On the approximation of Fourier series by Abel means

This paper deals with the pointwise saturation problem for the Abel means of Fourier series. Most of the known results on pointwise approximation in connection with summation processes of Fourier series are Jackson-type theorems. (See the books of N. Achieser, P. P. Korovkin, and I. P. Natanson on approximation theory. We wish also to refer to the results due to G. Alexits and his school, cf. [I] and [5]). Here we shall study a Bernstein-type problem for the Abel means, i.e., a converse of the Jackson-type problems. But we shall study it only in case of saturated approximation. Although this type of question has been studied for several years, only a few results are available. BajSanski-BojaniC [3] proved a pointwise “0”-theorem for the Bernstein polynomials, and recently V. A. Andrienko [2] studied this problem for the FejCr means of Fourier series. A generalization of his result was given by the author [4]. On the other hand, these problems are closely connected with theorems concerning generalized derivatives of functions which have their origin in Schwas we denote