Abstract
The present manuscript aims to extend a Tauberian theorem previously established for the Cesàro and weighted mean summability methods of single sequences in ordered spaces to the logarithmic summability method, also known as the (ℓ,1,1) method, for double sequences. In order to achieve this, we present several Tauberian conditions which address the O_L-oscillatory behavior of a double sequence (s_mn ) with respect to logarithmic summability in certain senses. These conditions facilitate the transition from (ℓ,1,1), (ℓ,1,0), and (ℓ,0,1) summability to P-convergence in ordered spaces.
Keywords
Double sequences ordered linear spaces slowly decreasing sequences with respect to (l,1,1) Tauberian conditions Tauberian theorems logarithmic summability method
Ethical Statement
Ethical approval is not applicable for this article.
References
- Ishiguro, K., On the summability methods of logarithmic type, Proceedings of the Japan Academy, 38, 703-705, (1962).
- Ishiguro, K., A converse theorem on the summability methods, Proceedings of the Japan Academy, 39, 38-41, (1963).
- Hardy, G. H., Divergent Series, Clarendon Press, Oxford, (1949).
- Szász, O., Introduction to the theory of divergent series, University of Cincinnati, Ohio, (1952).
- Ishiguro, K., Tauberian theorems concerning the summability method of logarithmic type, Proceedings of the Japan Academy, 39, 156-159, (1963).
- Ishiguro, K., A note on the logarithmic means, Proceedings of the Japan Academy, 39, 575-577, (1963).
- Kwee, B., A Tauberian theorem for the logarithmic method of summation, Proceedings of the Cambridge Philosophical Society, 63, 401-405, (1966).
- Kwee, B., Some Tauberian theorems for the logarithmic method of summability, Canadian Journal of Mathematics, 20, 1324-1331, (1968).
- Kwee, B., On generalized logarithmic methods of summation, Journal of Mathematical Analysis and Applications, 35, 83-89, (1971).
- Rangachari, M. S., and Sitaraman, Y., Tauberian theorems for logarithmic summability (L), The Tohoku Mathematical Journal (2), 16, 257-269, (1964).
- Kaufman, B. L., Theorems of Tauberian type for logarithmic methods of summation, Izvestija Vysših Učebnyh Zavedeniĭ Matematika, 1, 56, 57-62, (1967).
- Kohanovskiĭ, A. P., Theorems of Tauberian type for a semicontinuous logarithmic method of summability of series, Ukrainskiĭ Matematičeskiĭ Žurnal, 26, 740-748, 861, (1974).
- Kohanovskiĭ, A. P., A condition for the equivalence of logarithmic summability methods, Ukrainskiĭ Matematičeskiĭ Žurnal, 27, 229-234, 285, (1975).
- Burljai, M. F., The logarithmic method for the summability of numerical double series, Izvestija Vysših Učebnyh Zavedeniĭ Matematika, 3, 166, 95-98, (1976).
- Móricz, F., Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences, Analysis, 24, 2, 127-145, (2004).
- Móricz, F., Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences, Studia Mathematica, 219, 2, 109-121, (2013).
- Alghamdi, M. A., Mursaleen, M. and Alotaibi, A., Logarithmic density and logarithmic statistical convergence, Advances in Difference Equations, 2013:227, 6, (2013).
- Totur, Ü. and Okur, M. A., On logarithmic averages of sequences and its applications, Kuwait Journal of Science, 43, 4, 56-67, (2016).
- Sezer, S. A. and Çanak, İ., Tauberian theorems for the summability methods of logarithmic type, Bulletin of the Malaysian Mathematical Sciences Society, 41, 4, 1977-1994, (2018).
- Sezer, S. A. and Çanak, İ., Tauberian conditions of slowly decreasing type for the logarithmic power series method, Proceedings of the National Academy of Sciences, India, Section A: Physical Sciences, 90, 1, 135-139, (2020).
- Çınar, N. and Çanak, İ., Necessary and sufficient Tauberian conditions under which statistically logarithmic convergence follows from statistically logarithmic summability, Journal of Classical Analysis, 21, 1, 29-34, (2023).
- Okur, M. A., General logarithmic control modulo and Tauberian remainder theorems, Communications, Faculty of Sciences, University of Ankara, Series A1: Mathematics and Statistics, 73, 2, 391-398, (2024).
- Maddox, I. J., A Tauberian theorem for ordered spaces, Analysis, 9, 3, 297-302, (1989).
- Çanak, I., A Tauberian theorem for a weighted mean method of summability in ordered spaces, National Academy Science Letters, 43, 6, 553–555, (2020).
- Pringsheim, A., Zur Theorie der zweifach unendlichen Zahlenfolgen, Mathematische Annalen, 53, 3, 289-321, (1900).
- Tauber, A., Ein Satz aus der Theorie der unendlichen Reihen, Monatshefte für Mathematik und Physik, 8, 1, 273-277, (1897).
- Knopp, K. Limitierungs-Umkehrsätze für Doppelfolgen, Mathematische Zeitschrift, 45, 573-589, (1939).
- Totur, Ü., Classical Tauberian theorems for the (C, 1, 1) summability method, Analele Ştiinţifice ale Universităţii “Alexandru Ioan Cuza” din Iaşi Serie Nouă Matematică, 61, 2, 401-414, (2015).
- Schmidt, R., Über divergente Folgen und lineare Mittelbildungen, Mathematische Zeitschrift, 22, 1, 89-152, (1925).
