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Mark Agranovsky

Publications and source records attributed to Mark Agranovsky.

At least 19 recordsLinked to original sources
arXiv2024

Convex bodies with algebraic section volume functions

The section volume function $A_K(\xi,t), \ \xi \in \mathbb R^n, \ t \in \mathbb R,$ of a body $K \subset \mathbb R^n$ evaluates the $(n-1)$-dimensional volume of the cross-section $K$ by the hyperplane $\{ x \cdot \xi=t \}.$ We are concerned with the question: can the shape of a body $K$ be detected from an algebraic type of its section function? We prove that among strictly convex bodies $K$ with $C^{\infty}$ boundaries, ellipsoids are completely described by the algebraic equation $qA_K^m+p=0,$ where $m \in \mathbb N$ and $q=q(\xi), \ p=p(\xi,t)$ are polynomials. The result is motivated by Arnold's problem on algebraically integrable domains (which, in turn, has its roots in Newton's Lemma about ovals) and generalizes known results on polynomially integrable domains.

math.MG↗
arXiv2022

An analog of polynomially integrable bodies in even-dimensional spaces

A bounded domain $K \subset \mathbb R^n$ is called polynomially integrable if the $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $\Pi$ polynomially depends on the distance from $\Pi$ to the origin. It was proved in [7] that there are no such domains with smooth boundary if $n$ is even, and if $n$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $n$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property.

math.FA↗
arXiv2022

On the exactness of the universal backprojection formula for the spherical means Radon transform

The spherical means Radon transform $\mathcal{M}f(x,r)$ is defined by the integral of a function $f$ in $\mathbb{R}^{n}$ over the sphere $S(x,r)$ of radius $r$ centered at a $x$, normalized by the area of the sphere. The problem of reconstructing $f$ from the data $\mathcal{M}f(x,r)$ where $x$ belongs to a hypersurface $\Gamma\subset\mathbb{R}^{n}$ and $r \in(0,\infty)$ has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When $\Gamma$ coincides with the boundary $\partial\Omega$ of a bounded (convex) domain $\Omega\subset\mathbb{R}^{n}$, a function supported within $\Omega$ can be uniquely recovered from its spherical means known on $\Gamma$. We are interested in explicit inversion formulas for such a reconstruction. If $\Gamma=\partial\Omega$, such formulas are only known for the case when $\Gamma$ is an ellipsoid (or one of its partial cases). This gives rise to the natural question: can explicit inversion formulas be found for other closed hypersurfaces $\Gamma$? In this article we prove, for the so-called "universal backprojection inversion formulas", that their extension to non-ellipsoidal domains $\Omega$ is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.

math.AP↗
arXiv2021

Domains with radical-polynomial X-ray transform

Let $K$ be a compact convex body in $\mathbb R^n.$ For any affine line $L,$ denote $\widehat{\chi}_K(L)=\int_{L}\chi_K(x)dl(x),$ where $dl$ is the arc length measure, the $X$-ray transform of the characteristic function $\chi_K,$ i.e., the length of the chord $K \cap L.$ We prove that if $K$ is bounded by a $C^{\infty}$ real algebraic hypersurface $\partial K$ and the $X$-ray transform $\widehat{\chi}_K(L)$ behaves, under small parallel translations of the line $L$ to the distance $t,$ as the $m$-th root of a polynomial of $t$, for some fixed $m \in \mathbb N,$ then $\partial K$ is an ellipsoid.

math.MG↗
arXiv2019

Injectivity of pairs of non-central Funk transforms

We study Funk-type transforms on the unit sphere in R^n associated with cross-sections of the sphere by lower-dimensional planes passing through an arbitrary fixed point inside the sphere or outside. Our main concern is injectivity of the corresponding paired transforms generated by two families of planes centered at distinct points. Necessary and sufficient conditions for the paired transforms to be injective are obtained, depending on geometrical configuration of the centers. Our method relies on the action of the automorphism group of the unit ball and the relevant billiard-like dynamics on the sphere.

