Application of negative order Cesàro summability methods to Fourier–Walsh series of functions from $L^{\infty }[0, 1]$
Volume 158 / 2019
Colloquium Mathematicum 158 (2019), 195-212
MSC: Primary 42A24; Secondary 42C10.
DOI: 10.4064/cm7624-10-2018
Published online: 29 July 2019
Abstract
The following results are proved:
$\bullet$ there exists a continuous function such that any subsequence of the Cesàro $(C, \alpha)$ $(-1 \lt \alpha \lt -1/2)$ means of the Fourier–Walsh series of this function diverges on a set of positive measure;
$\bullet$ the values of an arbitrary integrable function can be changed on a set of arbitrarily small measure so that the $1$-density subsequence of negative order Cesàro means of the Fourier–Walsh series of the modified function are convergent in $L^{\infty}$-norm.
