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Boris Rubin

Publications and source records attributed to Boris Rubin.

At least 37 records · Page 2Linked to original sources
arXiv2016

On Radon transforms between lines and hyperplanes

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in $\rn$. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John $d$-plane transform in $\rn$, the Grassmannian modification of the Kelvin transform, and the Erdélyi-Kober fractional integrals.

math.FA↗
arXiv2016

On the Determination of Star Bodies from Their Half-Sections

We obtain explicit inversion formulas for the Radon-like transform that assigns to a function on the unit sphere the integrals of that function over hemispheres lying in lower dimensional central cross-sections. The results are applied to determination of star bodies from the volumes of their central half-sections.

math.FA↗
arXiv2015

Null Spaces of Radon Transforms

We obtain new descriptions of the null spaces of several projectively equivalent transforms in integral geometry. The paper deals with the hyperplane Radon transform, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the Cormack-Quinto spherical mean transform for spheres through the origin. The consideration extends to the corresponding dual transforms and the relevant exterior/interior modifications. The method relies on new results for the Gegenbauer-Chebyshev integrals, which generalize Abel type fractional integrals on the positive half-line.

math.FA↗
arXiv2015

Gegenbauer-Chebyshev Integrals and Radon Transforms

We suggest new modifications of Helgason's support theorems and descriptions of the kernels for several projectively equivalent transforms of integral geometry. The paper deals with the hyperplane Radon transform and its dual, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the spherical mean transform for spheres through the origin. The assumptions for functions are formulated in integral terms. The proofs rely on the properties of the Gegenbauer-Chebyshev integrals which generalize Abel type fractional integrals on the positive half-line.

math.FA↗
arXiv2014

New Inversion Formulas for the Horospherical Transform

The following two inversion methods for Radon-like transforms are widely used in integral geometry and related harmonic analysis. The first method invokes mean value operators in accordance with the classical Funk-Radon-Helgason scheme. The second one employs integrals of the potential type and polynomials of the Beltrami-Laplace operator. Applicability of these methods to the horospherical transform in the hyperbolic space was an open problem. In the present paper we solve this problem for $L^p$ functions in the maximal range of the parameter $p$ and for compactly supported smooth functions, respectively. The main tools are harmonic analysis in the hyperbolic space and associated fractional integrals.

math.FA↗
arXiv2013

Overdetermined Transforms in Integral Geometry

A simple example of an $n$-dimensional admissible complex of planes is given for the overdetermined $k$-plane transform in $\mathbb{R}^n$. For the corresponding restricted $k$-plane transform sharp existence conditions are obtained and explicit inversion formulas are discussed in the general context of $L^p$ functions. Similar questions are studied for overdetermined Radon type transforms on the sphere and the hyperbolic space. A theorem describing the range of the restricted $k$-plane transform on the space of rapidly decreasing smooth functions is proved.

math.FA↗
arXiv2012

Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms

Semyanistyi's fractional integrals have come to analysis from integral geometry. They take functions on $R^n$ to functions on hyperplanes, commute with rotations, and have a nice behavior with respect to dilations. We obtain sharp inequalities for these integrals and the corresponding Radon transforms acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Similar results are obtained for fractional integrals associated to $k$-plane transforms for any $1\le k<n$.

math.FA↗
arXiv2012

On the Funk-Radon-Helgason Inversion Method in Integral Geometry

The paper deals with totally geodesic Radon transforms on constant curvature spaces. We study applicability of the historically the first Funk-Radon-Helgason method of mean value operators to reconstruction of continuous and $L^p$ functions from their Radon transforms. New inversion formulas involving Erdélyi-Kober type fractional integrals are obtained. Particular emphasis is placed on the choice of the differentiation operator in the spirit of the recent Helgason's formula.

math.FA↗
arXiv2012

Weighted norm inequalities for k-plane transforms

We obtain sharp inequalities for the k-plane transform, the "j-plane to k-plane" transform, and the corresponding dual transforms, acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Some generalizations and open problems are discussed.

math.FA↗
arXiv2011

Inversion Formulas for the Spherical Means in Constant Curvature Spaces

The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct a function $f$ supported in an $n$-dimensional ball $B$, if the spherical means of $f$ are known over all geodesic spheres centered on the boundary of $B$.} We propose a new unified approach based on the idea of analytic continuation. This approach gives explicit inversion formulas not only for the Euclidean space $\bbr^n$ (as in the original set-up) but also for arbitrary constant curvature space $X$, including the $n$-dimensional sphere and the hyperbolic space. The results are applied to inverse problems for a large class of Euler-Poisson-Darboux equations in constant curvature spaces of arbitrary dimension.

math.CA↗
arXiv2011

The Generalized Mader's Inversion Formulas for the Radon Transforms

In 1927 Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $\bbr^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader's formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the $k$-plane transform was presented in the recent book "Integral geometry and Radon transform" by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.

math.CV↗
arXiv2011

Funk, Cosine, and Sine Transforms on Stiefel and Grassmann manifolds, II

We investigate analytic continuation of the matrix cosine and sine transforms introduced in Part I and depending on a complex parameter $\a$. It is shown that the cosine transform corresponding to $\a=0$ is a constant multiple of the Funk-Radon transform in integral geometry for a pair of Stiefel (or Grassmann) manifolds. The same case for the sine transform gives the identity operator. These results and the relevant composition formula for the cosine transforms were established in Part I in the sense of distributions. Now we have them pointwise. Some new problems are formulated.

math.FA↗
arXiv2010

Funk, Cosine, and Sine Transforms on Stiefel and Grassmann manifolds

The Funk, cosine, and sine transforms on the unit sphere are indispensable tools in integral geometry. They are also known to be interesting objects in harmonic analysis. The aim of the paper is to extend basic facts about these transforms to the more general context for Stiefel or Grassmann manifolds. The main topics are composition formulas, the Fourier functional relations for the corresponding homogeneous distributions, analytic continuation, and explicit inversion formulas.

math.FA↗
arXiv2009

The Radon Transform on the Heisenberg Group and the Transversal Radon Transform

The notion of the Radon transform on the Heisenberg group was introduced by R. Strichartz and inspired by D. Geller and E.M. Stein's related work. The more general transversal Radon transform integrates functions on the m-dimensional real Euclidean space over hyperplanes meeting the last coordinate axis. We obtain new boundedness results and explicit inversion formulas for both transforms on $L^p$ functions in the full range of the parameter $p$. We also show that these transforms are isomorphisms of the corresponding Semyanistyi-Lizorkin spaces of smooth functions. In the framework of these spaces we obtain inversion formulas, which are pointwise analogues of the corresponding formulas by R. Strichartz.

math.FA↗
arXiv2009

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$ and negative if $n>4$. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic $n$-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic $n$-dimensional space has an affirmative answer if and only if $n =2$. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations are discussed.

math.FA↗
arXiv2007

Invariant Functions on Grassmannians

It is known, that every function on the unit sphere in $\bbr^n$, which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces dimension of its actual argument, hold for every compact symmetric space and can be obtained in the framework of Lie-theoretic consideration. In the present article, this phenomenon is given precise meaning for functions on the Grassmann manifold $G_{n,i}$ of $i$-dimensional subspaces of $\bbr^n$, which are invariant under orthogonal transformations preserving complementary coordinate subspaces of arbitrary fixed dimension. The corresponding integral formulas are obtained. Our method relies on bi-Stiefel decomposition and does not invoke Lie theory.

math.FA↗