ACTA issues

On rearrangements of orthogonal systems

O. A. Ziza

Acta Sci. Math. (Szeged) 68:1-2(2002), 229-236
2838/2009

 Abstract. Let $\{f_n\}$ be an orthonormal system in $L^2[0,1]$ (ONS). It is called a system of convergence if the orthogonal series in $L^2$ (OS) $\sum c_nf_n(x), x\in[0,1], \{c_n\} \in l^2$, 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 by 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 $\sum c_nf_{\tau_n}(x)$ is $(\varphi,\lambda )$-summable a.e. for any $c_n$. AMS Subject Classification (1991): 42C15, 40A30 Received January 29, 2001. (Registered under 2838/2009.)