In this paper we obtain degree of approximation of functions in L p by operators associated with their Fourier series using integral modulus of continuity. These results generalize many know results and are proved under less stringent conditions on the infinite matrix.

In the present paper we introduce a new class of sequences called GM (β, r), which is a generalization of the class considered by Tikhonov in [13]. Moreover, we obtain sufficient and necessary conditions for uniform convergence of sine series with (β, r)-general monotone coefficients.

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.

The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [2], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double sine series, which are necessary as well in case the coefficients are non-negative. We shall generalize those results defining a new class of double sequences for the coefficients.Mathematics subject classification number: 42A20, 42A32, 42B99.

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