On orthorecursive expansion by a certain function systemстатья
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Аннотация:The extension of Parseval's theorem given by B.S. Stechkin and S.B. Stechkin 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 L2[0,1] to this element. The approach adopted enables us to study questions of uniform convergence, pointwise convergence and convergence in the Lp metrics of expansions by the signum system of functions not only in L2[0,1], but also in Lp(X,\Xi,\mu), where (X,\Xi,\mu) is an arbitrary measurable space with a finite measure. We prove the convergence in the Lp metric of the expansion of any Lp function, 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.