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.

math.FA↗
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

Locally polynomially integrable surfaces and finite stationary phase expansions

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by hyperplanes, sufficiently close to the tangent hyperplanes to $M,$ depend polynomially on the distance of the hyperplanes to the origin. It is conjectured that only quadrics in odd dimensional spaces possess such a property. The main result of this article partially confirms the conjecture. The study of integrable domains and surfaces is motivated by a conjecture of V.I. Arnold about algebraically integrable domains. The result and the proof are related to study oscillating integrals for which the asymptotic stationary phase expansions consist of finite number of terms.

math.MG↗
arXiv2017

On algebraically integrable domains in Euclidean spaces

Let $D$ be a bounded domain $D$ in $\mathbb R^n $ with infinitely smooth boundary and $n$ is odd. We prove that if the volume cut off from the domain by a hyperplane is an algebraic function of the hyperplane, free of real singular points, then the domain is an ellipsoid. This partially answers a question of V.I. Arnold: whether odd-dimensional ellipsoids are the only algebraically integrable domains?

math.MG↗
arXiv2015

Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in $\mathbb R^3.$

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If $S$ is an immersed $C^1-$ manifold then $S$ is a Coxeter system of planes. Full description of common nodal sets of the Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean transform and confirms, for the case of ruled surfaces in $\mathbb R^3,$ a conjecture of E.T. Quinto and the author .

math.AP↗
arXiv2010

The support theorem for the single radius spherical mean transform

Let f(x) belong to L^p(R^n) and R>0. The transform is considered that integrates the function f over (almost) all spheres of radius R in R^n. This operator is known to be non-injective (as one can see by taking Fourier transform). However, the counterexamples that can be easily constructed using Bessel functions of the 1st kind, only belong to L^p if p>2n/(n-1). It has been shown previously by S. Thangavelu that for p not exceeding the critical number 2n/(n-1), the transform is indeed injective. In this article, the support theorem is proven that strengthens this injectivity result. Namely, if K is a convex bounded domain in R^n, the index p is not above 2n/(n-1), and (almost) all the integrals of $f$ over spheres of radius $R$ not intersecting K are equal to zero, then f is supported in the closure of the domain K. In fact, convexity in this case is too strong a condition, and the result holds for any what we call an R-convex domain.

math-ph↗
arXiv2010

Boundary Forelli theorem for the sphere in $\mathbb C^n$ and $n+1$ bundles of complex lines

Let $B^n$ be the unit ball in $\mathbb C^n$ and let the points $a_1,...,a_{n+1} \in B^n $ are affinely independent. If $f \in C(\partial B^n)$ and for any complex line $L$, containing at least one of the points $a_j$, the restriction $f|_{L \cap \partial B^n}$ extends holomorphically in the disc $L \cap B^n$, then $f$ is the boundary value of a holomorphic function in $B^n$. The condition for the points $a_j$ is sharp. The result confirms a conjecture from the preprint arXiv:0910.3592 by the author.

math.CV↗
arXiv2009

Range conditions for a spherical mean transform and global extension of solutions of Darboux equation

The transform under study is defined by integration of functions over spheres centered on a sphere. Such transform is of interest due to its applications in analysis and (thermoacoustic) tomography. The range of this transform has been described recently. Besides natural smoothness and support conditions, two type of conditions were involved: orthogonality condition and, in even dimension, moment condition. However, the moment condition was later derived from the other ones, which implies that orthogonality condition completely characterizes the range. We present a direct proof of this fact by proving existence of global extension for solutions of a certain boundary value problem for Darboux equation, associated with the spherical means. This extendibility phenomenon seems to be of independent interest.

math.AP↗
arXiv2009

Range conditions for a spherical mean transform

The paper is devoted to the range description of the Radon type transform that averages a function over all spheres centered on a given sphere. Such transforms arise naturally in thermoacoustic tomography, a novel method of medical imaging. Range descriptions have recently been obtained for such transforms, and consisted of smoothness and support conditions, moment conditions, and some additional orthogonality conditions of spectral nature. It has been noticed that in odd dimensions, surprisingly, the moment conditions are superfluous and can be eliminated. It is shown in this text that in fact the same happens in any dimension.

math.AP↗
arXiv2007

Complex dimensions of real manifolds, attached analytic discs and parametric argument principle

Let $Ω$ be a smooth real analytic submanifold of a complex manifold $X$. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold $Ω$ admits, 2) lower bounds for dimensions of complex tangent spaces of $Ω$, 3) a generalization of the argument principle for smooth families of holomorphic mappings from the standard complex disc to $X$. In particular, we obtain characterization of complex manifolds and their boundaries in terms of attached analytic discs. The special case when $Ω$ is the graph, leads to new characterizations of holomorphic and $CR$ functions, and in particular, to solutions of some open problems about such functions.

math.CV↗