an:06630901 Zbl 1354.42047 Astashkin, S. V.; Semenov, E. M. Lebesgue constants of the Walsh system and Banach limits EN Sib. Math. J. 57, No. 3, 398-410 (2016); translation from Sib. Mat. Zh. 57, No. 3, 512-526 (2016). 00357650 2016
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42C10 Walsh functions; Rademacher functions; Lebesgue constants; Banach limit; almost convergent sequence The authors make a careful analysis of the Lebesgue constants for $$W_k$$, the Walsh system in $$[0,1]$$, given by $$L_n(W)=\int_0^1 |\sum_{k=1}^n W_k(t)|dt$$, $$n\in \mathbb N$$. Refining some estimates due to \textit{N. J. Fine} [Trans. Am. Math. Soc. 65, 372--414 (1949; Zbl 0036.03604)], they manage to compute $$\max_{1\leq n\leq 2^{2m+1}} L_n(W)$$ for $$m\in \mathbb N$$, which allows them, using a result by \textit{G. G. Lorentz} [Acta Math. 80, 167--190 (1948; Zbl 0031.29501)], to get that the sequence $$\{\frac{L_n(W)}{\log_2 n}\}$$ is not almost convergent. They also consider the step functions $$f_n(t)=\frac{1}{n}L_{[2^n(1+t)]}(W)$$ and show that $$\lim_{n\to \infty} f_n(t)=\frac{1}{4}$$ for almost all $$t\in [0,1]$$, $$\lim_{n\to \infty} f_n(t)=0$$ for all dyadic rational $$t\in [0,1]$$ and that there exists a dense set $$A\subset [0,1]$$ such that $$\liminf_{n\to\infty} f_n(t)=0$$ and $$\limsup_{n\to\infty} f_n(t)=\frac{1}{3}$$ for $$t\in A$$. Oscar Blasco (Valencia) Zbl 0036.03604; Zbl 0031.29501