Lebesgue constants of the Walsh system and Banach limits (Q311237)

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 \(t\in A\).