Abstract
We set out the connection between convergence of the Haar series and differentiation with respect to nets of a function. This connection allows us to give a new proof of certain earlier theorems on Haar series, and also to prove a number of new generalizations.
Similar content being viewed by others
Literature cited
V. A. Skvortsov, “Calculation of the coefficients of an everywhere convergent Haar series,” Mat. Sb.,75, No. 3, 349–360 (1968).
S. Saks, Theory of the Integral [Russian translation], Moscow (1949).
F. G. Arutyunyan, “On series in the Haar system,” Dokl. Akad. Nauk Armyanskoi SSR,42, No. 3, 134–140 (1968).
A. A. Talalyan and F. G. Arutyunyan, “Convergence to + ∞ in the Haar system,” Mat. Sb.,66, No. 2, 240–247 (1965).
M. B. Petrovskaya, “Some uniqueness theorems for series in the Haar system,” Vestnik Mosk. Un-ta, (1), No. 5, 15–28 (1964).
V. A. Skvortsov, “A Cantor-type theorem for the Haar system,” Vestnik Mosk. Un-ta, (1), No. 5, 3–6 (1964).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 33–40, July, 1968.
In conclusion we remark that the results we have proved here for the Haar series can be extended to the series in Haar-type systems considered at the end of [4].
Rights and permissions
About this article
Cite this article
Skvortsov, V.A. Differentiation with respect to nets and the Haar series. Mathematical Notes of the Academy of Sciences of the USSR 4, 509–513 (1968). https://doi.org/10.1007/BF01429811
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01429811