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arxiv: 2401.06871 · v3 · submitted 2024-01-12 · 🧮 math.AP · math.CV· math.DS

Hyperbolic Fourier series and the Klein-Gordon equation

H. Hedenmalm , A. Montes-Rodriguez This is my paper

Pith reviewed 2026-05-24 04:06 UTC · model grok-4.3

classification 🧮 math.AP math.CVmath.DS
keywords hyperbolic Fourier seriesKlein-Gordon equationbiorthogonal systemdistributional convergenceFourier interpolationweak-star completenessDirac delta representationlattice interpolation
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The pith

Distributions on the real line admit unique hyperbolic Fourier series via a biorthogonal system to exponentials on R and 1/R.

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

The paper establishes that the combined system of functions e^{i π m x} and e^{-i π n / x} is weak-star complete in L^∞(R). This completeness yields a biorthogonal system whose coefficients grow at most polynomially, allowing any distribution on the compactified real line to be written uniquely as a hyperbolic Fourier series that converges in the distributional sense. The same construction produces interpolating solutions for the Klein-Gordon equation that vanish on a lattice-cross except at one prescribed point, thereby recovering a general solution from its values on that cross.

Core claim

The system {e_m, e_n†} admits a biorthogonal system {A_n, B_n} with at most polynomial growth, so that the unit point mass admits the expansion δ_x(t) = A_0(x) + ∑_{n≠0} (A_n(x) e^{i π n t} + B_n(x) e^{-i π n / t}) with distributional convergence; more generally every distribution f on the compactified line possesses a unique hyperbolic Fourier series f(t) = a_0(f) + ∑_{n≠0} (a_n(f) e^{i π n t} + b_n(f) e^{-i π n / t}). The same biorthogonal system supplies solutions of ∂_x ∂_y u + u = 0 that vanish on the lattice-cross {(π k, 0), (0, π l)} except at one point and thereby permit restoration of any solution from its values on the cross.

What carries the argument

The biorthogonal system {A_n, B_n} to {e_m, e_n†}, which supplies the coefficients of the hyperbolic Fourier series and the interpolating solutions for the Klein-Gordon equation.

If this is right

  • Every distribution on the compactified real line possesses a unique hyperbolic Fourier series converging in the distributional sense.
  • The Dirac delta at an arbitrary point x admits an explicit representation with polynomially bounded coefficients.
  • Solutions of the Klein-Gordon equation are uniquely recoverable from their values on the lattice-cross of points (π k, 0) and (0, π l).
  • The same biorthogonal system yields interpolating solutions that vanish on the entire lattice-cross except at one prescribed point.

Where Pith is reading between the lines

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

  • The construction may supply analogous series for other hyperbolic PDEs whose characteristics involve reciprocal variables.
  • Truncations of the series could be tested numerically for approximation quality on compactly supported test distributions.
  • The link to radial Fourier interpolation suggests the series could be used to recover radial functions from partial data on spheres of varying radii.

Load-bearing premise

The weak-star completeness of {e_m, e_n†} in L^∞(R) together with the existence of a biorthogonal system whose coefficients grow at most polynomially.

What would settle it

A distribution on the compactified real line whose representation by the series either fails to converge distributionally or requires coefficients that grow faster than any polynomial.

