arXiv subjects

B. Rubin

Publications and source records attributed to B. Rubin.

8 recordsLinked to original sources

arXiv2018

Reconstruction of functions on the sphere from their integrals over hyperplane sections

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 transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of cosine transforms, Semyanisyi's integrals, and modified stereorgraphic projection with the pole at $A$. Assumptions for functions are close to minimal.

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arXiv2017

Radon Transforms over Lower-Dimensional Horospheres in Real Hyperbolic Space

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 $d=n-1$ agrees with the well-known Gelfand-Graev transform.

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arXiv2012

Analytic and Group-Theoretic Aspects of the Cosine Transform

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 and the corresponding Funk transform are extended to the "higher-rank" case for functions on Stiefel and Grassmann manifolds. The main topics are the analytic continuation and the structure of polar sets, the connection with the Fourier transform on the space of rectangular matrices, inversion formulas and spectral analysis, and the group-theoretic realization as an intertwining operator between generalized principal series representations of SL(n, R).

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arXiv2006

Composite Cosine Transforms

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 distinctive features of higher-rank objects in full generality. We call these new transforms the composite cosine transforms, by taking into account that their kernels agree with the composite power function of the cone of positive definite symmetric matrices. We show that injectivity of the composite cosine transforms can be studied using standard tools of the Fourier analysis on matrix spaces. In the framework of this approach, we introduce associated generalized zeta integrals and give new simple proofs to the relevant functional relations. Our technique is based on application of the higher-rank Radon transform on matrix spaces.

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arXiv2005

The Composite Cosine Transform on the Stiefel Manifold and Generalized Zeta Integrals

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 power function of the cone of positive definite symmetric matrices. Our aim is to describe the set of all $\lam \in C^m$ for which $T^\lam$ is injective on the space of integrable functions. We obtain the precise description of this set in some important cases, in particular, for $\lam$-cosine transforms on Grassmann manifolds. The main tools are the classical Fourier analysis of functions of matrix argument and the relevant zeta integrals.

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arXiv2004

Multiscaled wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices

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 rank-one case $m=1$. We prove an analog of Calderon's reproducing formula for $L^2$-functions and obtain explicit inversion formulas for the Riesz potentials and Radon transforms on $M$. We also introduce continuous ridgelet transforms associated to matrix planes in $M$. An inversion formula for these transforms follows from that for the Radon transform.

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arXiv2004

The Radon transform of functions of matrix argument

The monograph contains a systematic treatment of a circle of problems in analysis and integral geometry related to inversion of the Radon transform on the space of real rectangular matrices. This transform assigns to a function $f$ on the matrix space the integrals of $f$ over the so-called matrix planes, the linear manifolds determined by the corresponding matrix equations. Different inversion methods are discussed. They rely on close connection between the Radon transform, the Fourier transform, the Garding-Gindikin fractional integrals, and matrix modifications of the Riesz potentials. A special emphasis is made on new higher rank phenomena, in particular, on possibly minimal conditions under which the Radon transform is well defined and can be explicitly inverted. Apart of the space of Schwartz functions, we also employ $L^p$-spaces and the space of continuous functions. Many classical results for the Radon transform on $R^n$ are generalized to the higher rank case.

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arXiv2004

An analogue of the Fuglede formula in integral geometry on matrix spaces

The well known formula of B. Fuglede expresses the mean value of the Radon k-plane transform on $R^n$ as a Riesz potential. We extend this formula to the space of $n \times m$ real matrices and show that the corresponding matrix k-plane transform $f \to \hat f$ is injective if and only if $n-k \ge m$. Different inversion formulas for this transform are obtained. We assume that $f \in L^p$ or $f$ is a continuous function satisfying certain "minimal" conditions at infinity.

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