Circle embeddings with restrictions on Fourier coefficients

Abstract

This paper continues the investigation of the relation between the geometry of a circle embedding and the values of its Fourier coefficients. First, we answer a question of Kovalev and Yang concerning the support of the Fourier transform of a starlike embedding. An important special case of circle embeddings are homeomorphisms of the circle onto itself. Under a one-sided bound on the Fourier support, such homeomorphisms are rational functions related to Blaschke products. We study the structure of rational circle homeomorphisms and show that they form a connected set in the uniform topology.

Introduction

Every continuous map f of the unit circle $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$ to the complex plane extends to a harmonic map F of the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$, and the Taylor coefficients of F are given by the Fourier coefficients of f, namely $\hat{f}(n)=\frac{1}{2\pi }{\int }_0^{2\pi }f(e^{i\theta })e^{-in\theta }d\theta$. This simple but important relation has consequences both for geometric function theory and for the theory of minimal surfaces [3], [7], [9], especially when f is an embedding, i.e., an injective continuous map. For example, when f is a sense-preserving embedding with a convex image, the Radó-Kneser-Choquet theorem [3, p. 29] states that F is a sense-preserving diffeomorphism and in particular $\hat{f}(1)\ne 1$. On the other hand, for every integer N there exists a sense-preserving embedding $f:\mathbb{T}\to \mathbb{C}$ such that $\hat{f}(n)=0$ whenever $|n|\le N$ [10, Theorem 5.1]. Thus, some restrictions on the shape of $f(\mathbb{T})$ are necessary to obtain a non-vanishing result for $\hat{f}$.

By a circle embedding we mean an injective continuous map $f:\mathbb{T}\to \mathbb{C}$. In the special case $f(\mathbb{T})=\mathbb{T}$ the map f is called a circle homeomorphism. The curve $f(\mathbb{T})$ is called shar-shaped about $w_0\in \mathbb{C}$ if the argument of $f(e^{i\theta })-w_0$ is a monotone function of θ. In this case f is called a starlike embedding. In this paper we answer Question 5.1 in [10] by proving that for a starlike embedding f, at least one of the coefficients $\hat{f}(1)$ and $\hat{f}(-1)$ is nonzero.

If the Fourier series of a sense-preserving circle homeomorphism $f:\mathbb{T}\to \mathbb{T}$ terminates in either direction, then f is the restriction of the quotient of two Blaschke products [10, Proposition 3.2]. It is difficult to describe exactly which ratios of Blaschke products restrict to circle homeomorphisms. We give some sufficient conditions in Section 5. In Section 4, specifically Theorem 4.1, Theorem 4.3 we show that the set of rational circle homeomorphisms is connected in the uniform topology.