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Strongly Summable Bivariate Measurable Functions of Weight g

Yıl 2020, Cilt: 15 Sayı: 1, 80 - 89, 31.05.2020
https://doi.org/10.29233/sdufeffd.694376

Öz

In 1965 Borwein presented the concept of strongly summable single valued functions. Using Borwein's results, in 2019 Patterson et all. introduced the notion of multidimensional linear functions connected with double strongly Cesaro summability theory. The aim of this study is to extend Patterson et all's results to strongly summable bivariate functions with respect to weight of g. To achieve this by considering a real valued non-negative bivariate measurable function defined on the interval (1,∞)×(1,∞) the concepts of double [W_(λ,μ)^g ]_f -strongly summable and [S_(λ,μ)^g ]_f-double statistical convergence of weight g will be introduced, where g:[0,∞)×[0,∞)→[0,∞) such that g(x_m,x_n )→∞ as x_m→∞ and x_n→∞. Also g is factorable. In addition, the relationship between these two concepts will be examined and some algebraic characterization of real valued lebesgue measurable bivariate functions will be also presented.

Kaynakça

  • [1] M. Balcerzak, P. Das, and M. Filipczak, J. Swaczyna, “Generalized kinds of density and the associated ideals,” Acta Math. Hungar., 147(1), 97-115, 2015.
  • [2] D. Borwein, “Linear functionals with strong Cesáro summability,” J. Lond. Math. Soc., 40, 628-634, 1965.
  • [3] R. C. Buck, “Generalized asymptotic density,” Amer. J. Math., 75 (1953), 335--346.
  • [4] J. S. Connor, “The statistical and strongly Cesáro convergence of sequences,” Analysis (Munich), 8(1-2) , 47-63, 1988.
  • [5] J. Connor, and M. Ganichev, V. Kadets, “A characterization of Banach Spaces with separable duals via week statistical convergence,” J. Math. Anal. Appl., 244, 251-261, 2000.
  • [6] R. Çolak, “Statistical convergence of order ,” Modern methods in Analysis and its Applications, New Delhi, India, Anamaya Pub., 121-129, 2010.
  • [7] R. Çolak, and C. A. Bektaş, “ -statistical convergence of order ,” A Acta Math. Sci. Ser. A Chin. Ed., 31B(3), 953-959, 2011.
  • [8] O. Duman, M. K. Khan, and C. Orhan, “A-statistical convergence of approximating operators,” Math. Inequal. Appl., 6, 689-699, 2003.
  • [9] P. Erdös, G. Tenenbaum, “Sur les densities de certaines suites d'entiers,” J. Lond. Math. Soc., 59(3), 417-438, 1989.
  • [10] M. Et, S. A. Mohiuddine, and A. Alotaibi, “On -statistical convergence and strongly -summable functions of order ,” J. Inequal. Appl., 2-8, 2013.
  • [11] H. Fast, “Sur la convergence statistique”. Colloq. Math., 2, 241-244, 1951.
  • [12] R. Freedman, and J. J. Sember, “Densities and summability,” Pacific J. Math., 95(2), 293-305, 1981.
  • [13] J. A. Fridy, “On statistical convergence”, Analysis (Munich), 5, 301-313, 1985.
  • [14] J.A. Fridy, and M. K. Khan, “Tauberian theorems via statistical convergence,” J. Math. Anal. Appl., 228, 73-95, 1988.
  • [15] A. D. Gadjiev, and C. Orhan, “Some approximation theorems via statistical convergence,” Rocky Mountain J. Math., 32(1), 508-520, 2002.
  • [16] L. Liendler, “ber die verallgemeinerte de la Vallee-Poussinsche Summierbarkeit allgemeiner Orthogonalreihen,” (German), Acta Math. Hungar., 16, 375-387, 1965.
  • [17] H. I. Miller, “A measure theoretical subsequence characterization of statistical convergence,” Trans. Amer. Math. Soc., 347, 1811-1819, 1995.
  • [18] F. Moricz, “Statistical convergence of multiple sequences,” Arch. Math., 81, 82-89, 2003.
  • [19] M. Mursaleen, “ -statistical convergence,” Math. Slovaca, 50(1), 111-115, 2000.
  • [20] M. Mursaleen, C. Cakan, and S. A. Mohiuddine, E. Savas, “Generalized statistical convergence and statistical core of double sequences,”Acta Math. Sin. (Engl. Ser.), 26(11), 2131-2144, 2010.
  • [21] M. Mursaleen, and O. H. Edely, “Statistical convergence of double sequences,” J. Math. Anal. Appl., 288, 223-231, 2003.
  • [22] M. Mursaleen, and C. Belen, “On statistical Lacunary summability of double sequences,” Filomat, 28(2), 231-239, 2014.
  • [23] M. Mursaleen, and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,” Chaos Solitons Fractals, 41, 2414-2421, 2009.
  • [24] F. Nuray, “ -strongly summable and -statistically convergent functions,” Iran. J. Sci. Technol. Trans. A Sci., A4(34), 335-339, 2010.
  • [25] S. Pehlivan, and M. A. Mamedov, “Statistical cluster points and turnpike,” Optimization, 48, 93-106, 2000.
  • [26] A. Pringsheim, “Zur theorie der zweifach unendlichen zahlen folgen,” Math. Ann., 53, 289-321, 1900.
  • [27] R. F. Patterson, R. Savaş, and E. Savaş, “Multidimensional linear functional connected with double strongly cesaro summability,” Indian J. Pure, App. Math., (accepted-preprint).
  • [28] R. Savas, “ double statistical convergence of function”, Filomat, 33(2), 519-524, 2019.
  • [29] R. Savaş, and R. F. Patterson, “I-lacunary strongly summability for multidimensional measurable,” preprint.
  • [30] I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Amer. Math. Monthly, 66, 361-375, 1959.
  • [31] H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloq. Math, 2, 73-74, 1951.
  • [32] A. Zygmund, Trigonometric Series. Cambridge, Cambridge University Press, 1979.

