Abstract:
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, $0<\alpha_X\le\beta_X<1$. 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 $1/p<\alpha_X\le\beta_X<1$.
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