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On rearrangements of orthogonal systems. (English) Zbl 0997.42018

Summary: Let \(\{f_n\}\) be an orthonormal system (ONS) in \(L^2[0,1]\). It is called a system of convergence if the orthogonal series (OS) in \(L^2\), \(\sum c_n f_n(x)\), \(x\in [0,1]\), \(\{c_n\}\in \ell^2\), is convergent a.e. for any \(c_n\). The following Kolmogorov-Men’shov problem is classical: for an arbitrary ONS \(\{f_n\}\), does there exist a rearrangement \(\{f_{\tau_n}\}\) that is a system of convergence? The answer is not known. In this note we consider a similar problem in which the convergence of OS after a rearrangement is replaced by the summability methods of the class \(\Phi\Lambda\). This class contains a number of well-known special summability methods. We find conditions on a method \((\varphi,\lambda)\in \Phi\Lambda\) sufficient for the existence, for any ONS, of a rearrangement \(\{f_{\tau_n}\}\) such that the OS \(\sum c_n f_{\tau_n}(x)\) is \((\varphi,\lambda)\)-summable a.e. for any \(c_n\).

MSC:

42C20 Other transformations of harmonic type
40A30 Convergence and divergence of series and sequences of functions
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