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Lebesgue constants of the Walsh system. (English. Russian original) Zbl 1329.42026
Dokl. Math. 91, No. 3, 344-346 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 462, No. 5, 509-511 (2015).
Let \(W=(W_n)_{n=0}^\infty\) be the Walsh system, \(L_n(W)\) the Lebesgue constants of the Walsh system. It is known that \[ \frac{1}{4}\mathrm{Var}(n)\leq L_n(W)\leq \mathrm{Var}(n) \] where \(\mathrm{Var}(n)\) is the binary variation of \(n\). The authors study the asymptotic behavior of \(L_n(W)\). We give two results as examples.
1) For any \(k\in\mathbb N\) \[ \sum_{n=2^k+1}^{2^{k+1}}L_n(W)=2^k\left(\frac{k}{4}+1\right). \] 2) The following relation holds: \[ \lim\limits_{k\to\infty}\frac{1}{k}\sum_{n=2}^k\frac{L_n(W)}{\log_2n}=\frac{1}{4}. \]

MSC:
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A75 Harmonic analysis on specific compact groups
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References:
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