On the Fourier Coefficients of Powers of a Finite Blaschke Product
Copyright policy )On the Fourier Coefficients of Powers of a Finite Blaschke Product
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 $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. 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 Schäffer’s question on norms of inverses.
- CNRS@CREATE Ltd (Singapore) Singapore
- Centrale Marseille France
- Université de Toulon France
- Université Côte d'Azur France
- Centre de Physique Théorique France
van der Corput lemma, Mathematics - Complex Variables, Convergence and absolute convergence of Fourier and trigonometric series, invertible linear mapping, 1991 Mathematics Subject Classification. 30J10 42A16 41A60 15A60 Fourier coefficients finite Blaschke products van der Corput lemma condition number of matrices, [MATH] Mathematics [math], 15A60 Fourier coefficients, 1991 Mathematics Subject Classification. 30J10, Fourier coefficients, Fourier series of functions with special properties, special Fourier series, finite Blaschke products, inequalities, Banach spaces of continuous, differentiable or analytic functions, 41A60, FOS: Mathematics, Complex Variables (math.CV), 42A16, 30J10, 42A16, 41A60, 15A60, condition number of matrices, norms of inverses
van der Corput lemma, Mathematics - Complex Variables, Convergence and absolute convergence of Fourier and trigonometric series, invertible linear mapping, 1991 Mathematics Subject Classification. 30J10 42A16 41A60 15A60 Fourier coefficients finite Blaschke products van der Corput lemma condition number of matrices, [MATH] Mathematics [math], 15A60 Fourier coefficients, 1991 Mathematics Subject Classification. 30J10, Fourier coefficients, Fourier series of functions with special properties, special Fourier series, finite Blaschke products, inequalities, Banach spaces of continuous, differentiable or analytic functions, 41A60, FOS: Mathematics, Complex Variables (math.CV), 42A16, 30J10, 42A16, 41A60, 15A60, condition number of matrices, norms of inverses
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