On the means for centralized systems of functions (Q1823413)

The system of functions \(\phi =\{\phi_ n\}\) defined and Lebesgue integrable on [0,1] is called centralized system of functions if \(\int_{\Lambda}\phi_{n+1}(x)dx=0\) for each n and for each \(\Lambda\in {\mathcal F}_ n(\phi)\), where \({\mathcal F}_ n(\phi)\) is the minimal \(\sigma\)-algebra with respect to which the functions \(\phi_ 1,...,\phi_ n\) are measurable. Adding the conditions \(\int^{1}_{0}\phi_ 1(x)dx=0\) and \(\int^{1}_{0}\phi^ 2_ n(x)dx=1\) the system \(\phi\) becomes an orthonormal system. In the present paper the author refers to an article by K. Tandori in which the means of stochastic independent functions are considered. The author (guided by K. Tandori) transposes the results obtained by K. Tandori for stochastic independent functions to the case of centralized system of functions. Even the method of proving is taken from the cited paper by \textit{K. Tandori} [Acta Math. Hungar. 45, 397-423 (1985; Zbl 0583.42010)] and other papers treating the same problem for orthogonal functions.