Jan Boman

Novikov’s inversion formula for the attenuated Radon transform—A new approach

We study the inversion of weighted Radon transforms in two dimensions, Rρƒ(L)=ƒL =ƒ(·), where the weight function ρ(L, x), L a line and x ∈ L, has a special form. It was an important breakthrough when R.G. Novikov recently gave an explicit formula for the inverse of Rρ when ρ has the form(1.2); in this case Rρ is called the attenuated Radon transform. Here we prove similar results for a somewhat larger class of ρ using completely different and quite elementary methods.

Finite rank singular perturbations and distributions with discontinuous testfunctions

Point interactions for the n-th derivative operator in one dimension are investigated. Every such perturbed operator coincides with a selfadjoint extension of the n-th derivative operator restricte ...

Support theorems for Radon transforms on real analytic line complexes in three-space

In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in R 3 . Let f be a distribution of compact support on R 3 . Assume Y is a real analytic admissible line complex and Y 0 is an open connected subset of Y with one line in Y 0 disjoint from supp f. Under weak geometric assumptions, if the Radon transform of f is zero for all lines in Y 0 , then supp f intersects no line in Y 0 . These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support

Plane intersections of rotational ellipsoids

This result could have been proved by the Greeks more than two thousand years ago, so we certainly did not expect it to be a new theorem. However, since we found it (1996) we have asked a large number of experts, we have searched the literature, and we have posted queries on the internet, all without finding anyone who has heard of the theorem. We thus feel comfortable in concluding that the theorem is, if not unknown, at least not well known. If a reader of this note knows a reference to the theorem or to related facts, we would appreciate being informed. Before we give a mathematical proof of the theorem, we will explain in principle how the theorem was discovered experimentally. An extremely short light pulse is emitted in a wide range of directions from a point A. The duration of the light pulse is about 1 picosecond = 10−12 seconds, which produces a light pulse of length about 0.3 mm. As the light hits an arbitrary fixed plane π , it is scattered in all directions. At a point B, different from A, a “photograph” is taken with extremely short exposure time. The outcome of the experiment is that a bright circle is seen on the photographic plate. In other words, a bright curve—which is actually an ellipse—is seen on the plane π , and this curve looks circular as viewed from B. (The technique used to carry out such experiments is described in [2].) How can we interpret this experiment as supporting our theorem? Well, let us assume that photons are emitted precisely at time t = 0 from the point A and that the “photograph” is taken at time t = t1 > 0. Then photons originating at A can contribute to the picture only if their travel time is exactly t1, and hence the distance that they have travelled is exactly s = ct1, where c is the speed of light. This means that for any photon that has made one bounce at some point P on the plane π on its way from A to B, the sum of the distances AP and P B must be equal to s. What is the locus of the set of all such points P? If for a moment we forget about the plane π , the locus of all the points P in space such that the sum of the distances AP and P B is equal to s is a rotational ellipsoid E with axis of symmetry through A and B and focal points at A and B. Accordingly the set of bounce points P on the plane π is the curve in which π intersects E . This is the bright curve that is viewed by our camera located at the point B. It is an elementary fact that this curve is an ellipse, and as already explained, the assertion of the theorem is that this curve looks circular when viewed from B.

On Comparison Theorems for Generalized Moduli of Continuity

In this note we will present sharpened versions of two theorems of Harold Shapiro on comparison between generalized moduli of continuity. By specialization of the measures defining the moduli of continuity in question our theorem gives a sharp form of the Jackson and Bernstein theorems. In particular our theorem implies the known fact that the order of best approximation by trigonometric polynomials for any continuous and periodic function f satisfies En (f) = O((log n)−1) if and only if the modulus of continuity of f is O(|log t|−1).

On stable region-of-interest reconstruction in tomography: examples of non-existence of bounded inverse

In applications of computerized tomography (CT) it is often of interest to reconstruct a function in a proper subset, the region of interest (ROI), of its support from a proper subset of a full CT-scan. In several recent works it has been shown that stable ROI reconstruction is possible in certain limited data situations. In this paper we investigate the limits of what is possible in that direction by proving that reconstruction in an interior ROI is severely unstable for a certain set of data that contains integrals over all lines that intersect the ROI and is large enough for the unknown function to be uniquely determined.

On Local Injectivity for Generalized Radon Transforms

We consider a class of weighted plane generalized Radon transforms Rf(γ)=∫f(x,u(ξ,η,x))m(ξ,η,x) dx, where the curve γ=γ (ξ,η) is defined by y=u(ξ,η,x), and m(ξ,η,x) is a given positive weight function. We prove local injectivity for this transform across a given curve γ 0 near a given point (x 0,y 0) on γ 0 for classes of curves and weight functions that are invariant under arbitrary smooth coordinate transformations in the plane.

Mikael passare: Curriculum vitae

1959-01-01. Kjell Alrik Mikael Pettersson is born in Vasteras, Sweden. Mother: Britt Gunvor Emilia Pettersson, later with the family name Elfstrom. Father: Werner Siems. Stepfathers: Kjell Pettersson and Hans Elfstrom.

Siciak's theorem on separate analyticity

We give a simple proof of an important special case of the famous theorem of Josef Siciak on separate analyticity.

Appearance of conical sections

Experiments with ultrashort laser pulses have indicated that the intersection of a rotational symmetric ellipsoidal by a flat surface at any angle appears circular when studied form one focal point of the ellipsoidal. This statement is mathematically proved for the general case. The spherical coordinate system of an observer that travels at ultrahigh velocity appears transformed into an ellipsoidal coordinate system but this fact is hidden from sight by said statement. If this was not so, his absolute velocity would be visible to the traveling observer in contradiction to Einsteins postulate.