On the Euler method of summability and concerning Tauberian theorems

Yıl 2021, Cilt: 42 Sayı: 1, 141 - 144, 29.03.2021

Sefa Anıl Sezer İbrahim Çanak

https://doi.org/10.17776/csj.834216

Öz

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.

Anahtar Kelimeler

Euler summability, Cesàro summability, Tauberian theorem, order condition

Kaynakça

• [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.