DIFFERENTIATION WITH RESPECT AND THE HAAR SERIES V. A. Skvortsov TO NETS UDC 517.5 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 ear- lier theorems on Haar series, and also to prove a number of new generalizations. w 1. We are going to .establish the connection between convergence of the Haar series and differentia- tion with respect to nets of a function of a point and of a function of a segment. We consider the series in the Haar system: ~--J,~=l a~;~ (x). (1) Let us set ~ (x) = a.nXn (x), ~ (x) = dr. Then (1) takes the form ~ ~ (~). We denote the sum of this series at the points of convergence by r and the sum of the series by r (x). r n~l ~n (x) It is easy to prove the following Lemrpa (see [1]). LEMMA 1. For any series (1) the series (2) converges at each binary-rational point x to the sum (2) (3) We denote the set of binary-rational points in the segment [0,1] by R. Points that are not binary- rational will be called binary-irrational; the set of such points will be denoted by I. The points of R of the form i/2J, when i = 1, 2 ..... 2J will be said to be the elements of the net Rj of order j, j = 0, 1, 2 ..... The union of the nets Rj of all orders coincides with R. Intervals of the following form occur in the definition of the Haar system ((i -- t)/2 j, i/2J), ] ~ 0, t, 2 .... ; i --~ t, 2 ..... 2j. (4) To each point x E I corresponds a unique sequence of embedded intervals of the form (4), for each of which the point x is an interior point. We denote tMs sequence by <Uk} = {(ak, bk)), k = 0, 1, 2 ..... Here rues n k = 1/2 k. To x E R corresponds two sequences (left and right) of intervals of the form (4) for which x is the common end-point. Definition 1. Let x E [0,1] and let {(a k, bk) } be a sequence of embedded intervals of the form (4) that contracts to x (when x E R we consider both of these sequences). Then r is said to be R-continuous at x if M. V. Lomonosov Moscow State University. Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 33-40, July, 1968. Original article submitted November 16, 1967. 509