PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Limited-angle data sample the visible Fourier wedge and zero frequency

Statement

Assume the Axiom of Countable Choice. Let $Θ \subset S^{1}$ be a nonempty proper open antipodally symmetric set whose complement contains a nonempty open arc, and let $W_{Θ} = {σ θ : θ \in Θ, σ \in R ∖ {0}}$ . For $f \in S (R^{2})$ and every measured normal $θ \in Θ$ , the one-dimensional Fourier transform in the offset variable obeys $\hat{R_{S} f} (θ, σ) = \hat{f} (σ θ)$ for all $σ \in R$ . In particular, zero restricted data imply that $\hat{f}$ vanishes on $W_{Θ}$ . If in addition $f \in C_{c}^{\infty} (R^{2})$ , zero restricted data imply $f = 0$ everywhere. For compactly supported inputs, the stated Countable Choice assumption identifies the Lebesgue line integrals here with the compact-support Riemann convention; the compact-support Fourier-slice and uniqueness results in that convention are proved without Countable Choice.

Statement references

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1

Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography, §1.3.2

Marco Mazzucchelli, Mikko Salo, and Leo Tzou, A general support theorem for analytic double fibration transforms, §§1.2–1.3

Facts & Assumptions

Given: The set $Θ$ , $W_{Θ}$ , and a function $f \in S (R^{2})$ ; in the compact-support conclusion also $f \in C_{c}^{\infty} (R^{2})$ and $R_{S} f (θ, s) = 0$ for every $θ \in Θ$ and $s \in R$ .

[F1]

Visible covectors for limited-angle data: the measured Fourier wedge is $W_{Θ} = {σ θ : θ \in Θ, σ \neq 0}$ .

[F2]

Radon transform on Schwartz functions: $R_{S} f (θ, s) = \int_{R} f (s θ + t θ^{⊥}) d t$ with Lebesgue measure; under Countable Choice, this agrees on compactly supported smooth inputs with the compact-support Riemann line integral.

[F3]

Fourier-slice identity for Schwartz inputs: for $f \in S$ , $\hat{R_{S} f} (θ, σ) = \hat{f} (σ θ)$ .

[F4]

The Fourier-slice identity for the Euclidean Radon transform: for $f \in C_{c}^{\infty}$ and the IP-36 normalization, the same compact-support Fourier-slice identity holds.

[F5]

Compactly supported smooth functions are unique from any open set of Radon angles: compactly supported smooth functions with zero data on a nonempty open set of line normals vanish everywhere; the proof uses the Riemann Fourier-Laplace transform and no Countable Choice.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used by [F2] to identify the compact-support line-integral conventions and by [F3] for the general Schwartz Fourier-slice formula. The compact-support Fourier-slice and uniqueness results [F4]–[F5] themselves are choice-free in the Riemann convention.

Proof

1.1givenF1F2F3A1

For each $θ \in Θ$ and $σ \in R$ , [F3] gives the Fourier-slice identity. If the restricted data vanish, their one-dimensional Fourier transforms vanish, so $\hat{f} (σ θ) = 0$ . Every point of $W_{Θ}$ has this form, hence $\hat{f}$ vanishes on $W_{Θ}$ .

2.1F2F4F5given∎

Now assume $f \in C_{c}^{\infty}$ and its restricted data are zero. By [F2] and the stated Countable Choice assumption, those data use the compact-support Riemann line-integral convention. The compact-support Fourier-slice identity [F4] gives $\hat{f} (σ θ) = 0$ for every $θ \in Θ$ and $σ \in R$ , so the transform vanishes on the measured wedge. The compact-support uniqueness result [F5] then yields $f = 0$ everywhere through its separate choice-free analytic-continuation proof.

Depends on

Used by

Sharp deterministic stability for limited-angle reconstruction Open problem

Dependency tree · two levels

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Sources

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1 (standard reference, not scraped)
Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography (standard reference, not scraped)
Marco Mazzucchelli, Mikko Salo, and Leo Tzou, A general support theorem for analytic double fibration transforms (standard reference, not scraped)