Abstract
Bu çalışma daha önce sıralı uzaylardaki tek katlı dizilerin Cesàro ve ağırlıklı ortalama toplanabilirlik yöntemleri için oluşturulmuş Tauber tipi teoremleri, iki katlı diziler için logaritmik toplanabilirlik yöntemine, diğer adıyla (ℓ,1,1) yöntemine genişletmeyi amaçlar. Bu amaçla, çeşitli anlamlarda logaritmik toplanabilirliğe göre iki katlı bir (s_mn ) dizinin O_L-salınım davranışını ele alan birkaç Tauber tipi koşul sunuyoruz. Bu koşullar, sıralı uzaylarda dizinin (ℓ,1,1), (ℓ,1,0) ve (ℓ,0,1) toplanabilirliğinden P-yakınsaklığına geçişine olanak sağlar.
Keywords
Çift diziler sıralı doğrusal uzaylar (l,1,1) metoduna göre yavaş azalan diziler Tauber koşullar Tauber teoremler logaritmik toplanabilme metodu
References
- Ishiguro, K., On the summability methods of logarithmic type, Proceedings of the Japan Academy, 38, 703-705, (1962).
- Ishiguro, K., A converse theorem on the summability methods, Proceedings of the Japan Academy, 39, 38-41, (1963).
- Hardy, G. H., Divergent Series, Clarendon Press, Oxford, (1949).
- Szász, O., Introduction to the theory of divergent series, University of Cincinnati, Ohio, (1952).
- Ishiguro, K., Tauberian theorems concerning the summability method of logarithmic type, Proceedings of the Japan Academy, 39, 156-159, (1963).
- Ishiguro, K., A note on the logarithmic means, Proceedings of the Japan Academy, 39, 575-577, (1963).
- Kwee, B., A Tauberian theorem for the logarithmic method of summation, Proceedings of the Cambridge Philosophical Society, 63, 401-405, (1966).
- Kwee, B., Some Tauberian theorems for the logarithmic method of summability, Canadian Journal of Mathematics, 20, 1324-1331, (1968).
- Kwee, B., On generalized logarithmic methods of summation, Journal of Mathematical Analysis and Applications, 35, 83-89, (1971).
- Rangachari, M. S., and Sitaraman, Y., Tauberian theorems for logarithmic summability (L), The Tohoku Mathematical Journal (2), 16, 257-269, (1964).
- Kaufman, B. L., Theorems of Tauberian type for logarithmic methods of summation, Izvestija Vysših Učebnyh Zavedeniĭ Matematika, 1, 56, 57-62, (1967).
- Kohanovskiĭ, A. P., Theorems of Tauberian type for a semicontinuous logarithmic method of summability of series, Ukrainskiĭ Matematičeskiĭ Žurnal, 26, 740-748, 861, (1974).
- Kohanovskiĭ, A. P., A condition for the equivalence of logarithmic summability methods, Ukrainskiĭ Matematičeskiĭ Žurnal, 27, 229-234, 285, (1975).
- Burljai, M. F., The logarithmic method for the summability of numerical double series, Izvestija Vysših Učebnyh Zavedeniĭ Matematika, 3, 166, 95-98, (1976).
- Móricz, F., Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences, Analysis, 24, 2, 127-145, (2004).
- Móricz, F., Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences, Studia Mathematica, 219, 2, 109-121, (2013).
- Alghamdi, M. A., Mursaleen, M. and Alotaibi, A., Logarithmic density and logarithmic statistical convergence, Advances in Difference Equations, 2013:227, 6, (2013).
- Totur, Ü. and Okur, M. A., On logarithmic averages of sequences and its applications, Kuwait Journal of Science, 43, 4, 56-67, (2016).
- Sezer, S. A. and Çanak, İ., Tauberian theorems for the summability methods of logarithmic type, Bulletin of the Malaysian Mathematical Sciences Society, 41, 4, 1977-1994, (2018).
- Sezer, S. A. and Çanak, İ., Tauberian conditions of slowly decreasing type for the logarithmic power series method, Proceedings of the National Academy of Sciences, India, Section A: Physical Sciences, 90, 1, 135-139, (2020).
- Çınar, N. and Çanak, İ., Necessary and sufficient Tauberian conditions under which statistically logarithmic convergence follows from statistically logarithmic summability, Journal of Classical Analysis, 21, 1, 29-34, (2023).
- Okur, M. A., General logarithmic control modulo and Tauberian remainder theorems, Communications, Faculty of Sciences, University of Ankara, Series A1: Mathematics and Statistics, 73, 2, 391-398, (2024).
- Maddox, I. J., A Tauberian theorem for ordered spaces, Analysis, 9, 3, 297-302, (1989).
- Çanak, I., A Tauberian theorem for a weighted mean method of summability in ordered spaces, National Academy Science Letters, 43, 6, 553–555, (2020).
- Pringsheim, A., Zur Theorie der zweifach unendlichen Zahlenfolgen, Mathematische Annalen, 53, 3, 289-321, (1900).
- Tauber, A., Ein Satz aus der Theorie der unendlichen Reihen, Monatshefte für Mathematik und Physik, 8, 1, 273-277, (1897).
- Knopp, K. Limitierungs-Umkehrsätze für Doppelfolgen, Mathematische Zeitschrift, 45, 573-589, (1939).
- Totur, Ü., Classical Tauberian theorems for the (C, 1, 1) summability method, Analele Ştiinţifice ale Universităţii “Alexandru Ioan Cuza” din Iaşi Serie Nouă Matematică, 61, 2, 401-414, (2015).
- Schmidt, R., Über divergente Folgen und lineare Mittelbildungen, Mathematische Zeitschrift, 22, 1, 89-152, (1925).
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | January 16, 2025 |
| Publication Date | January 20, 2025 |
| Submission Date | September 26, 2024 |
| Acceptance Date | December 11, 2024 |
| Published in Issue | Year 2025 Volume: 27 Issue: 1 |