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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 theBlaschke factor associated to $\lambda\in\mathbb{D}$, $k\in[0,\infty)$ and $n$is a large integer. We distinguish several regions of different asymptoticbehavior of those coefficients in terms of $k$ and $n$. Given$\beta\in((1-\lambda)/(1+\lambda),(1+\lambda)/(1-\lambda))$ their decay isoscillatory 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 nearthe $k$-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 ofasymptotic analysis of Laplace-type integrals. More precisely, the integraldefining the $k^{\rm th}$ Fourier coefficient of $b_\lambda^n$ is perfectlysuited for an application of the method of stationary phase when$k\in\left(n(1-\lambda)/(1+\lambda),n(1+\lambda)/(1-\lambda)\right)$ andrequires the use of the method of the steepest descent when$k\notin[n(1-\lambda)/(1+\lambda),n(1+\lambda)/(1-\lambda)]$. Uniform versionsof those standard methods are required when $k$ approaches one of theboundaries $n(1-\lambda)/(1+\lambda),$ $n(1+\lambda)/(1-\lambda)$. As anapplication, we construct strongly annular functions with Taylor coefficientssatisfying sharp summation properties.

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