read the original abstract

In an effort to extend classical Fourier theory, Hedenmalm and Montes-Rodr\'{\i}guez (2011) found that the function system \[ e_m(x)=e^{i\pi mx},\quad e_n^\dagger(x)=e_n(-1/x)=e^{-i\pi n/x} \] is weak-star complete in $L^{\infty}(\mathbb{R})$ when $m,n$ range over the integers with $n\ne0$. It turns out that the system can be used to provide unique representation of functions and more generally distributions on the real line $\mathbb{R}$. For instance, we may represent uniquely the unit point mass at a point $x\in\mathbb{R}$: \[ \delta_x(t)=A_0(x)+\sum_{n\ne0}\big(A_n(x)\,e^{i\pi nt} +B_n(x)\,e^{-i\pi n/t}\big), \] with at most polynomial growth of the coefficients, so that the sum converges in the sense of distribution theory. In a natural sense, the system $\{A_n,B_n\}_n$ is biorthogonal to the initial system $\{e_n,e_n^\dagger\}_n$ on the real line. More generally, for a distribution $f$ on the compactified real line, we may decompose it in a \emph{hyperbolic Fourier series} \[ f(t)=a_0(f)+\sum_{n\ne0}\big(a_n(f)\,e^{i\pi nt}+b_n(f)\,e^{-i\pi n/t}\big), \] understood to converge in the sense of distribution theory. Such hyperbolic Fourier series arise from two different considerations. One is the Fourier interpolation problem of recovering a radial function $\phi$ on $\mathbb{R}^d$ from partial information on $\phi$ and its Fourier transform $\hat \phi$, studied by Radchenko and Viazovska (2019). Another consideration is the interpolation theory of the Klein-Gordon equation $\partial_x\partial_y u+u=0$. For instance, the biorthogonal system $\{A_n,B_n\}_n$ leads to a collection of solutions that vanish along the lattice-cross of points $(\pi k,0)$ and $(0,\pi l)$ save for one of these points. These interpolating solutions allow for restoration of a given solution $u$ from its values on the lattice-cross.

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 / 1 minor

Summary. The manuscript extends the 2011 weak-star completeness result of Hedenmalm and Montes-Rodríguez for the system {e_m(x)=e^{iπ m x}, e_n^†(x)=e^{-iπ n/x} (n≠0)} in L^∞(R) to claim unique representations of functions and distributions on R. It asserts that the Dirac delta admits the expansion δ_x(t)=A_0(x)+∑_{n≠0}(A_n(x) e^{iπ n t}+B_n(x) e^{-iπ n/t}) with coefficients of at most polynomial growth in n, converging in the distributional sense, and introduces the corresponding hyperbolic Fourier series for general distributions on the compactified line. These representations are applied to radial Fourier interpolation problems and to the construction of interpolating solutions for the Klein-Gordon equation ∂_x ∂_y u + u =0 that vanish on the lattice cross except at one point.

Significance. If the existence of a polynomially bounded biorthogonal system is established, the work supplies a concrete distributional representation tool that could be useful for interpolation questions in Fourier analysis and for initial-value problems on the Klein-Gordon equation. The connection to the Radchenko-Viazovska interpolation problem and the explicit construction of lattice-cross interpolants for the wave equation are potentially valuable applications, though they rest on the same unverified growth control.

major comments (2)
  1. [Abstract] Abstract (and § on distributional representation): the central claim that the biorthogonal coefficients A_n(x), B_n(x) admit at most polynomial growth in n (for each fixed x) is invoked to guarantee that the series converges in the sense of distributions when tested against Schwartz functions. The 2011 weak-star completeness result only yields uniqueness of any existing expansion; it does not construct the dual system or supply the required growth bound. No independent estimate, explicit formula, or reference to a proof of polynomial growth appears in the provided text.
  2. [Klein-Gordon interpolation] Klein-Gordon interpolation section: the assertion that the biorthogonal system produces solutions vanishing on the lattice cross (πk,0) and (0,πl) except at one point relies on the same distributional representation. Without a verified growth estimate on the coefficients, the convergence of the resulting series solution to the Klein-Gordon equation in the appropriate function space remains unestablished.
minor comments (1)
  1. [Abstract] The abstract states that the system is 'weak-star complete in L^∞(R)' but does not restate the precise statement of the 2011 theorem or indicate which parts of the present argument are new versus cited.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. The primary issues raised concern the justification for the polynomial growth of the biorthogonal coefficients, which underpins the distributional representations and their applications. We provide point-by-point responses below and commit to revisions that address these concerns by supplying the missing proofs.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and § on distributional representation): the central claim that the biorthogonal coefficients A_n(x), B_n(x) admit at most polynomial growth in n (for each fixed x) is invoked to guarantee that the series converges in the sense of distributions when tested against Schwartz functions. The 2011 weak-star completeness result only yields uniqueness of any existing expansion; it does not construct the dual system or supply the required growth bound. No independent estimate, explicit formula, or reference to a proof of polynomial growth appears in the provided text.

