Harmonic Analysis and Convexity (eBook)

Alexander Koldobsky, Alexander Volberg (Herausgeber)

eBook Download: EPUB

2023
480 Seiten
De Gruyter (Verlag)
978-3-11-077543-3 (ISBN)

Infos

In recent years, the interaction between harmonic analysis and convex geometry has increased which has resulted in solutions to several long-standing problems. This collection is based on the topics discussed during the Research Semester on Harmonic Analysis and Convexity at the Institute for Computational and Experimental Research in Mathematics in Providence RI in Fall 2022.

The volume brings together experts working in related fields to report on the status of major problems in the area including the isomorphic Busemann-Petty and slicing problems for arbitrary measures, extremal problems for Fourier extension and extremal problems for classical singular integrals of martingale type, among others.

Alexander Koldobsky, University of Missouri, Colombia, USA; Alexander Volberg, Michigan State University, USA.

Textauszug

Algebraically integrable bodies and related properties of the Radon transform

Mark Agranovsky

Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel

Jan Boman

Department of Mathematics, Stockholm University, Stockholm, Sweden

Alexander Koldobsky

Department of Mathematics, University of Missouri-Columbia, Columbia, USA

Victor Vassiliev

Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel

Vladyslav Yaskin

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada

Abstract

Generalizing Lemma 28 from Newton’s “Principia” [25], Arnold [10] asked for a complete characterization of algebraically integrable domains. In this chapter we describe the current state of Arnold’s problems. We also consider closely related problems involving the Radon transform of indicator functions.

Keywords: Integrability, analytic continuation, monodromy, Picard–Lefschetz theory, Radon transform, convex body, Fourier transform,

MSC 2020: 14D05, 42B10, 44A12, 44A99, 52A20,

Acknowledgement

The third named author was supported in part by the U. S. National Science Foundation Grant DMS-2054068. The fifth author was supported in part by NSERC. This material is partially based on the work supported by the U. S. National Science Foundation grant DMS-1929284. The third and fifth authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Harmonic Analysis and Convexity semester program. The main part of the work of the fourth author was done at the Steklov Mathematical Institute, Moscow.

1 Introduction

The questions considered in this survey belong to the area of geometric tomography (see the book [17]), which lies at the crossroads between convex geometry and integral geometry and can be defined as the study of geometric properties of solids based on data about their sections and projections.

We study algebraic properties of two important volumetric characteristics in geometric tomography. For a body (compact set with non-empty interior) K in Rn, ξ∈Sn−1, and t∈R, the cutoff functions of K represent the n-dimensional volume of the parts of K cut by the hyperplane perpendicular to ξ at distance t from the origin:

(1.1)VK+(ξ,t)=Voln(K∩{x∈Rn:⟨x,ξ⟩≤t})=∫K∩{x∈Rn:⟨x,ξ⟩≤t}dx,VK−(ξ,t)=Voln(K∩{x∈Rn:⟨x,ξ⟩≥t})=∫K∩{x∈Rn:⟨x,ξ⟩≥t}dx.

The section function of K is the (n−1)-dimensional volume of the section of K by the same hyperplane:

(1.2)AK(ξ,t)=Voln−1(K∩{x∈Rn:⟨x,ξ⟩=t})=R(χK)(ξ,t)=∫K∩{x∈Rn:⟨x,ξ⟩=t}dx.

Here R stands for the Radon transform, χK is the indicator (characteristic function) of K, ⟨x,ξ⟩ is the inner (scalar) product in Rn, and dx is the Lebesgue measure on Rn or {x:⟨x,ξ⟩=t}, correspondingly. Clearly, the cutoff functions and the section function are related via differentiation in t.

Most of our problems take root in Lemma 28 about ovals from Newton’s Principia [25]; see also the discussion in [8], [9], [34]. Newton proved that if K is a convex infinitely smooth domain in R2, then the cutoff function of K cannot appear as the solution of a polynomial equation involving the parameters of the cutting hyperplane. Formalizing the question and extending it to higher dimensions, Arnold [10] asked whether there exist domains with smooth boundaries in Rn (apart from ellipsoids for odd n) for which the cutoff functions VK± are branches of an algebraic function. Recall that a function f(ξ,t) is algebraic if there exists a non-zero polynomial Φ(ξ,t,w) of n+2 variables such that

Φ(ξ,t,f(ξ,t))≡0.

Definition 1.1 (cf. [10], [8], [29]).

A domain K is algebraically integrable if the two-valued cutoff function VK±(ξ,t) coincides with some branches of an algebraic function.

In Section 2, we present the current state of Arnold’s problems. In particular, it was proved in [29] that there are no algebraically integrable bodies with infinitely smooth boundaries in even dimensions. However, the odd-dimensional case is still open.

In Sections 3–7, we consider similar questions that are motivated by Arnold’s problem and address the single-valued section function AK(ξ,t) rather than the multi-valued cutoff function VK(ξ,t). Therefore, we study geometric properties of bodies K from the point of view of algebraic properties of their Radon transform AK. The following definition is similar to Definition 1.1.

Definition 1.2.

Let K be a body in Rn. We say that K has algebraic Radon transform if there exists a function Ψ∈C(Sn−1)[t,w] which is an element of the polynomial ring of two variables over the algebra C(Sn−1) (i. e., it is a polynomial with respect to t, w with coefficients which are continuous functions of ξ) and satisfies the equation

Ψ(ξ,t,AK(ξ,t))=0

for every t such that the hyperplane ⟨ξ,x⟩=t intersects K.

The essential difference between the two definitions is that in Definition 1.2 we do not assume that Ψ is a polynomial in ξ, as we do in Definition 1.1, so the section function AK(ξ,t) is algebraic only with respect to the variable t.

Note that if K is algebraically integrable (i. e., the cutoff function VK± is algebraic), then the section function AK(ξ,t) is also algebraic as the derivative

AK(ξ,t)=±ddtVK±(ξ,t)

of an algebraic function. Thus, the class of domains with algebraic Radon transform contains algebraically integrable domains.

Our basic example is the unit ball Bn in Rn. In this case

ABn(ξ,t)=πn−12Γ(n+12)(1−t2)n−12.

If n is odd, then ABn(ξ,t) is a polynomial in t. Applying an affine transformation to Bn we obtain that AK(ξ,t) is a polynomial in t if n is odd and K is an ellipsoid.

In this chapter, we consider classes of bodies K satisfying Definition 1.2 with the defining polynomial Ψ of a certain form. The property of ellipsoids in odd-dimensional spaces mentioned above gives rise to the following.

Definition 1.3 ([1]).

Let K be a domain in Rn. We call K polynomially integrable if the Radon transform AK(ξ,t) of χK is a polynomial with respect to t when the corresponding hyperplane intersects K.

In the case of polynomially integrable domains, the equation Ψ(ξ,t,w)=0 in Definition 1.2 has the form...

Produkt-Details

• Erscheint lt. Verlag: 24.7.2023
• Reihe/Serie: Advances in Analysis and Geometry
• Reihe/Serie: Advances in Analysis and Geometry
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Zusatzinfo: 25 b/w ill.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Schlagworte: convex geometry • Geometric properties • geometric tomography • Harmonic Analysis • optimal algorithms
• ISBN-10: 3-11-077543-3 / 3110775433
• ISBN-13: 978-3-11-077543-3 / 9783110775433