ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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A point source as a distributional Radon limit

Example

Let $η (u) = 0$ for $u \leq 0$ and $η (u) = e^{- 1 / u}$ for $u > 0$ , set $ψ (x) = η (1 - ∣ x ∣^{2})$ on $R^{2}$ , let $J = \int_{R^{2}} ψ (x) d x$ , and define $ϕ = J^{- 1} ψ$ . For $ε > 0$ put $ϕ_{ε} (x) = ε^{- 2} ϕ (x / ε)$ . The functions $ϕ_{ε}$ are radial smooth unit-mass point-source approximations. For $ρ (s) = \int_{R} ϕ (s θ + t θ^{⊥}) d t$ , which is independent of $θ$ , their full line data satisfy $R ϕ_{ε} (θ, s) = ε^{- 1} ρ (s / ε) .$ For each fixed $θ \in S^{1}$ and every test function $h \in C_{c}^{\infty} (R)$ , the offset distributions converge to the point mass: $\lim_{ε ↓ 0} \int_{R} R ϕ_{ε} (θ, s) h (s) d s = h (0) .$ Thus the point-source sinogram is $δ_{0} (s)$ in the offset variable, for every line normal.

Facts & Assumptions

Given: The functions $η$ , $ψ$ , $ϕ$ , and $ϕ_{ε}$ in the Example, with $J$ to be shown positive and finite.

[F1]

The Euclidean Radon transform on R²: $R f (θ, s) = \int_{R} f (s θ + t θ^{⊥}) d t$ .

[F2]

A continuous real function on a compact Jordan measurable set is Riemann integrable over that set: continuous functions on compact Jordan sets are Riemann integrable.

[F3]

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ : the Riemann integral is monotone and additive across coordinate cuts, and linear on integrable functions.

[F4]

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections: an integrable function on a product rectangle has equal product and iterated Riemann integrals when its sections are integrable.

[F5]

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage: compactly supported Riemann integrals obey the change-of-variables formula for an injective $C^{1}$ diffeomorphism with its absolute Jacobian.

Verification

1.1givenalgebra

For $u > 0$ , repeated differentiation gives $η^{(k)} (u) = P_{k} (1 / u) e^{- 1 / u}$ for a polynomial $P_{k}$ . The exponential-series bound $e^{v} \geq v^{m + 1} / (m + 1)!$ for $v > 0$ shows that each such expression and its difference quotient tend to $0$ as $u ↓ 0$ ; induction on $k$ proves that the zero extension is smooth. Hence $ψ$ is smooth, nonnegative, vanishes for $∣ x ∣ \geq 1$ , and is positive near $x = 0$ .

2.1givenF2F3F5step 1.1

By [F2], $ψ$ is Riemann integrable on a bounding rectangle; $0 \leq ψ \leq 1$ and its support lies in $[- 1, 1]^{2}$ , so [F3] gives $J < \infty$ . On $[- 1 / 4, 1 / 4]^{2}$ one has $1 - ∣ x ∣^{2} \geq 7 / 8$ , hence $ψ \geq e^{- 8 / 7}$ ; monotonicity in [F3] gives $J \geq e^{- 8 / 7} / 4 > 0$ . Thus $ϕ$ is defined, radial, smooth, compactly supported, nonnegative, and has integral one. Applying [F5] to $x = ε y$ gives $\int_{R^{2}} ϕ_{ε} (x) d x = ε^{- 2} ε^{2} \int_{R^{2}} ϕ (y) d y = 1$ for every $ε > 0$ .

3.1F1F2F4F5step 2.1

The scaling $t = ε u$ in the line integral [F1] gives $R ϕ_{ε} (θ, s) = ε^{- 1} ρ (s / ε)$ . Radiality gives $ϕ (s θ + t θ^{⊥}) = J^{- 1} η (1 - s^{2} - t^{2})$ , so $ρ$ is independent of $θ$ and vanishes for $∣ s ∣ > 1$ . It is continuous: for $s^{'} \to s$ , $∣ ρ (s^{'}) - ρ (s) ∣ \leq 2 \sup_{∣ t ∣ \leq 1} ∣ ϕ (s^{'} θ + t θ^{⊥}) - ϕ (s θ + t θ^{⊥}) ∣ ⟶ 0,$ since $ϕ$ is uniformly continuous. Thus $ρ$ is compactly supported and Riemann integrable by [F2]. Riemann-Fubini [F4] and the orthogonal change of variables $(s, t) \mapsto s θ + t θ^{⊥}$ , whose absolute Jacobian is $1$ , give $\int_{R} ρ (s) d s = \int_{R^{2}} ϕ (x) d x = 1$ .

4.1givenF2F5step 3.1algebra

By step 3.1, $ρ$ is continuous and supported on $[- 1, 1]$ , so $R ϕ_{ε} (θ, \cdot) h$ is compactly supported and continuous. If $h$ is complex-valued, apply [F2] and the compact-support change of variables [F5] separately to its real and imaginary parts; thus with $s = ε u$ we get $\int_{R} R ϕ_{ε} (θ, s) h (s) d s = \int_{- 1}^{1} ρ (u) h (ε u) d u$ . Uniform continuity of $h$ on $[- 1, 1]$ implies $\sup_{∣ u ∣ \leq 1} ∣ h (ε u) - h (0) ∣ \to 0$ . Since $ρ \geq 0$ and $\int ρ = 1$ by step 3.1, the integral tends to $h (0)$ .

5.1step 4.1∎

The functional $h \mapsto h (0)$ is the point mass $δ_{0}$ by definition of the Dirac distribution. Step 4.1 proves the asserted distributional limit for every fixed normal $θ$ .

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