arXiv subjects

Mark L. Agranovsky

Publications and source records attributed to Mark L. Agranovsky.

4 recordsLinked to original sources

arXiv2017

On polynomially integrable domains in Euclidean spaces

Let $D$ be a bounded domain in $\mathbb R^n,$ with smooth boundary. Denote $V_D(ω,t), \ ω\in S^{n-1}, t \in \mathbb R,$ the Radon transform of the characteristic function $χ_{D}$ of the domain $D,$ i.e., the $(n-1)-$ dimensional volume of the intersection $D$ with the hyperplane $\{x \in \mathbb R^n: <ω,x>=t \}.$ If the domain $D$ is an ellipsoid, then the function $V_D$ is algebraic and if, in addition, the dimension $n$ is odd, then $V(ω,t)$ is a polynomial with respect to $t.$ Whether odd-dimensional ellipsoids are the only bounded smooth domains with such a property? The article is devoted to partial verification and discussion of this question.

math.FA↗

arXiv2009

Characterization of polyanalytic functions by meromorphic extensions into chains of circles

One-parameter smooth families of circles in the complex plane with the following property are described: a function is polyanalytic if and only if it has meromorphic extension inside any circle from the family, with the only singularity-a pole at the center.

math.DG↗

arXiv2009

Holomorphic extension from the unit sphere in $\mathbb C^n$ into complex lines passing through a finite set

Let $B^n$ be the $n$-dimensional unit complex ball and let $a$ and $b$ be two distinct points in its closure. Let $f$ be a real-analytic function on the complex unit sphere $\partial B^n.$ Suppose that for any complex line $L,$ meeting the two points set $\{a,b\},$ the function $f$ admits one-dimensional holomorphic extension in the cross-section $L \cap B^n.$ Then $f$ is the boundary value of a function holomorphic in $B^n$. Two points can not be replaced by a single point. The proof essentially uses recent result of the author about characterization of polyanalytic functions in the complex plane.

math.CV↗

arXiv1999

Geometry of Stationary Sets for the Wave Equation in R^n, The Case of Finitely Supported Initial Data, An Announcement

We consider the Cauchy problem for the wave equation in the whole space, R^n, with initial data which are distributions supported on finite sets. The main result is a precise description of the geometry of the sets of stationary points of the solutions to the wave equation.

math.AP↗