Differentiation with respect to nets and the Haar series

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.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].