In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.

The results corresponding to some theorems of S. Lal [Appl. Math. and Comput. 209 (2009), 346-350] and the results of W. Łenski and B. Szal [Banach Center Publ., 95, (2011), 339-351] are shown. The better degrees of pointwise approximation than these in mentioned papers by another assumptions on summability methods for considered functions are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated.

We consider the pointwise and normwise approximation of function by some special matrix means of its Fourier series. The results corresponding to the theorem of Łenski and Szal in [5] and the results of Saini and Singh in [6] are shown. Some special cases as corollaries are also formulated. 2010 Mathematics Subject Classification: 42A24.

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Abstract:The results concerninig pointwise approximation and product of summability methods corresponding to the theorems of Xh. Z. Krasniqi [Poincare J. Anal. Appl., 2014, 1, 1-8] and W. Łenski and B. Szal [Math. Slovaca, 2016, 66(6), 1-12] are generalized. Some special cases are also formulated as corollaries.