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Pré-Publication, Document De Travail Année : 2021

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

Alexander Borichev
  • Fonction : Auteur
Karine Fouchet
  • Fonction : Auteur
Rachid Zarouf

Résumé

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.
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hal-03275608 , version 1 (01-07-2021)

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Alexander Borichev, Karine Fouchet, Rachid Zarouf. On the Fourier coefficients of powers of a Blaschke factor and strongly annular fonctions. 2021. ⟨hal-03275608⟩
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