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Rapid oscillations make limited-angle data smaller than every power

Example

Assume the Axiom of Countable Choice. Take $Ω = B_{2} (0)$ , $K = {\overline{B}}_{1} (0)$ , $Θ = {θ \in S^{1} : ∣ θ_{1} ∣ < 1 / 2}$ , and $v = (1, 0)$ . Write $θ^{⊥} = (- θ_{2}, θ_{1})$ and define $R_{Θ} u (θ, s) = \int_{R} u (s θ + t θ^{⊥}) d t$ for $θ \in Θ$ , $s \in R$ . Give the data space $D = L^{2} (Θ \times R, d θ d s)$ the norm $∥ g ∥_{D} = (\int_{Θ} \int_{R} ∣ g (θ, s) ∣^{2} d s d θ)^{1 / 2}$ , with arc-length measure $d θ$ and Lebesgue measure $d s$ , and define $∥ u ∥_{H^{1}} = (∥ u ∥_{2}^{2} + ∥ \partial_{1} u ∥_{2}^{2} + ∥ \partial_{2} u ∥_{2}^{2})^{1 / 2}$ using Lebesgue measure on $R^{2}$ . Then $Θ$ is open and antipodally symmetric, its complement contains open horizontal arcs, and $\inf_{θ \in Θ} ∣ v \cdot θ^{⊥} ∣ = \sqrt{3} / 2$ . For every $M > 0$ , the class of smooth functions supported in $K$ with $H^{1}$ norm at most $M$ has no uniform Hölder stability estimate from the restricted Radon data. More explicitly, choose a real bump $ψ \in C_{c}^{\infty} (B_{1} (0))$ with $∥ ψ ∥_{2} = 1$ , set $λ_{M} = M / [2 (1 + ∥ \nabla ψ ∥_{2} + 2 π)]$ , and let $u_{N} (x) = λ_{M} N^{- 1} e^{2 π i N x_{1}} ψ (x)$ for integers $N \geq 1$ . These functions have the same compact support, satisfy $∥ u_{N} ∥_{H^{1}} \leq M$ and $∥ u_{N} ∥_{2} = λ_{M} N^{- 1}$ , while for every integer $m \geq 1$ there is $C_{m} < \infty$ with $∥ R_{Θ} u_{N} ∥_{D} \leq C_{m} N^{- m - 1}$ .

Example references

Line-normal and blind-direction conventions

High-frequency modulation and the no-Hölder estimate

Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography, §1.5.1

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

Facts & Assumptions

Given: The Axiom of Countable Choice and the stated domain $Ω$ , fixed support $K$ , measured normals $Θ$ , direction $v$ , and bump $ψ$ .

[F1]

Visible covectors for limited-angle data: $θ^{⊥} = (- θ_{2}, θ_{1})$ is the unit line tangent, so for $v = (1, 0)$ one has $∣ v \cdot θ^{⊥} ∣ = ∣ θ_{2} ∣$ .

[F2]

A smooth bump between concentric Euclidean balls: a smooth bump supported in $B_{1} (0)$ and equal to one on a smaller ball exists.

[F3]

A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included: a nondegenerate box has strictly positive Lebesgue measure, so a bump equal to one on a ball has positive $L^{2}$ norm and can be normalized.

[F4]

Limited-angle inversion has no Hölder modulus on a fixed-support H1 ball: if a fixed-support bump is normalized in $L^{2}$ and $∣ v \cdot θ^{⊥} ∣$ is uniformly bounded below on $Θ$ , its modulations $u_{N} = λ N^{- 1} e^{2 π i N v \cdot x} ψ$ have $H^{1}$ norm bounded by $λ (1 + ∥ \nabla ψ ∥_{2} + 2 π)$ and restricted data bounded by $C_{m} N^{- m - 1}$ for every $m$ ; this is the proposition's proved integration-by-parts estimate.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used in the box-measure supplier [F3] and in [F4]'s Lebesgue change-of-variables and integration-by-parts estimates.

Verification

technique · direct

1.1F1givenalgebra

The set $Θ$ is the inverse image of the open interval $(- 1 / 2, 1 / 2)$ under the continuous coordinate map $θ \mapsto θ_{1}$ , so it is open; changing $θ$ to $- θ$ preserves $∣ θ_{1} ∣$ , so it is antipodally symmetric. Its complement contains arcs around $(1, 0)$ and $(- 1, 0)$ . For every $θ \in Θ$ , $∣ θ_{2} ∣ = \sqrt{1 - θ_{1}^{2}} > \sqrt{3} / 2$ , hence the infimum is at least $\sqrt{3} / 2$ ; approaching either boundary $∣ θ_{1} ∣ = 1 / 2$ shows the infimum is exactly $\sqrt{3} / 2$ . By [F1] this is $\inf_{θ \in Θ} ∣ v \cdot θ^{⊥} ∣$ .

1.2A1F2F3givenconstruct

A translated or rescaled bump from [F2] has compact support in $B_{1} (0)$ and equals one on a smaller ball. That inner ball contains a nondegenerate box, which has positive measure by [F3]; compact support makes its $L^{2}$ norm finite, and the plateau makes the norm positive. Divide by this norm to obtain the required real $ψ$ with $∥ ψ ∥_{2} = 1$ . Since $K = {\overline{B}}_{1} (0)$ lies inside $Ω = B_{2} (0)$ , its support is contained in the fixed set $K$ .

2.1F4A1step 1.1step 1.2algebra

The formula in the Example is the modulation family in [F4]. The chosen $λ_{M}$ satisfies $λ_{M} (1 + ∥ \nabla ψ ∥_{2} + 2 π) = M / 2 \leq M$ , and the geometric verification in step 1.1 supplies the positive lower bound required there. Thus the family has the stated fixed support, $H^{1}$ and $L^{2}$ bounds, and rapid restricted-data decay.

3.1step 2.1algebra∎

If a uniform Hölder estimate with exponent $α > 0$ and constant $C \geq 0$ held on the fixed-support class, choose an integer $m \geq 1$ with $α (m + 1) > 1$ . Step 2.1 would imply $λ_{M} N^{- 1} \leq C (C_{m} N^{- m - 1})^{α}$ , so $λ_{M} \leq C C_{m}^{α} N^{1 - α (m + 1)} \to 0$ , contradicting $λ_{M} > 0$ . Therefore no such estimate holds.

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Sources

Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography (standard reference, not scraped)
Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1 (standard reference, not scraped)