    Authors: The referee correctly notes that the 2011 weak-star completeness result provides uniqueness but does not address existence or growth bounds. Our manuscript asserts the polynomial growth without a detailed proof in the submitted version. To rectify this, we will include in the revision a new subsection that constructs the biorthogonal coefficients explicitly and proves their polynomial growth in n for fixed x. This construction will rely on solving the moment problem associated with the system and using estimates derived from the completeness property combined with analytic continuation arguments. With this addition, the distributional convergence will be fully justified. revision: yes

  2. Referee: [Klein-Gordon interpolation] Klein-Gordon interpolation section: the assertion that the biorthogonal system produces solutions vanishing on the lattice cross (πk,0) and (0,πl) except at one point relies on the same distributional representation. Without a verified growth estimate on the coefficients, the convergence of the resulting series solution to the Klein-Gordon equation in the appropriate function space remains unestablished.

    Authors: We agree that the validity of the interpolating solutions for the Klein-Gordon equation hinges on the convergence guaranteed by the polynomial growth. In the revised manuscript, after establishing the growth bound in the distributional representation part, we will explicitly verify that the series solutions converge in the appropriate space (e.g., suitable Sobolev or distribution spaces) and satisfy the Klein-Gordon equation with the desired vanishing properties on the lattice cross. This will be done by term-by-term differentiation justified by the growth control. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to 2011 completeness result; new representations and Klein-Gordon applications developed independently

specific steps
  1. self citation load bearing [Abstract, opening paragraph]
    "Hedenmalm and Montes-Rodríguez (2011) found that the function system e_m(x)=e^{iπ m x}, e_n^†(x)=e_n(-1/x)=e^{-iπ n/x} is weak-star complete in L^∞(R) when m,n range over the integers with n≠0. It turns out that the system can be used to provide unique representation of functions and more generally distributions on the real line R. For instance, we may represent uniquely the unit point mass at a point x∈R: δ_x(t)=A_0(x)+∑_{n≠0}(A_n(x) e^{iπ n t} + B_n(x) e^{-iπ n / t}), with at most polynomial growth of the coefficients, so that the sum converges in the sense of distribution theory."

    Uniqueness of the claimed representation is justified solely by invoking the weak-star completeness result from the authors' own 2011 paper. While the manuscript asserts existence of the biorthogonal coefficients with polynomial growth and distributional convergence, the provided text does not exhibit an independent construction or growth estimate here; the self-citation therefore supplies the uniqueness half of the central representation claim.

full rationale

The paper cites its own prior 2011 work only for weak-star completeness of {e_m, e_n^†} in L^∞(R), which underpins uniqueness of any existing expansion. The existence of the biorthogonal system {A_n, B_n} with at most polynomial growth, the distributional convergence for deltas and general distributions, the definition of hyperbolic Fourier series, and the Klein-Gordon interpolation solutions are all developed as new content in this manuscript. No reduction of a central claim to a self-citation chain, self-definition, or fitted input occurs. This is a standard minor self-citation that does not affect the independence of the main results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The representation rests on the weak-star completeness of the given function system and on the polynomial-growth bound for the biorthogonal coefficients; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The system {e_m(x), e_n^dagger(x)} is weak-star complete in L^infty(R) for m,n integers, n≠0.
    Invoked as the foundation for unique representation; taken from the authors' 2011 paper.

pith-pipeline@v0.9.0 · 5991 in / 1352 out tokens · 36696 ms · 2026-05-24T04:06:14.292833+00:00 · methodology

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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 echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    δ_x(t)=A_0(x)+∑(A_n(x)e^{iπ n t}+B_n(x)e^{-iπ n /t}), with at most polynomial growth of the coefficients, so that the sum converges in the sense of distribution theory.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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