On the Fourier coefficients of powers of a finite Blaschke product

Alexander Borichev; Karine Fouchet; Rachid Zarouf

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arXiv · 2401.17520

On the Fourier coefficients of powers of a finite Blaschke product

Alexander Borichev·Karine Fouchet·Rachid Zarouf

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Abstract

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell^{\infty}$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as a power of $n$. This is motivated by Sch\"affer's question on norms of inverses.

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Alexander Borichev, Karine Fouchet, Rachid Zarouf. 2024-01-31. On the Fourier coefficients of powers of a finite Blaschke product. https://arxiv.org/abs/2401.17520

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