On the Fourier coefficients of powers of a Blaschke factor and strongly annular fonctions

TL;DR

This paper derives detailed asymptotic formulas for the Fourier coefficients of powers of a Blaschke factor, revealing oscillatory, exponential, and Airy-type behaviors depending on the region, and applies these results to construct strongly annular functions with specific coefficient properties.

Contribution

The paper provides the first comprehensive asymptotic analysis of Fourier coefficients of Blaschke powers, including boundary behaviors, using advanced integral methods.

Findings

Oscillatory decay in certain coefficient regions

Exponential decay in others

Airy-type transition behavior near boundary points

Abstract

We compute asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_\lambda^n$, where $b_\lambda(z)=\frac{z-\lambda}{1-\lambda z}$ is the Blaschke factor associated to $\lambda\in\mathbb{D}$, $k\in[0,\infty)$ and $n$ is a large integer. We distinguish several regions of different asymptotic behavior of those coefficients in terms of $k$ and $n$. Given $\beta\in((1-\lambda)/(1+\lambda),(1+\lambda)/(1-\lambda))$ their decay is oscillatory for $k\in[\beta n,n/\beta]$. Given $\alpha\in(0,(1-\lambda)/(1+\lambda))$ their decay is exponential for $k\in[0,n\alpha]\cup[n/\alpha,\infty).$ Airy-type behavior is happening near the $k$-transition points $n(1-\lambda)/(1+\lambda)$ and $n(1+\lambda)/(1-\lambda)$. The asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_{\lambda}^{n}$ are derived using standard tools of asymptotic analysis of Laplace-type integrals. More precisely, the integral defining the $k^{\rm th}$ Fourier coefficient of $b_\lambda^n$ is perfectly suited for an application of the method of stationary phase when $k\in\left(n(1-\lambda)/(1+\lambda),n(1+\lambda)/(1-\lambda)\right)$ and requires the use of the method of the steepest descent when $k\notin[n(1-\lambda)/(1+\lambda),n(1+\lambda)/(1-\lambda)]$. Uniform versions of those standard methods are required when $k$ approaches one of the boundaries $n(1-\lambda)/(1+\lambda),$ $n(1+\lambda)/(1-\lambda)$. As an application, we construct strongly annular functions with Taylor coefficients satisfying sharp summation properties.