Affine Walsh-type systems of functions in symmetric spaces

Affine Walsh-type systems of functions in symmetric spaces are investigated. It is shown that such a system can only be an unconditional basis in $L^2$. On the other hand the Besselian affine system generated by a function $f$ in the Zygmund-Orlicz space $\operatorname{Exp}L^p$, $p>0$, is an $\mathrm{RUC}$-system in a symmetric space $X$ if and only if $(\operatorname{Exp}L^q)^0\subset X\subset L^2$, where $(\operatorname{Exp}L^q)^0$ is the closure of $L^\infty$ in $\operatorname{Exp}L^q$ and $q=2p/(p+2)$.
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