CONTINUOUS FUNCTIONS WITH A FOURIER EXPANSION IN HAAR FUNCTIONS IN WHICH THE COEFFICIENTS ARE MONOTONICALLY DECREASING V. G. Krotov UDC 51.7.52 Let {Xm(t) } be the system of Haar functions orthonormalized in [0, 1] (for the definition see e.g., [1], pp. 14-16). When the properties of series in the Haar system with monotonic coefficients are investigated, these properties are found to differ fundamentally from those of trigonometric series with monotonic co- efficients. The most complete studies in this field are due to P. L. Ul'yanov [2]. In [3], B. I. Golubov found necessary and sufficient conditions for a function continuously differentiable in [0, 1] to have mono- tonically decreasing Fourier--Haar coefficients (see [3], Theorem 7). In [4], the present author gave the following theorem, providing the solution of the corresponding problem for continuous functions; Golubov's theorem is a particular consequence of this. THEOREM A. The necessary and sufficient conditions for a function f(t), continuous in [0, I], f(t) # eonst, to have monotonically decreasing Fourier coefficients am(f), m >- 2, when expanded in Haar func- tions, are that simultaneously: I) the derivative f'(t) of f(t) exists and is continuous everywhere in [0, i] with the possible exception of a countable set E c [0, i], in which it may have discontinuities of the first kind; 2) f'(t) is not decreasing and is negative in [0, i] ~E; 3) at ally points t, xE[0,1]~E, we have 2 -3/2 <- f'(t)/f'(x) <- 23/2. The bounds in 3) are strict. Because of its unwieldiness, the proof of necessity in Theorem A was omitted in [4], while it was pointed out (see [4], p. 122) that the sufficiency is proved in exactly the same way as in Golubov's theorem (see [3], p~ 1290). The strictness of the bounds in condition 3) of Theorem A also follows from Golubov's theorem. In the present article we give a proof of the necessity in Theorem A, the kernel of which is the fol- lowing construction, Let n = I, 2 .... and j = 1 ..... 2 n. We then set 5 (j) ~ ((j -- i)/2 n, j/2 n) and, after arbitrarily _< c 6n(j)" fixing p~ rnjE6(2J+-i1)(n>- 1, 1--<J 2n), we den~ ~ (TnJ, 7ni + 2-(n+l) Let the natural numbers n >- 1, 1 -< j <- 2n and the real number ~ > 0 be given. We construct by in- duction a sequence of natural numbers {nr}, nr+l > nr + 1 (r = O, 1 .... ) and a sequence of finite sets of natural numbers {i(kr)}, 1 <- jl (r) -< 2nr, k = 1 ..... Nr, satisfying the following conditions (with r, s = 0, I .... ): .(,) ~ A -- a~ ) (k ~ i ..... N~), (1) , (2) p=l ..... N~), V--1 N~ 2~-~ -- ~_.~N, (2 ",-n,~ + l), (3) I Translated from Sibirskii Matematicheskii Zhurnal, Vol. 15, No. 2, pp. 439-444, March-April, 1974. Original article submitted March 22, 1973. 9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 glest 17th Street, New York, N. Y. lO01i. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00. 316