We present an estimation of the ${H}_{k\u2080,{k}_{r}}^{q,\alpha}f$ and ${H}_{\lambda ,u}^{\varphi ,\alpha}f$ means as approximation versions of the Totik type generalization (see [5], [6]) of the result of G. H. Hardy, J. E. Littlewood. Some corollaries on the norm approximation are also given.

Considering the class of almost periodic functions integrable in the Stepanov sense we extend and generalize certain results of the first author, as well as of L. Leindler and P. Chandra.

Under some assumptions on the matrix of a summability method, whose rows are sequences of bounded variation, we obtain a generalization and an improvement of some results of Xie-Hua Sun and L. Leindler.

We present an estimate of the (C,1)(E,1)-strong means with mixed powers of the Fourier series of a function $f\in {L}_{2\pi}$ as a generalization of the result obtained by M. Yildrim and F. Karakus. Some corollaries on the norm approximation are also given.

We show the results corresponding to some theorems of S. Lal and H. K. Nigam [Int. J. Math. Math. Sci. 27 (2001), 555-563] on the norm and pointwise approximation of conjugate functions and to the results of the authors [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] also on such approximations.

We show the general and precise conditions on the functions and modulus of continuity as well as on the entries of matrices generating the summability means and give the rates of approximation of functions from the generalized integral Lipschitz classes by double matrix means of their Fourier series. Consequently, we give some results on norm approximation. Thus we essentially extend and improve our earlier results [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] and the result of S. Lal [Appl....

Estimates of the strong means of Marcinkiewicz type with the Cesaro means of negative order in one of the variables instead of square partial sums are obtained by characteristics constructed on the basis of moduli of continuity.

We show the results corresponding to theorems of S. Lal [Appl. Math. Comput., 209 (2009) 346-350] on the rate of approximation of functions from the generalized integral Lipschitz classes by matrix summability means of their Fourier series as well as to the authors theorems [Acta Comment. Univ. Tartu. Math., 13 (2009), 11-24] also on such approximations.

We generalize and extend the some results of the paper [6]. Considering a wider class of function and more general means we obtain the results of the V. Totik type [8, 9].

We consider the class GM(₂β) in pointwise estimate of the deviations in strong mean of almost periodic functions from matrix means of partial sums of their Fourier series.

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