Reconstruction of Planar Domains from Partial Integral Measurements

TL;DR

This paper develops a method for reconstructing planar domains with piecewise-algebraic boundaries from their moments, and explores the concept of 'invisible sets' which are undetectable by certain moment measurements, revealing underlying rigidity and symmetry.

Contribution

It introduces a novel approach combining one-dimensional reconstruction with separation of variables for planar domains and analyzes the properties of invisible sets under incomplete moment data.

Findings

Reconstruction of domains with piecewise-polynomial boundaries is feasible using the proposed method.

Invisible sets exhibit rigidity and symmetry, making them undetectable under specific moment measurements.

The paper establishes connections between various types of invisibility, including zero quadrature domains and complex moments.

Abstract

We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewise-algebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [Bat]* and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects. * D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse Problems 25 (2009) 105001