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Decay estimates for the arithmetic means of coefficients connected with composition operators. (English) Zbl 1428.42012

Summary: Let \(f\in L^{\infty }(T)\) with \(\Vert f\Vert_{\infty }\le 1\). If \(f(0)\ne 0,n,k\in{\mathbb{Z}}\), and \(b_{n,n-k}=\int_E f(x)^n e^{-2\pi i(n-k)x}dx\), \(E=\{x\in T:|f(x)|=1\} \), we prove that the arithmetic means \(\frac{1}{N} \sum_{n=M}^{M+N}|b_{n,n-k}|^2\) decay like \(\{\log N\log_2 N \cdots \log_q N\}^{-1}\) as \(N\rightarrow \infty \), uniformly in \(k\in{\mathbb{Z}} \).

MSC:

42B05 Fourier series and coefficients in several variables
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References:

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