Abstract
We show that, for every number p ∈ (0, 1), there is g ∈ L 1[0, 1] (a universal function) that has monotone coefficients c k (g) and the Fourier–Walsh series convergent to g (in the norm of L 1[0, 1]) such that, for every f ∈ L p[0, 1], there are numbers δ k = ±1, 0 and an increasing sequence of positive integers N q such that the series ∑ +∞ k=0 δ k c k (g)W k (with {W k } theWalsh system) and the subsequence \(\sigma _{{N_q}}^{\left( \alpha \right)}\), α ∈ (−1, 0), of its Cesáro means converge to f in the metric of L p[0, 1].
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Yerevan. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 5, pp. 1021–1035, September–October, 2016; DOI: 10.17377/smzh.2016.57.508.
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Grigoryan, M.G., Sargsyan, A.A. On existence of a universal function for L p[0, 1] with p∈(0, 1). Sib Math J 57, 796–808 (2016). https://doi.org/10.1134/S0037446616050086
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DOI: https://doi.org/10.1134/S0037446616050086