On orthorecursive expansion by a~certain function system

V. V. Galatenko

Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 59-70

Résumé

The extension of Parseval's theorem given in [2] is interpreted from the viewpoint of expansion systems. To do this, we present the definition and basic properties of orthorecursive expansion systems (introduced by Lukashenko) and prove the equivalence of Stechkins' result and the convergence of the expansion by a certain system (the signum system) of any element in $L^2[0,1]$ to this element. The approach adopted enables us to study questions of uniform convergence, pointwise convergence and convergence in the $L^p$ metrics of expansions by the signum system of functions not only in $L^2 [0,1]$, but also in $L^p(X,\Xi,\mu)$, where $(X,\Xi,\mu)$ is an arbitrary measurable space with a finite measure. We prove the convergence in the $L^p$ metric of the expansion of any $L^p$ function, $1\leqslant p\leqslant\infty$, the uniform convergence of the expansion of any continuous function and the pointwise convergence of the expansion of any essentially unbounded function by the signum system to this function.