Boris Rubin

Louisiana State University | LSU · Department of Mathematics

Necessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with k-dimensional totally geodesic submanifolds of the unit sphere Sn in ℝn+1. This result generalizes the well known statement for the spherical means on Sn and is formulated in terms of zeros of Jacobi polynomials. The relevant harmoni...

We obtain sharp L p {L^{p}} - L q {L^{q}} estimates for fractional integrals generated by Radon transforms of the following three types: The classical Radon transform over the set of all hyperplanes in ℝ n {\mathbb{R}^{n}} , the Strichartz transversal transform over only those hyperplanes which meet the last coordinate axis, and the Radon transform...

We study injectivity of integral operators which map the Cauchy initial data for the Euler–Poisson–Darboux equation to the fixed time measurement of the solution of this equation. These operators generalize the well-known spherical means and are closely related to the shifted k-plane transforms, which assign to functions in Lp(Rn)\documentclass[12p...

We consider an integral transform which maps functions on the Euclidean half-space to integrals of these functions over hemispheres centered on the boundary hyperplane. The main results include sharp [Formula: see text]-[Formula: see text] estimates for this transform and new explicit inversion formulas under minimal assumptions for functions. The...

We apply the Fourier transform technique and a modified version of E. Stein’s interpolation theorem communicated by L. Grafakos, to obtain sharp L p {L^{p}} - L q {L^{q}} estimates for the Radon transform and more general convolution-type fractional integrals with the kernels having singularity on the paraboloids.

The inverse problem for the Euler-Poisson-Darboux equation deals with reconstruction of the Cauchy data for this equation from incomplete information about its solution. In the present article, this problem is studied in connection with the injectivity of the shifted $k$-plane transform, which assigns to functions in $L^p(\mathbb {R}^n)$ their mean...

The inverse problem for the Euler-Poisson-Darboux equation deals with reconstruction of the Cauchy data for this equation from incomplete information about its solution. In the present article, this problem is studied in connection with the injectivity of the shifted $k$-plane transform, which assigns to functions in $L^p(\mathbb {R}^n)$ their mean...

Necessary and sufficient conditions are obtained for injectivity of the shifted Funk-Radon transform associated with $k$-dimensional totally geodesic submanifolds of the unit sphere $S^n$ in $\mathbb{R}^{n+1}$. This result generalizes the well known statement for the spherical means on $S^n$ and is formulated in terms of zeros of Jacobi polynomials...

We study higher-rank Radon transforms of the form f(τ)→∫τ⊂ζf(τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\tau ) \rightarrow \int _{\tau \subset \zeta } f(\tau )...

We apply the Fourier transform technique and a modified version of E. Stein's interpolation theorem communicated by L. Grafakos, to obtain sharp $L^p$-$L^q$ estimates for the Radon transform and more general convolution-type fractional integrals with the kernels having singularity on the paraboloids.

We obtain sharp norm estimates for fractional integrals generated by Radon transforms of three types in the n-dimensional real Euclidean space. The method relies on recent interpolation results for analytic families of operators.

The sonar transform in geometric tomography maps functions on the Euclidean half-space to integrals of those functions over hemispheres centered on the boundary hyperplane. We obtain sharp $L^p$-$L^q$ estimates for this transform and new explicit inversion formulas under minimal assumptions for functions. The main results follow from intriguing con...

We introduce a new family of invariant differential operators associated with λ-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the λ-cosine transforms and yield new inversion formulas. Intermediate Funk-cosine transforms corresponding to integration over matrices of lower rank are studied. T...

We study higher-rank Radon transforms that take functions on $j$-dimensional totally geodesic submanifolds in the $n$-dimensional real constant curvature space to functions on similar submanifolds of dimension $k >j$. The corresponding dual transforms are also considered. The transforms are explored the Euclidean case (affine Grassmannian bundles),...

