On upper bounds of Fourier-Walsh coefficients

An upper bound is established for the upper bounds of the Fourier-Walsh coefficients a_n(f) whose modulus of continuity (,f) does not exceed a given modulus of continuity (). In the case of convex majorants of (), these bounds are attained for individual ordinal numbers n.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 725–736, December, 1969.