Unions of arcs from Fourier partial sums

TL;DR

This paper demonstrates that unions of arcs on the circle can be uniquely identified and reconstructed from their Fourier partial sums using complex analysis and Hilbert space techniques.

Contribution

It introduces a method to recover the endpoints of arcs from Fourier coefficients by solving polynomial equations, advancing arc reconstruction techniques.

Findings

Union of arcs is uniquely determined by Fourier partial sums.

Endpoints of arcs can be recovered from polynomial equations.

Method applies to unions of up to n arcs.

Abstract

Elementary complex analysis and Hilbert space methods show that a union of at most n arcs on the circle is uniquely determined by the nth Fourier partial sum of its characteristic function. The endpoints of the arcs can be recovered from the coefficients appearing in the partial sum by solving two polynomial equations.