On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials

Let H, H^L be classes of functionsf(x) whose modulus of continuity (f; t) and, respectively, integral modulus of continuity(f; t)_L do not exceed a given modulus of continuity(t), while H_v is a class of functionsf(x) whose variationfdoes not exceed a given number V > 0. Bounds are obtained for the upper limit of the best approximations in the metric of L by Haar-system polynomials on the classes just introduced (on the class H^L only when (t)=Kt). These bounds are exact for class H_V and, in case(t) is convex, also for the classes H and H^L.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 47–54, July, 1969.The author wishes to thank N. P. Korneichuk for having posed the problem and for his constant attention to this work.