Basis properties of affine Walsh systems in symmetric spaces

S. V. Astashkin ; P. A. Terekhin

Izvestiya. Mathematics , Tome 82 (2018) no. 3, pp. 451-476

Résumé

We study the basis properties of affine Walsh-type systems in symmetric spaces. We show that the ordinary Walsh system is a basis in a separable symmetric space $X$ if and only if the Boyd indices of $X$ are non-trivial, that is, $0\alpha_X\le\beta_X1$. In the more general situation when the generating function $f$ is the sum of a Rademacher series, we find exact conditions for the affine system $\{f_n\}_{n=0}^\infty$ to be equivalent to the Walsh system in an arbitrary separable s. s. with non-trivial Boyd indices. We also obtain sufficient conditions for the basis property. In particular, it follows from these conditions that for every $p\in(1,\infty)$ there is a function $f$ such that the affine Walsh system $\{f_n\}_{n=0}^{\infty}$ generated by $f$ is a basis in those and only those separable s. s. $X$ that satisfy $1/p\alpha_X\le\beta_X1$.

Détail

Keywords: basis, Walsh functions, Rademacher functions, Haar functions, symmetric space, affine Walsh-type system.