Conservative means of orthogonal series and the spaces \(L^ p[0;1]\), \(p\in(1;\infty)\) (Q1810061)

The main result of the paper states that under four assumptions on a \textit{conservative} (preserving the convergence of any convergent series rather than the sum as a \textit{regular} method does) summability method (or five if the method is irregular) of the orthogonal series is the Fourier series of a function in \(L^p[0,1].\) No serious discussion on the independence of the assumptions is carried out. Then the obtained results are applied to proving sufficient conditions for \((L^p,L^p)\)-multipliers. We remark that, on one hand, many historical details are discussed as well as a lengthy list of references is given. But, on the other hand, the author does not refer to a recent book on the subject by \textit{O. Ziza} [``Summability of orthogonal series'', URSS, Moscow (1999); in Russian with 50 pages English summary]{} and concerning the \((L^1,L^1)\)-multipliers the author is not aware of the reviewer's paper ``A family of function spaces and multipliers'' [ Isr. Math. Conf. Proc. 13, 141--149 (1999; Zbl 0948.42006)]{} in which more general results are proved than those cited.