We study the spherical slice transform [Formula: see text] which assigns to a function [Formula: see text] on the unit sphere [Formula: see text] in [Formula: see text] the integrals of [Formula: see text] over cross-sections of [Formula: see text] by [Formula: see text]-dimensional affine planes passing through the north pole [Formula: see text]....

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j , k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j -dimensional affine planes in ℝ n to functions on a similar manifold of k -dimensional planes by integration ove...

We study the spherical slice transform which assigns to a function on the $n$-dimensional unit sphere the integrals of that function over cross-sections of the sphere by $k$-dimensional affine planes passing through the north pole. These transforms are well known when $k=n$. We consider all $1< k < n+1$ and obtain an explicit formula connecting the...

We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphe...

We study non-geodesic Funk-type transforms on the unit sphere 𝕊n in ℝn+1 associated with cross-sections of 𝕊n by k-dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these transforms, inversion formulas, and connection with geodesic Funk transforms. We also show that, u...

We apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j -dimensional affine planes in ℝ n to functions on a similar manifold of k -dimensional planes by integration over the set of all j -planes that meet a given k -plane at a right angle. We obtain explicit inversion formulas f...

We introduce a new family of invariant differential operators associated with $\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the $\lambda$-cosine transforms and yield new inversion formulas. Intermediate Funk-cosine transforms corresponding to integration over matrices of lower ran...

We use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained for totally geodesic Funk transforms on the sphere and the correpsonding lambda-cosine transforms.

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point A which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk...

The vertical slice transform in spherical integral geometry takes a function on the unit sphere S ⁿ to integrals of that function over spherical slices parallel to the last coordinate axis. This transform was investigated for n = 2 in connection with inverse problems of spherical tomography. The present article gives a survey of some methods which...

We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphe...

We study non-geodesic Funk-type transforms on the unit sphere in the real Euclidean space associated with cross-sections of the sphere by lower dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these transforms, inversion formulas, and connection with geodesic Funk tra...

We study a Radon-like transform that takes functions on the Grassmannian of $j$-dimensional affine planes in $\Bbb R ^n$ to functions on a similar manifold of $k$-dimensional planes by integration over the set of all $j$-planes that meet a given $k$-plane at a right angle. The case $j=0$ gives the classical Radon-John $k$-plane transform. For any $...

The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k -fold Lebesgue measure on ℝ ⁿ in terms of the corresponding measures on k -dimensional linear subspaces of ℝ ⁿ . We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayl...

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Fu...

The vertical slice transform takes a function on the n-dimensional unit sphere to integrals of that function over spherical slices parallel to the last coordinate axis. This transform arises in thermoacoustic tomography. We obtain new inversion formulas for the vertical slice transform and its singular value decomposition. The results can be applie...

The Blaschke-Petkantschin formula is a variant of the polar decomposition of the $k$-fold Lebesgue measure on $\mathbb {R}^n$ in terms of the corresponding measures on $k$-dimensional linear subspaces of $\mathbb {R}^n$. We suggest a new elementary proof of this formula and discuss its connection with the celebrated Drury's identity that plays a ke...

The d-plane Radon-John transform takes functions on Rⁿ to functions on the set of all d-dimensional planes in Rⁿ by integration over these planes. We study the action of this transform on degenerate functions of the form f (x) = f0 (r) Yk (θ), where r = |x| > 0, θ = x/|x|, and Yk is a spherical harmonic of degree k. It is shown that the results for...

The article is devoted to remarkable interrelation between the norm estimates for $k$-plane transforms in weighted and unweighted $L^p$ spaces and geometric integral inequalities for cross-sections of measurable sets in $\mathbb{R}^n$. We also consider more general $j$-plane to $k$-plane transforms on affine Grassmannians and their compact modifica...

We transfer the results of Part I related to the modified support theorem and the kernel description of the hyperplane Radon transform to 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...

We study horospherical Radon transforms that integrate functions on the $n$-dimensional real hyperbolic space over horospheres of arbitrary fixed dimension $1\le d\le n-1$. Exact existence conditions and new explicit inversion formulas are obtained for these transforms acting on smooth functions and functions belonging to $L^p$. The case $d=n-1$ ag...

