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

Decay estimates for the arithmetic means of coefficients connected with composition operators Let $$f\in L^{\infty }(T)$$ f ∈ L ∞ ( T ) with $$\Vert f\Vert _{\infty }\le 1$$ ‖ f ‖ ∞ ≤ 1 . If $$f(0)\ne 0,$$ f ( 0 ) ≠ 0 , $$n,k\in {\mathbb {Z}}$$ n , k ∈ Z , and $$b_{n,n-k}=\int _{E}f(x)^{n}e^{-2\pi i(n-k)x}dx$$ b n , n - k = ∫ E f ( x ) n e - 2 π i ( n - k ) x d x , $$E=\{x\in T:|f(x)|=1\}$$ E = { x ∈ T : | f ( x ) | = 1 } , we prove that the arithmetic means $$\frac{1}{N} \sum _{n=M}^{M+N}|b_{n,n-k}|^{2}$$ 1 N ∑ n = M M + N | b n , n - k | 2 decay like $$\{\log N\log _{2}N\cdot \cdot \cdot \log _{q}N\}^{-1}$$ { log N log 2 N · · · log q N } - 1 as $$N\rightarrow \infty$$ N → ∞ , uniformly in $$k\in {\mathbb {Z}}$$ k ∈ Z . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

, Volume 9 (4) – Dec 6, 2018
8 pages

/lp/springer-journals/decay-estimates-for-the-arithmetic-means-of-coefficients-connected-MjYRakPS1X
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-018-0272-2
Publisher site
See Article on Publisher Site

### Abstract

Let $$f\in L^{\infty }(T)$$ f ∈ L ∞ ( T ) with $$\Vert f\Vert _{\infty }\le 1$$ ‖ f ‖ ∞ ≤ 1 . If $$f(0)\ne 0,$$ f ( 0 ) ≠ 0 , $$n,k\in {\mathbb {Z}}$$ n , k ∈ Z , and $$b_{n,n-k}=\int _{E}f(x)^{n}e^{-2\pi i(n-k)x}dx$$ b n , n - k = ∫ E f ( x ) n e - 2 π i ( n - k ) x d x , $$E=\{x\in T:|f(x)|=1\}$$ E = { x ∈ T : | f ( x ) | = 1 } , we prove that the arithmetic means $$\frac{1}{N} \sum _{n=M}^{M+N}|b_{n,n-k}|^{2}$$ 1 N ∑ n = M M + N | b n , n - k | 2 decay like $$\{\log N\log _{2}N\cdot \cdot \cdot \log _{q}N\}^{-1}$$ { log N log 2 N · · · log q N } - 1 as $$N\rightarrow \infty$$ N → ∞ , uniformly in $$k\in {\mathbb {Z}}$$ k ∈ Z .

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 6, 2018

### References

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