Restricting a Schauder basis to a set of positive measure

Abstract:

Let $\{ {f_n}\}$ be an orthonormal system of functions on [0, 1] containing a subsystem $\{ {f_{{n_k}}}\}$ for which (a) ${f_{{n_k}}} \to 0$ weakly in ${L_2}$, and (b) given $E \subset [0,1]$, $|E| > 0$, ${\operatorname {Lim}}\;{\operatorname {Inf}}{\smallint _E}|{f_{{n_k}}}(x)|dx > 0$. There then exists a subsystem $\{ {g_n}\}$ of $\{ {f_n}\}$ such that for any set E as above, the linear span of $\{ {g_n}\}$ in ${L_1}(E)$ is not dense. For every set E as above, there is an element of ${L_p}(E)$, $1 < p < \infty$, whose Walsh series expansion converges conditionally and an element of ${L_1}(E)$ whose Haar series expansion converges conditionally.