We suggest new versions of Helgason's support theorems and related characterizations of the kernel (the null space) for the classical 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 ass...

We obtain new inversion formulas for the Radon transform and the corresponding dual transform acting on affine Grassmann manifolds of planes in $R^n$. The consideration is performed in full generality on continuous functions and functions belonging to $L^p$ spaces.

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.

We derive new inversion formulas for the Radon transform between lines and hyperplanes in Rn and the corresponding dual transform. In this setting, the Radon transform is noninjective and the consideration is restricted to the so-called quasiradial functions that are constant on symmetric clusters of lines. For the dual transform, which is injectiv...

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 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. Applicab...

We review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in ℝn . These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel mea...

Disruption of four different influenza viruses (strains A/PR8, A1/Ann Arbor/1/57, A2/Japan/170/62 and B/Maryland 1/59) by Tween-ether or sodium deoxycholate leads to polydisperse populations of strain-specific immunoprecipitating antigens which occur both associated with or separately from hemagglutinins and which seem to be heterogeneous with resp...

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...

This is a brief survey of recent results by the authors devoted to one of the most important operators of integral geometry. Basic facts about the analytic family of cosine transforms on the unit sphere in R n and the corre-sponding Funk transform are extended to the "higher-rank" case for functions on Stiefel and Grassmann manifolds. The main topi...

Many Radon-like transforms are members of suitable operator families indexed by a complex parameter. V. I. Semyanistyjs fractional integrals [Sov. Math., Dokl. 1, 1114–1117 (1961); translation from Dokl. Akad. Nauk SSSR 134, 536–539 (1960; Zbl 0096.30905)] associated to the classical Radon transform on ℝ n are a typical example of such a family. We...

In 1927, Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on R 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...

The following problem arises in thermoacoustic tomography and has intimate connection with PDEs and integral geometry. Reconstruct a function f supported in an n-dimensional ball B given the spherical means of f over all geodesic spheres centered on the boundary of B. We propose a new approach to this problem, which yields explicit reconstruction f...

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 weig...

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 Erd\'elyi-Kober type fractional integrals are o...

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.

We extend the Funk–Radon–Helgason inversion method of mean value operators to the Radon transform of continuous and Lp functions which are integrated over matrix planes in the space of real rectangular matrices. Necessary and sufficient conditions of existence of for such f and explicit inversion formulas are obtained. New higher-rank phenomena rel...

New proofs are given to some approximate and explicit inversion formulas for the Riesz potentials. The results are applied to reconstruction of functions from their integrals over k-dimensional planes in ℝn .

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 geo...

We present a brief discussion of the interrelations between integral geometry and harmonic analysis and then proceed to the d-dimensional totally geodesic Radon transform f, assuming f ¡Ê L p (Hn), where Hn is the n-dimensional real hyperbolic space, and 1 ¡Ü d ¡Ü n - 1. We show that f is well defined if and only if 1 ¡Ü p < (n -1)/(d - 1) and pro...

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 formu...

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...

New simple proofs are given to some elementary approximate and explicit inversion formulas for Riesz potentials. The results are applied to reconstruction of functions from their integrals over Euclidean planes in integral geometry.

The classical Busemann–Petty problem (1956) asks, whether origin-symmetric convex bodies in Rn with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n⩽4 and negative if n>4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison function...

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...

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 explic...

We generalize Y. Nievergelt's inversion method for the Radon transform on lines in the 2-plane to the $k$-plane Radon transform of continuous and $L^p$ functions on $R^n$ for all $1\leq k<n$. Comment: 9 pages

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto l...

The paper contains a simple proof of the Finch–Patch–Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. Inversion of that transform is important for thermoacoustic tomography and represents a challenging mathematical problem. The argument relies on the idea of analytic continuation and known properties of the Erdélyi...

