[Paper Review] Support theorems for the Radon transform and Cramér-Wold theorems

Jan Boman, Filip Lindskog|ArXiv.org|Feb 29, 2008

Bayesian Methods and Mixture Models22 references4 citations

TL;DR

This paper extends the Cramér-Wold theorem to measures with infinite mass near the origin by leveraging injectivity theorems for the Radon transform in distribution theory. It establishes new uniqueness and convergence results for measures and sequences of measures supported outside compact convex sets or in convex cones, proving sharpness of assumptions via counterexamples.

ABSTRACT

This article presents extensions of the Cram{é}r-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distribution theory and Fourier analysis we show that the presented injectivity results for the Radon transform lead to Cram{é}r-Wold type results for measures. One purpose of this article is to contribute to making known to probabilists interesting results for the Radon transform that have been developed essentially during the 1980ies and 1990ies.

Motivation & Objective

• Address the lack of uniqueness in Cramér-Wold theorems when measures have infinite mass near the origin, particularly in extreme value theory and limit theorems.
• Extend classical Cramér-Wold results to signed and non-negative measures that may not be probability measures.
• Provide conditions under which a measure is uniquely determined by its values on halfspaces not containing the origin, especially in the exterior of convex sets or convex cones.
• Establish convergence criteria for sequences of such measures based on halfspace distribution functions.
• Demonstrate the sharpness of assumptions through counterexamples involving homogeneous distributions and radial functions.

Proposed method

• Use distribution theory and Fourier analysis to extend the Radon transform to measures and distributions, enabling analysis of non-integrable or infinite-mass measures.
• Apply the exterior Radon transform to reconstruct measures outside a compact convex set K, using injectivity theorems (Theorems B–D) for homogeneous or cone-supported measures.
• Construct homogeneous extensions of functions and measures defined on R^d \ {0} to R^d using distributional derivatives and spherical polar coordinates.
• Define measures as continuous linear functionals on test functions, allowing treatment of signed and infinite-mass measures via distribution theory.
• Derive the key identity (6.2) for extending homogeneous distributions via iterated distributional derivatives, ensuring homogeneity and consistency with the original function on R^d \ {0}.
• Use the Radon transform's injectivity properties to prove Cramér-Wold-type theorems for measures and sequences, particularly in cases where the measure is supported in a convex cone containing no full line.

Experimental results

Research questions

• RQ1 Can the Cramér-Wold theorem be extended to measures with infinite mass near the origin, particularly when the origin is in the support?
• RQ2 Under what conditions is a measure uniquely determined by its values on all closed halfspaces disjoint from the origin?
• RQ3 What are the necessary and sufficient conditions for weak convergence of sequences of such measures based on halfspace distribution functions?
• RQ4 How do assumptions on homogeneity, decay, or cone support affect the injectivity of the exterior Radon transform?
• RQ5 Are the assumptions in the extended Cramér-Wold theorems sharp, and can counterexamples be constructed to show their necessity?

Key findings

• Theorem 4 and Theorem 4′ establish Cramér-Wold-type uniqueness and convergence for measures supported in a closed convex cone containing no complete straight line, which is a novel contribution.
• Theorem 3b and Theorem 3′ provide uniqueness for non-negative measures under cone support assumptions, even without homogeneity, extending previous results.
• Counterexamples in Section 5 show that the assumptions in Theorems B–D are sharp, including constructions using homogeneous distributions of non-integral degree.
• For any non-integral γ < -d, a homogeneous distribution f on R^d \ {0} of degree γ admits a unique homogeneous extension to R^d, which is constructed via iterated distributional derivatives.
• Measures μ in M_loc(R^d \ {0}) with ||μ_ε||_M ≤ Cε^{-m} can be extended to distributions on R^d, enabling analysis of singular or slowly decaying measures.
• The conjecture in Basrak, Davis, and Mikosch (2002) regarding scaling limits of probability measures is affirmed via Corollary 2, which follows from Theorem 4′.