On the closedness theorem in the theory of trigonometric series

Abstract

The presented, highly simple proof, although it uses the lemma of Prof. V. A. Steklov, expressed by formula (41), still differs significantly from the various proofs of the same theorem given by Prof. V. A. Steklov and is closer in its idea to the proof of Prof. Hurwitz (based on the Cesàro-Reyer method of arithmetic means), having, it seems to me, the advantage of greater simplicity, since it is based on the application of a known theorem on term-by-term integration of trigonometric series, which itself represents a special case of the closure theorem.

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