We introduce a new analytic family of intertwining operators which include the Radon transform over matrix planes and its inverse. These operators generalize integral transformations introduced by Semyanistyi (Dokl. Akad. Nauk SSSR 134:536–539, [1960]) in his research related to the hyperplane Radon transform in ℝn . We obtain an extended version o...

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 o...

The paper contains a simple proof of the Finch-Patch-Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. This transform arises in thermoacoustic tomography. Applications are given to the Cauchy problem for the Euler-Poisson-Darboux equation with initial data on the cylindrical surface. The argument relies on the idea...

We introduce a new concept of the so-called {\it composite wavelet transforms}. These transforms are generated by two components, namely, a kernel function and a wavelet function (or a measure). The composite wavelet transforms and the relevant Calder\'{o}n-type reproducing formulas constitute a unified approach to explicit inversion of the Riesz,...

For the operator M(t)-alpha, t > 0, alpha + n/2 not-equal-to 0, -1, -2, ..., defined on Fourier transforms of Schwartz functions omega is-a-member-of S(R(n)) by the relation [GRAPHICS] [GRAPHICS] the question of extension to a bounded linear operator M(t)-alpha: L(p)r --> L(q)s is considered, where L(p)r and L(q)s are Lebesgue spaces of Bessel pote...

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. The paper contains a complete solution to the problem when the body with smaller sections is invariant under rotations, preserving mutually orthogonal coordinate sub...

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied...

Riesz potentials on the space of rectangular n×m matrices arise in diverse “higher rank” problems of harmonic analysis, representation theory, and integral geometry. In the rank-one case m=1 they coincide with the classical operators of Marcel Riesz. We develop new tools and obtain a number of new results for Riesz potentials of functions of matrix...

Let Mn,m be the space of real n×m matrices which can be identified with the Euclidean space Rnm. We introduce continuous wavelet transforms on Mn,m with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on Mn,m and coincide with classical ones in the rank-one cas...

The cosine transforms of functions on the unit sphere play an important role in convex geometry, the Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting mathematical object useful for applications. We introduce more general integral transforms that reveal distin...

The spherical Radon transform on the unit sphere can be regarded as a member of the analytic family of suitably normalized generalized cosine transforms. We derive new formulas for these transforms and apply them to study classes of intersections bodies in convex geometry.

The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies with lower-dimensional smaller sections necessarily have smaller volume. We study the weighted version of this problem corresponding to the physical situation when bodies are endowed with mass distribution and the relevant sections are measured with attenuation.

We introduce a family of wavelet-like transforms associated to certain admissible semigroups of operators acting on Lp-spaces, and prove the corresponding reproducing formula of Caldern’s type. The new transforms constitute a unified approach to inversion of a wide class of integral operators in analysis and applications. We illustrate the general...

We introduce a new integral transform $T^\lam f$, $\lam \in C^m$, on the Stiefel manifold of orthonormal $m$-frames in $R^n$ which generalizes the $\lam$-cosine transform on the Grassmann manifold of $m$-dimensional linear subspaces of $R^n$. We call it the composite cosine transform, by taking into account that its kernel agrees with the composite...

Thek-plane Radon transform assigns to a functionsf(x) on ℝn the collection of integralsf(τ)=∫τ f over allk-dimensional planesτ. We give a systematic treatment of two inversion methods for this transform, namely, the method of Riesz potentials, and the method of spherical means. We develop new analytic tools which allow to invertf(τ) under minimal a...

We develop an analytic approach to the Radon transform ^ f ( )= R f(), where f() is a function on the ane Grassmann manifold G(n;k) of k-dimensional planes in Rn ,a nd is a k0-dimensional plane in the similar manifold G(n;k0) ;k 0 >k .F orf 2 Lp(G(n;k)), we prove that this transform is nite almost everywhere on G(n;k0) if and only if 1 p< (n k)=(k0...

Let $M$ be the space of real $n\times m$ matrices which can be identified with the Euclidean space $R^{nm}$. We introduce continuous wavelet transforms on $M$ with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on $M$ and coincide with classical ones in the ra...

We present generalizations of the Busemann–Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of...