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

  • Faruk F. Abi-KhuzamEmail author


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 \(k\in {\mathbb {Z}} \).


Toeplitz Fourier coefficient Decay estimates 

Mathematics Subject Classification

Primary 42B05 Secondary 42B33 



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.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsAmerican University of BeirutBeirutLebanon

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