hardy class possess Hermite coefficients whose size is controlled by the weight functions; when the weights are logarithmic this control transfers from the known theory of Pilipović spaces and, via Hermite-function asymptotics, yields explicit decay for the time-dependent solutions of the harmonic-oscillator Schrödinger equation. For other weights the Hermite projections themselves decay exponentially, as shown by Laguerre expansions and the short-time Fourier transform. The work also improves the subcritical case of the Hardy uncertainty principle.

What carries the argument

The weighted Hardy class, consisting of functions and their Fourier transforms that decay like Gaussians modulated by weight functions; the resulting decay estimates for the Hermite coefficients serve as the bridge to the Schrödinger decay and the projection bounds.

If this is right

Solutions of the harmonic-oscillator Schrödinger equation decay in time at a rate determined by the weight function.
Hermite projection operators decay exponentially when acting on the weighted spaces.
A partial strengthening holds for the subcritical Hardy uncertainty principle.

Where Pith is reading between the lines

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

The coefficient decay may extend to other quadratic potentials or to certain nonlinear Schrödinger equations whose linear part is the harmonic oscillator.
The use of the short-time Fourier transform suggests that similar estimates could apply to broader classes of time-frequency localized operators.

Load-bearing premise

The equivalence, under logarithmic weights, between the weighted Hardy spaces and Pilipović spaces is needed to carry over the coefficient decay estimates.

What would settle it

A concrete function belonging to one of the weighted Hardy spaces whose Hermite coefficients fail to obey the predicted decay rate, or a solution of the harmonic-oscillator Schrödinger equation whose time decay violates the claimed bound.

read the original abstract

In this paper, we analyze a function space consisting of functions for which both the function and its Fourier transform exhibit Gaussian decay together with exponential growth governed by suitable weight functions. First, we examine logarithmic-type weights, in which case these function spaces are equivalent to Pilipovi\'c spaces. In this setting, we establish a decay estimate for the Hermite coefficients of functions. Furthermore, by combining these estimates with the asymptotic behavior of Hermite functions, we prove a decay rate for solutions to the harmonic oscillator Schr\"odinger equation. Second, we consider a class of weights and prove the exponential decay of the Hermite projection operators on these spaces by analyzing Laguerre expansions and the short-time Fourier transform. Additionally, we revisit the subcritical Hardy uncertainty principle and obtain a partial improvement toward a conjecture posed by Vemuri.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit Hermite coefficient decay rates for logarithmic weights in these Gaussian-decay spaces, applies them to the harmonic oscillator Schrödinger equation, and adds a partial improvement on Vemuri's conjecture via projection decay estimates.

read the letter

The main things here are the decay rates for Hermite coefficients when the weights are logarithmic, the resulting control on solutions to the harmonic oscillator Schrödinger equation, and the separate exponential decay result for projection operators that yields a partial strengthening of Vemuri's conjecture on the subcritical Hardy uncertainty principle. The logarithmic case rests on showing equivalence to Pilipović spaces, after which the coefficient bounds follow from known asymptotics of Hermite functions. The other weight class uses Laguerre expansions and short-time Fourier transform properties to get the projection decay directly. Both threads are presented as new and not reducible to earlier work on these spaces. The applications are straightforward once the coefficient estimates are available, and the tools are standard ones in the area. The paper does a clean job sequencing the steps without obvious circularity. The equivalence step for the logarithmic weights is the part that carries the most weight, so a referee will want to see the details of that identification. If it holds, the rest of the argument looks routine and the claims should go through. The projection decay argument appears less delicate since it leans on established properties of Laguerre polynomials and the STFT. This is specialized harmonic analysis work aimed at people who already care about weighted Pilipović-type spaces, uncertainty principles, or decay estimates for the harmonic oscillator. A reader working on related PDE or Fourier analysis questions could pick up the explicit rates and use them. It is grounded enough and specific enough to merit a serious referee who can check the equivalence and the new estimates in detail rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper studies function spaces in which both a function and its Fourier transform exhibit Gaussian decay modulated by weight functions. For logarithmic weights these spaces are equivalent to Pilipović spaces, yielding decay estimates on Hermite coefficients; these estimates are combined with the known asymptotics of Hermite functions to obtain decay rates for solutions of the harmonic-oscillator Schrödinger equation. For a broader class of weights the authors prove exponential decay of the Hermite projection operators by means of Laguerre expansions and the short-time Fourier transform. The paper also supplies a partial improvement of the subcritical Hardy uncertainty principle that advances toward a conjecture of Vemuri.

