On the absolute Riesz summability of orthogonal series

On Absolute Almost Generalized Nörlund Summability of Orthogonal Series

“…After several articles in which we have dealt with absolute summability of an orthogonal series we proved some general results which include all of the theorems that had been proved previously by Okuyama [23], Okuyama and Tsuchikura [20], and gave some new consequences, as well [6]- [11]. This notion motivated us to consider not simply absolute almost generalized Nörlund summability of an orthogonal series but its absolute almost generalized Nörlund summability of order k, 1 ≤ k ≤ 2.…”

“…The absolute Nörlund summability, the absolute generalized Nörlund summability, the absolute Riesz summability, absolute generalized Cesàro summability, absolute Euler summability of an orthogonal series has been studied by many authors. For example, one can see the work of Tandori [27], Leindler [12]- [15], Okuyama and Tsuchikura [20], Okuyama [18]- [22], Szalay [26], Billard [2], Grepaqevskaya [4], Spevakov and Kudrajatsev [25]. In 2002 Okuyama [23] using generalized Nörlund means has proved two theorems which give sufficient conditions in terms of the coefficients of an orthogonal series under which it is absolute generalized Nörlund summable almost everywhere.…”

An application ofq-mathematics on absolute summability of orthogonal series

“…In one hand, as a recent result can be mentioned those of Y. Okuyama (see Section 2) who has proved two theorems which give sufficient conditions in terms of the coefficients of an orthogonal series under which such series would be absolute generalized Nörlund summable almost everywhere. In the second hand, an interested reader could find some new results, see [7]- [9], where are given some statements which include all of the results previously proved by Y. Okuyama and T. Tsuchikura [1,2], and also are given some new consequences. In order to make an advance study in this direction, here we introduce the q-absolute Cesàro summability which distinguishes essentially from ordinary absolute Cesàro summability introduced earlier.…”

On absolute generalized Hausdorff summability of orthogonal series

“…On the other hand, by many authors such notions are employed to study an interesting topic in theory of orthogonal series the so-called their absolute summability. For example, one can see the work of Tandori [19], Leindler [8][9][10][11], Okuyama and Tsuchikura [15], Okuyama [13,14], Szalay [17], Billard [1], Grepaqevskaya [4], Spevakov and Kudrajatsev [16]. In 2002 Okuyama [14] using generalized Nörlund means has proved two theorems which give sufficient conditions in terms of the coefficients of an orthogonal series under which it is absolute generalized Nörlund summable almost everywhere.…”