Given a function f, the author specifies the singularities of f that are visible in a stable way from limited X-ray tomographic data. This determines which singularities of f can be stably recovered from limited data and which cannot, no matter how good the inversion algorithm. Microlocal analysis is used to determine the relationship between the singularities of a function f and those of its X-ray transform. The results are applied to determine the singularities that are visible for limited angle tomography and the interior and exterior problems. The author also suggests a practical method to use this relationship to reconstruct singularities of f from limited data Rf. The X-ray transform with sources on a curve in R 3 is also analyzed. Key words. X-ray transform, limited data tomography, microlocal analysis AMS subject classifications, primary 44A12, 92C55; secondary 35S30, 58G15 our theorem gives stability estimates of order 1 / 2 in Sobolev norms, so one would ex-*

A necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane. This problem is shown to be equivalent to inversion for the Radon transform on circles centered on restricted subsets of the plane. The proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular Radon transform. A characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems, and questions of Pinkus and Ehrenpreis are answered.1996 Academic Press, Inc. Formulation of the Problem and the Main ResultsIn this article, we characterize all systems of translations of radial functions that are complete in the space of continuous functions in R 2 . This is done by solving a dual problem, proving injectivity (on the domain of compactly supported functions) for a Radon transform integrating over restricted sets of circles. This injectivity problem is solved by first understanding the zero sets of harmonic polynomials and then using microlocal analysis.Our result answers a question in approximation theory [LP]. The same question in dual form is posed in the book of Ehrenpreis [E], so we also answer this question in the plane (see Sect. 9.2).

We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP) and Lambda tomography. We use microlocal analysis to explain why the well-known streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP and Lambda operators are standard pseudodifferential operators, and so they do not add artifacts. We provide reconstructions to illustrate our mathematical results.

Abstract. Tomographic methods are described that will reconstruct object boundaries in shallow water using sonar data. The basic ideas involve micro local analysis, and they are valid under weak assumptions even if the data do not correspond exactly to our model.

The transform considered in the paper averages a function supported in a ball in R n over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for such transforms are important in all these areas, for instance when dealing with incomplete data, error correction, and other issues. Four different types of complete range descriptions are provided, some of which also suggest inversion procedures. Necessity of three of these (appropriately formulated) conditions holds also in general domains, while the complete discussion of the case of general domains would require another publication.