Significance. If the arguments are correct, the work furnishes concrete decay rates that connect weighted Hardy-type spaces to classical harmonic-analysis tools and to the harmonic-oscillator evolution, while offering a modest but explicit advance on an open uncertainty-principle conjecture. The reliance on standard asymptotics, Laguerre expansions, and the short-time Fourier transform, together with the explicit equivalence to Pilipović spaces, supplies reproducible and falsifiable statements in the form of precise decay exponents.

major comments (1)

[section on logarithmic weights / equivalence to Pilipović spaces] The equivalence between the weighted Gaussian-decay spaces and Pilipović spaces for logarithmic weights is invoked to transfer the coefficient-decay estimates; the manuscript must supply a self-contained proof of this equivalence (including the precise constants that appear in the norm comparison) in the section treating logarithmic weights, because the subsequent decay rates for the Schrödinger equation rest directly on it.

minor comments (2)

[Introduction and statements of main results] The precise decay exponents obtained for the Schrödinger solutions and for the Hermite projections should be stated explicitly in the introduction or in a summary theorem, together with a brief comparison to the best previously known rates.
[Notation and definitions] Notation for the weight functions (especially the distinction between logarithmic and general weights) should be introduced once and used consistently; a short table summarizing the admissible weights and the corresponding results would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address the single major comment below and will revise the paper to incorporate the requested addition.

read point-by-point responses

Referee: [section on logarithmic weights / equivalence to Pilipović spaces] The equivalence between the weighted Gaussian-decay spaces and Pilipović spaces for logarithmic weights is invoked to transfer the coefficient-decay estimates; the manuscript must supply a self-contained proof of this equivalence (including the precise constants that appear in the norm comparison) in the section treating logarithmic weights, because the subsequent decay rates for the Schrödinger equation rest directly on it.

Authors: We agree that the equivalence to Pilipović spaces for logarithmic weights is invoked rather than proved in detail in the current version, and that a self-contained argument with explicit norm constants is needed to make the subsequent decay estimates for the harmonic-oscillator Schrödinger equation fully rigorous. In the revised manuscript we will insert a complete proof of the equivalence (including the comparison of norms with explicit constants) directly in the logarithmic-weights section, so that the coefficient-decay estimates and their application to the Schrödinger equation no longer rely on external references. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent standard tools and equivalences

full rationale

The paper's chain proceeds by first establishing (or invoking) equivalence of the weighted Gaussian-decay spaces to Pilipović spaces for logarithmic weights, then deriving Hermite-coefficient decay estimates from that equivalence, and finally combining the estimates with the known asymptotic decay of Hermite functions to control solutions of the harmonic-oscillator Schrödinger equation. The second part uses Laguerre expansions and the short-time Fourier transform—standard, externally verifiable tools—to obtain exponential decay of projections for other weights. The partial improvement on the Vemuri conjecture is obtained by revisiting the subcritical Hardy uncertainty principle with these estimates. None of these steps reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or redefinition of the target quantity; all load-bearing inputs are either classical asymptotics or independently established equivalences.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claims rest on standard properties of Hermite and Laguerre functions together with the asserted equivalence of the weighted spaces to Pilipović spaces for logarithmic weights; no new entities or fitted numerical parameters are introduced.

axioms (3)

standard math Hermite functions possess known asymptotic decay and growth properties at infinity
Invoked to convert coefficient bounds into pointwise or solution decay for the Schrödinger equation.
domain assumption The weighted Gaussian-decay spaces coincide with Pilipović spaces when the weights are logarithmic
Stated explicitly as the setting in which the coefficient decay estimates are first obtained.
domain assumption Laguerre expansions and the short-time Fourier transform are well-defined and admit the required estimates on the chosen weight class
Used to establish exponential decay of the Hermite projection operators.

pith-pipeline@v0.9.0 · 5448 in / 1600 out tokens · 45569 ms · 2026-05-11T02:01:15.566909+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
we establish a decay estimate for the Hermite coefficients... by combining these estimates with the asymptotic behavior of Hermite functions, we prove a decay rate for solutions to the harmonic oscillator Schrödinger equation
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear
the spaces E^d_F(1,1,λ,(log+x)^{1/(1-2s)}) are equivalent to the Pilipović spaces

Reference graph

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