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

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.