zbMATH — the first resource for mathematics

Lebesgue constants of the Walsh system and Banach limits. (English. Russian original) Zbl 1354.42047
Sib. Math. J. 57, No. 3, 398-410 (2016); translation from Sib. Mat. Zh. 57, No. 3, 512-526 (2016).
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 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 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 \(t\in A\).

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Full Text: DOI
[1] Bari N. K., Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).
[2] Kashin, B. S.; Saakyan, A. A., Orthogonal series, (1999) · Zbl 1188.42010
[3] Golubov B. I., Efimov A. V., and Skvortsov V. A., Walsh Series and Transforms. The Theory and Applications [in Russian], Nauka, Moscow (1987). · Zbl 0692.42009
[4] Fine, N. J., On the Walsh functions, Trans. Amer. Math. Soc., 65, 372-414, (1949) · Zbl 0036.03604
[5] Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math., 80, 167-190, (1948) · Zbl 0031.29501
[6] Sucheston, L., Banach limits, Amer. Math. Monthly, 74, 308-311, (1967) · Zbl 0148.12202
[7] Semenov, E. M.; Sukochev, F. A., Invariant Banach limits, J. Func. Anal., 259, 1517-1541, (2010) · Zbl 1205.46012
[8] Astashkin, S. V.; Semenov, E. M., Lebesgue constants of the Walsh system, Dokl. Math., 91, 344-346, (2015) · Zbl 1329.42026
[9] Kwapien S. and Woyczijnski W. A., Random Series and Stochastic Integrals. Single and Multiple, Birkhäuser, Boston (1992).
[10] Astashkin, S. V., Rademacher chaos in symmetric spaces. 2, East J. Approx., 6, 71-86, (2000) · Zbl 1084.42508
[11] Lorentz, G. G., Relations between function spaces, Proc. Amer. Math. Soc., 12, 127-132, (1961) · Zbl 0124.31704
[12] Kreĭn, S. G.; Petunin, Yu. I.; Semenov, E. M., Interpolation of linear operators, (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.