Banach Journal of Mathematical Analysis

On the existence of universal series by the generalized Walsh system

Sergo A. Episkoposian

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In this paper, we prove the following: let ω(t) be a continuous function with ω(+0)=0 and increasing in [0,). Then there exists a series of the form

k=1ckψk(x)withk=1ck2ω(|ck|)<with the following property: for each ε>0 a weight function μ(x), 0<μ(x)1, |{x[0,1):μ(x)1}|<ε can be constructed so that the series is universal in the weighted space Lμ1[0,1) both with respect to rearrangements and subseries.

Article information

Banach J. Math. Anal., Volume 10, Number 2 (2016), 415-429.

Received: 12 March 2015
Accepted: 22 July 2015
First available in Project Euclid: 19 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42A65: Completeness of sets of functions
Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

generalized Walsh system weighted space universal series


Episkoposian, Sergo A. On the existence of universal series by the generalized Walsh system. Banach J. Math. Anal. 10 (2016), no. 2, 415--429. doi:10.1215/17358787-3589331.

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