Winding numbers and Fourier series

TL;DR

This paper explores classical analysis topics related to winding numbers and Fourier series, discussing historical context, recent developments, and new results on summation processes for continuous functions of constant magnitude.

Contribution

It provides an expository overview of the subject with new insights into summation processes for Fourier series of constant magnitude functions.

Findings

Historical overview of winding numbers and Fourier series

Recent developments in the field

New results on summation processes

Abstract

This is an expository talk on a topic of classical analysis, arising from the VMO theory of the topological degree due to Br\'ezis and Nirenberg (1995). We sketch the history of the subject and some of its recent developments. The paper is organized as a sequence of questions. Most of them, in particular the last one, deal with Fourier series of continuous functions of constant absolute value. One of them contains new results on the comparison of summation processes.