Decay estimates for the arithmetic means of coefficients connected with composition operators

Abstract

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 \(\{\log N\log _{2}N\cdot \cdot \cdot \log _{q}N\}^{-1}\) as \(N\rightarrow \infty \), uniformly in \(k\in {\mathbb {Z}} \).

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Acknowledgements

I am grateful to the referee for all his comments, and in particular, his suggestion to use the function \(\log |z{\bar{w}}-r\phi (z)\overline{\phi (w)}|\) in the proof of the key inequality, which led to considerable improvement in the presentation of the results of this paper.

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Correspondence to Faruk F. Abi-Khuzam.

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Abi-Khuzam, F.F. Decay estimates for the arithmetic means of coefficients connected with composition operators. Anal.Math.Phys. 9, 1753–1760 (2019). https://doi.org/10.1007/s13324-018-0272-2

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Keywords

  • Toeplitz
  • Fourier coefficient
  • Decay estimates

Mathematics Subject Classification

  • Primary 42B05
  • Secondary 42B33