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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
##### Keywords:
Lebesgue constants; Walsh system
Full Text:
##### References:
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