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Ziza, Oksana Andreevna. Summability of Orthogonal Series.\break Moscow, Editorial URSS, 1999. 281 p. In Russian, with English Summary, pp. 206–255.
The book is devoted to the theory of summability of orthogonal series from L2[0,1]. The questions of summability of orthogonal series almost everywhere in [0,1] are investigated. Several main topics of this theory are considered: the comparison of summability of orthogonal series, the strong and the very strong summability, the Weyl multipliers, the absolute summability, estimates of the rate of summability. In each of the topics, at first classical theorems are formulated; their proofs can be found in the well-known monographs by S. Kaczmarz and H. Steinhaus (1935) and by G. Alexits (1961). A detailed survey of results appeared later in the literature is given. After this the methods of the class ФЛ are considered in detail. This is a wide class containing a lot of concrete well-known methods. The theorems proved by the author for this class give as corollaries a number of known results as well as many new results. This approach gives a general point of view for considering the above-mentioned topics for these methods.
The Summary contains the survey of the literature as well as the formulations and the discussion of the results proved in the Russian text.
The literature consists of more than 300 items.
The book is addressed to mathematicians, the specialists in the function theory and other branches of mathematical analysis. Some sections can be interesting for specialists in the numerical analysis. The book is available to graduate students of departments of pure and applied mathematics.
The author is a professor in Moscow State Institute of Electronics and Mathematics. Moscow, Russia.
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