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Boris Rubin

Publications and source records attributed to Boris Rubin.

At least 19 recordsLinked to original sources
arXiv2025

One-sided Fractional Integrals and Riesz Potentials on a Spherical Cap

We introduce fractional integrals on the $n$-dimensional spherical cap, study their boundednes in weighted $L^p$ spaces and obtain explicit inversion formulas. The results are applied to the inversion problem for Riesz potentials on a spherical cap. In the case $n=2$, this problem arises in electrostatics, elasticity, and some other applied areas. The main tools are stereographic projection, hypersingular integrals of the Marchaud type, Poisson integrals, and spherical harmonic decompositions.

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arXiv2024

Fractional Integrals and Tangency Problems in Integral Geometry

Many known Radon-type transforms of symmetric (radial or zonal) functions are represented by one-dimensional Riemann-Liouville fractional integrals or their modifications. The present article contains new examples of such transforms in the Euclidean, spherical, and hyperbolic settings, when integration is performed over lower-dimensional geodesic spheres or cross-sections, which are tangent to a given surface. Simple inversion formulas are obtained and admissible singularities at the tangency points are studied. Possible applications to the half-ball screening in mathematical tomography and some difficulties related to the general (not necessarily symmetric) case are discussed.

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arXiv2022

The Inverse Problem for the Euler-Poisson-Darboux Equation and Shifted $k$-Plane Transforms

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 values over all k-planes at a fixed distance from the given $k$-planes. Several generalizations, including the Radon transform over strips of fixed width in $\mathbb {R}^2$ and a similar transform over tubes of fixed diameter in $\mathbb {R}^3$, are considered.

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arXiv2022

On the Injectivity of the Shifted Funk-Radon Transform and Related Harmonic Analysis

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. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk-Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.

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arXiv2022

On Fractional Integrals Generated by Radon Transforms over Paraboloids

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.

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arXiv2022

Fractional Integrals Associated with Radon Transforms

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.

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arXiv2022

A Note on the Sonar Transform and Related Radon Transforms

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 connection between the sonar transform, the Radon transform over paraboloids, and the transversal Radon transform, which integrates functions over hyperplanes, meeting the last coordinate axis.

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arXiv2021

Higher-Rank Radon Transforms on Constant Curvature Spaces

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), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.

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arXiv2021

On the Spherical Slice Transform

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 spherical slice transform with the classical Radon-John transform over $(k-1)$-dimensional planes in the $n$-dimensional Euclidean space. Using this connection, known facts for the Radon-John transform, like inversion formulas, support theorems, representation on zonal functions, and others, can be reformulated for the spherical slice transform.

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arXiv2020

The $\lambda$-Cosine Transforms, Differential Operators, and Funk Transforms on Stiefel and Grassmann Manifolds

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 rank are studied. The main tools are polar decomposition and Fourier analysis on matrix space.

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arXiv2019

Non-geodesic Spherical Funk Transforms with One and Two Centers

We study non-geodesic Funk-type transforms associated with cross-sections of the n-sphere 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, unlike the case of planes through a single common center, the integrals over spherical sections by planes through two distinct centers provide the corresponding reconstruction problem a unique solution.

math.FA↗
arXiv2019

On Two Families of Funk-Type Transforms

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 sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon-John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center.

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arXiv2019

Radon Transforms for Mutually Orthogonal Affine Planes

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 $j$ and $k$, our transform has a mixed structure combining the $k$-plane transform and the dual $j$-plane transform. The main results include action of such transforms on rotation invariant functions, sharp existence conditions, intertwining properties, connection with Riesz potentials and inversion formulas in a large class of functions. The consideration is inspired by the previous works of F. Gonzalez and S. Helgason who studied the case $j+k=n-1$, $n$ odd, on smooth compactly supported functions.

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arXiv2018

On the Blaschke-Petkantschin Formula and Drury's Identity

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 key role in the study of mapping properties of the Radon-John $k$-plane transforms. We give a new derivation of this identity and provide it with precise information about constant factors and the class of admissible functions.

math.MG↗
arXiv2018

The Vertical Slice Transform in Spherical Tomography

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 applied to the inverse problem for the Euler-Poisson-Darboux equation associated to the corresponding spherical means.

math.FA↗
arXiv2017

Norm Estimates for $k$-Plane Transforms and Geometric Inequalities

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 modifications. The article contains a series of new integral-geometric inequalities with sharp constants, explicit equalities, conjectures, and open problems.

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arXiv2016

New Inversion Formulas for Radon Transforms on Affine Grassmannians

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.

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