This article, published in 1933, first formulated the theorem (known in radio engineering as the Kotelnikov theorem) about the exact representation of a function with a limited spectrum of its frames produced at individual points. It is widely used in radio physics, optics, in the theory of digital signal processing. Article 1933 . is reproduced according to the publication carried out for the 70th anniversary of the Kotelnikov theorem and the 95th anniversary of Vladimir Alexandrovich Kotelnikov by the Institute of Radio Engineering and Electronics of the Moscow Energy Institute (MEI) (Technical University) in 2003. under the leadership of the Director of IRE IEI (TU) H. H. Udalova, with minor changes: the formula entry is given in the format adopted in the UVN, the page numbering of footnotes is replaced by a continuous one, spelling and syntax are brought into line with modern norms. Author's style saved.