TAUBERIAN THEOREMS FOR THE PRODUCT OF BOREL AND LOGARITHMIC METHODS OF SUMMABILITY

Yıl 2019, Cilt: 9 Sayı: 1, 165 - 171, 01.03.2019

S. A. Sezer

Öz

In this paper, we show that if a sequence is summable by the product method B `; k , then it is also summable by the logarithmic method `; k , provided two-sided conditions of Hardy-type are satised. We also obtain some classical Tauberian theorems and their generalizations as special cases of our main theorems.

Anahtar Kelimeler

Tauberian theorems, Borel summability, logarithmic summability, Hardytype conditions.

Kaynakça

• Borwein, D., (1958), A logarithmic method of summability, J. Lond. Math. Soc., 33, pp. 212–220.
• Braha, N. L., (2016), A Tauberian theorem for the generalized N¨orlund-Euler summability method
• J. Inequal. Spec. Funct., 7 (4), pp. 137–142. C¸ anak ˙I., (2014), Some extended Tauberian theorems for (A)(k)(C, α) summability method, Acta Sci., Technol., 36 (4), pp. 679–683.
• C¸ anak, ˙I., Erdem, Y. and Totur, (2010), ¨U., Some Tauberian theorems for (A)(C, α) summability method, Math. Comput. Modelling, 52 (5-6), pp. 738–743.
• Erdem, Y. and C¸ anak, ˙I., (2016), A Tauberian theorem for the product of Abel and Ces`aro summa- bility methods, Georgian Math. J., 23 (3), pp. 343–350.
• Erdem, Y., C¸ anak, ˙I. and Allahverdiev, B., (2015), Two theorems on the product of Abel and Ces`aro summability methods, C. R. Acad. Bulg. Sci., 68 (3), pp. 287–294.
• Hardy, G. H., (1949), Divergent Series, Clarendon Press, Oxford.
• Ishiguro, K., (1963), Tauberian theorems concerning the summability methods of logarithmic type, Proc. Japan Acad., 39, pp. 156–159.
• Kwee, B., (1967), A Tauberian theorem for the logarithmic method of summation, Proc. Cambridge Philos. Soc., 63 (2), pp. 401–405.
• Kwee, B., (1968), Some Tauberian theorems for the logarithmic method of summability, Canad. J. Math., 20, pp. 1324–1331.
• Kwee, B., (1989), The relation between the Borel and Riesz methods of summation, Bull. London Math. Soc., 21 (4), pp. 387–393.
• Loku, V. and Braha, N. L., (2017), Tauberian theorems by weighted summability method, Armen. J. Math., 9 (1), pp. 35–42.
• Mure¸san, M., (2008), A Concrete Approach to Classical Analysis, Springer, Berlin.
• M´oricz, F., (2004), Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences, Analysis (Munich), 24 (2), pp. 127–145.
• Parameswaran, M. R., (1957), Some product theorems in summability, Math. Z., 68, pp. 19–26.
• Parameswaran, M. R., (1994), New Tauberian theorems from old, Canad. J. Math., 46 (2), pp. 380–
• Ramanujan, M. S., (1958), On products of summability methods, Math. Z., 69, pp. 423–428.
• Sezer, S. A. and C¸ anak, ˙I., (2018), Tauberian theorems for the summability methods of logarithmic type, Bull. Malays. Math. Sci. Soc., 41 (4), pp. 1977–1994.
• Sz´asz, O., (1952), On products of summability methods, Proc. Amer. Math. Soc., 3, pp. 257–263.
• Totur, ¨U. and Okur, M. A., (2016), On logarithmic averages of sequences and its applications, Kuwait J. Sci., 43 (4), pp. 56–67.