Abi-Khuzam, Faruk F. Decay estimates for the arithmetic means of coefficients connected with composition operators. (English) Zbl 1428.42012 Anal. Math. Phys. 9, No. 4, 1753-1760 (2019). 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 Keywords:Toeplitz; Fourier coefficient; decay estimates PDFBibTeX XMLCite \textit{F. F. Abi-Khuzam}, Anal. Math. Phys. 9, No. 4, 1753--1760 (2019; Zbl 1428.42012) Full Text: DOI References: [1] Abi-Khuzam, F., Shayya, B.: A remark on the WAT conjecture. Preprint (2009) · Zbl 1049.42005 [2] Nazarov, F.; Shapiro, J., On the Toeplitzness of composition operators, Complex Var. Elliptic Equ., 52, 193-210 (2007) · Zbl 1122.47021 [3] Rudin, W., Real and Complex Analysis (1986), New York: McGraw-Hill Company, New York [4] Shapiro, J., Every composition operator is (mean) asymptotically Toeplitz, J. Math. Anal. Appl., 333, 523-529 (2007) · Zbl 1119.47023 [5] Shayya, B., The WAT conjecture on the Torus, Proc. Am. Math. Soc, 139, 3633-3643 (2011) · Zbl 1227.42009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.