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Russians are known on having very strong literature in Mathematics and Sciences overall. So as I am willing to improve my knowledge and problem solving in real-analysis, especially in Fourier series, so I wonder what are the books written and used by the Russians( I have a somewhat good level in Russian)?

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    $\begingroup$ see here books.google.de/books/about/… i hope this will help you $\endgroup$ – Dr. Sonnhard Graubner Jan 10 '17 at 18:07
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    $\begingroup$ Ok, I just saw that Dr Graubner has already posted Tolstov's book which comes as no surprise given its status. Sorry for that.. At least I hope the other two will be of some interest. $\endgroup$ – MathematicianByMistake Jan 13 '17 at 22:18
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At the top of my head I would recommend Georgi Tolstov's "Fourier Series" (1960 or 1962 version) which is also available in English but rather easy to find in Russian as well. It provides for a very smooth and elegant introduction to the topic as far as my experience allows me to say. Descriptive and very well balanced. In any case it is considered a classic.

Another choice would be Anatolyi Aksionov's "Mathematical Analysis. Fourier Series" which is rather short (~90 pages) and part of a series on textbooks on analysis by the same author. I have not read it though.

A book on the topic written by a Russian, but in English (and perhaps the most recent one covering both of these "conditions") is Alexander Koldobsky's "Fourier Analysis In Convex Geometry" which is certainly more specialised and focuses on the application of Fourier analytic methods in solving the Busemann-Petty problem (for more on this check here).

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I hope it can help you :

Season $6-7$ of volume $3$ of The fundamentals of mathematical analysis by Grigorii Fichtenholz

фихтенгольц - курс дифференциального и интегрального исчисления том 3:

original language book

author: Фихтенгольц, Григорий Михайлович

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