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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2016 Volume 57, Number 5, Pages 1021–1035 (Mi smj2803)

This article is cited in 2 papers

On existence of a universal function for $L^p[0,1]$ with $p\in(0,1)$

M. G. Grigoryana, A. A. Sargsyanb

a Yerevan State University, Yerevan, Armenia
b Synchrotron Research Institute CANDLE, Yerevan, Armenia

Abstract: We show that, for every number $p\in(0,1)$, there is $g\in L^1[0,1]$ (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 $f\in L^p[0,1]$, there are numbers $\delta_k=\pm1,0$ and an increasing sequence of positive integers $N_q$ such that the series $\sum^{+\infty}_{k=0}\delta_kc_k(g)W_k$ (with $\{W_k\}$ the Walsh system) and the subsequence $\sigma^{(\alpha)}_{N_q}$, $\alpha\in(-1,0)$, of its Cesáro means converge to $f$ in the metric of $L^p[0,1]$.

Keywords: universal function, Fourier coefficient, Walsh system, convergence in a metric.

UDC: 517.51

Received: 21.04.2015

DOI: 10.17377/smzh.2016.57.508


 English version:
Siberian Mathematical Journal, 2016, 57:5, 796–808

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