We generalize and improve the results of A. Guven, D. Israfilov, Xh. Z. Krasniqi and T. N. Shakh-Emirov. We consider the general methods of summability of Fourier series of functions from L p(x) 2π with p (x) ≥ 1. For estimate of the error of approximation of functions by the matrix means we use a modulus of continuity constructed by the Steklov functions of the increments of considered functions without of absolute values.

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The results corresponding to some theorems of S. Lal [Appl. Math. Comput.

The degree of pointwise approximat.ion in the strong sense of 2nperiodic functions from LP (p = (1 + a)-1, a = . -i/2) is examined. An answer to the modified version of LEINDLER'S problem [ 4 ] is given.

We present an estimation of the H q k 0 ,k r f and H λϕ u f means as approximation versions of the Totik type generalization (see [6,7]) of the result of G. H. Hardy, J. E. Littlewood, considered by N. L. Pachulia in [5]. Some results on the norm approximation will also be given.

Abstract. There is introduced a modified local modulus of continuity as a measure of pointwise strong summability. The approximation versions of known results PuTraing Wang [6] and A. A. Zakhaxov [8] are obtained. IntroductionLet LP (1 < p < oo) [resp.C] be the class of all 27r-periodic real-valued functions integrable in the Lebesgue sense with p-th power [continuous] over Q = [-7T, 7r]