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

Abstract

We compute asymptotic formulas for the $k^{\mathrm{th}}$ 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 $k^{\mathrm{th}}$ Fourier coefficients of $b_{\lambda }^n$ are derived using standard tools of asymptotic analysis of Laplace-type integrals. More precisely, the integral defining the $k^{\mathrm{th}}$ 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.

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• Turan's problem 10 revisited
• Explicit error estimates for the stationary phase method I: The influence of amplitude singularities
• Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points