Research Article
Year 2021,
Volume: 42 Issue: 1, 141 - 144, 29.03.2021
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.
Year 2021,
Volume: 42 Issue: 1, 141 - 144, 29.03.2021
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 | Research Article |
| 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 |