Research Article
Abstract
References
- [1] Hardy G.H., Divergent series. 1st ed. Oxford: Clarendon Press, (1949).
- [2] Meyer-König W., Zeller K., Kronecker-Ausdruck und Kreisverfahren der Limitierungstheorie, Math. Z., 114 (1970), 300-302.
- [3] Schmidt R., Über Divergente Folgen und Lineare Mittelbildungen, Math. Z., 22 (1925), 89-152.
- [4] Korevaar J., Tauberian theory: A century of developments. 1st ed. Berlin: Springer-Verlag, (2004).
- [5] Peyerimhoff A., Lectures on summability, Lecture notes in mathematics. vol. 107 Berlin: Springer-Verlag, (1969).
- [6] Çanak İ., Braha N.L., Totur Ü., A Tauberian Theorem for the Generalized Nörlund Summability Method, Georgian Math. J., 27 (2020), 31-36.
- [7] Sezer S.A., Çanak İ., Tauberian Conditions of Slowly Decreasing Type for the Logarithmic Power Series Method, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 90 (2020), 135-139.
- [8] Sezer S.A., Çanak İ, Tauberian Conditions under which Convergence follows from Summability by the Discrete Power Series Method, Turk. J. Math., 43 (2019), 2898-2907.
- [9] Knopp K., Über das Eulersche Summierungs-verfahren II., Math. Z., 18 (1923), 125-156.
- [10] Dik M., Tauberian Theorems for Sequences with Moderately Oscillatory Control Modulo, Math. Morav., 5 (2001), 57-94.
- [11] Tam L., A Tauberian Theorem for the General Euler-Borel Summability Method, Can. J. Math., 44 (1992) 1100–1120.
On the Euler method of summability and concerning Tauberian theorems
Abstract
For any two regular summability methods (U) and (V), the condition under which V-limx_n=λ implies U-limx_n=λ is called a Tauberian condition and the corresponding theorem is called a Tauberian theorem. Usually in the theory of summability, the case in which the method U is equivalent to the ordinary convergence is taken into consideration. In this paper, we give new Tauberian conditions under which ordinary convergence or Cesàro summability of a sequence follows from its Euler summability by means of the product theorem of Knopp for the Euler and Cesàro summability methods.
References
- [1] Hardy G.H., Divergent series. 1st ed. Oxford: Clarendon Press, (1949).
- [2] Meyer-König W., Zeller K., Kronecker-Ausdruck und Kreisverfahren der Limitierungstheorie, Math. Z., 114 (1970), 300-302.
- [3] Schmidt R., Über Divergente Folgen und Lineare Mittelbildungen, Math. Z., 22 (1925), 89-152.
- [4] Korevaar J., Tauberian theory: A century of developments. 1st ed. Berlin: Springer-Verlag, (2004).
- [5] Peyerimhoff A., Lectures on summability, Lecture notes in mathematics. vol. 107 Berlin: Springer-Verlag, (1969).
- [6] Çanak İ., Braha N.L., Totur Ü., A Tauberian Theorem for the Generalized Nörlund Summability Method, Georgian Math. J., 27 (2020), 31-36.
- [7] Sezer S.A., Çanak İ., Tauberian Conditions of Slowly Decreasing Type for the Logarithmic Power Series Method, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 90 (2020), 135-139.
- [8] Sezer S.A., Çanak İ, Tauberian Conditions under which Convergence follows from Summability by the Discrete Power Series Method, Turk. J. Math., 43 (2019), 2898-2907.
- [9] Knopp K., Über das Eulersche Summierungs-verfahren II., Math. Z., 18 (1923), 125-156.
- [10] Dik M., Tauberian Theorems for Sequences with Moderately Oscillatory Control Modulo, Math. Morav., 5 (2001), 57-94.
- [11] Tam L., A Tauberian Theorem for the General Euler-Borel Summability Method, Can. J. Math., 44 (1992) 1100–1120.
There are 11 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | March 29, 2021 |
| Submission Date | December 1, 2020 |
| Acceptance Date | March 17, 2021 |
| Published in Issue | Year 2021 Volume: 42 Issue: 1 |