g Ağırlıklı Kuvvetli Toplanabilir İki Değişkenli Ölçülebilir Fonksiyonlar

Yıl 2020, Cilt: 15 Sayı: 1, 80 - 89, 31.05.2020
https://doi.org/10.29233/sdufeffd.694376

Öz

1965 yılında Borwein tek değişkenli kuvvetli toplanabilir fonksiyonları sunmuştur. Borwein'nin sonuçlarını kullanarak 2019 yılında Patterson ve diğerleri çift kuvvetli Cesaro toplanabilme teorisi ile bağlantılı olarak iki boyutlu lineer fonksiyonları tanımlamıştır. Bu makalenin amacı Patterson ve diğerlerinin sonuçlarının g ağırlığı ile ilişkili olarak kuvvetli toplanabilir iki değişkenli ölçülebilir fonksiyonlara genelleştirmektir. Bunu elde etmek için (1,∞)×(1,∞) aralığında tanımlı negatif olmayan reel değerli iki değişkenli ölçülebilir fonksiyonlar göz önüne alınarak eğer x_m→∞ ve x_n→∞ iken g(x_m,x_n )→∞ olacak şekilde g:[0,∞)×[0,∞)→[0,∞) ağırlıklı çift [W_(λ,μ)^g ]_f-kuvvetli toplanabilir ve [S_(λ,μ)^g ]_f-çift istatistiksel yakınsaklık kavramları sunulacaktır. Ayrıca, g fonksiyonu çarpanlarına ayrılabilir. Buna ek olarak, bu iki kavram arasındaki ilişki incelenecek ve reel değerli Lebesgue anlamında ölçülebilir iki değişkenli fonksiyonların bazı cebirsel özellikleri de verilecektir.

Kaynakça

  • [1] M. Balcerzak, P. Das, and M. Filipczak, J. Swaczyna, “Generalized kinds of density and the associated ideals,” Acta Math. Hungar., 147(1), 97-115, 2015.
  • [2] D. Borwein, “Linear functionals with strong Cesáro summability,” J. Lond. Math. Soc., 40, 628-634, 1965.
  • [3] R. C. Buck, “Generalized asymptotic density,” Amer. J. Math., 75 (1953), 335--346.
  • [4] J. S. Connor, “The statistical and strongly Cesáro convergence of sequences,” Analysis (Munich), 8(1-2) , 47-63, 1988.
  • [5] J. Connor, and M. Ganichev, V. Kadets, “A characterization of Banach Spaces with separable duals via week statistical convergence,” J. Math. Anal. Appl., 244, 251-261, 2000.
  • [6] R. Çolak, “Statistical convergence of order ,” Modern methods in Analysis and its Applications, New Delhi, India, Anamaya Pub., 121-129, 2010.
  • [7] R. Çolak, and C. A. Bektaş, “ -statistical convergence of order ,” A Acta Math. Sci. Ser. A Chin. Ed., 31B(3), 953-959, 2011.
  • [8] O. Duman, M. K. Khan, and C. Orhan, “A-statistical convergence of approximating operators,” Math. Inequal. Appl., 6, 689-699, 2003.
  • [9] P. Erdös, G. Tenenbaum, “Sur les densities de certaines suites d'entiers,” J. Lond. Math. Soc., 59(3), 417-438, 1989.
  • [10] M. Et, S. A. Mohiuddine, and A. Alotaibi, “On -statistical convergence and strongly -summable functions of order ,” J. Inequal. Appl., 2-8, 2013.
  • [11] H. Fast, “Sur la convergence statistique”. Colloq. Math., 2, 241-244, 1951.
  • [12] R. Freedman, and J. J. Sember, “Densities and summability,” Pacific J. Math., 95(2), 293-305, 1981.
  • [13] J. A. Fridy, “On statistical convergence”, Analysis (Munich), 5, 301-313, 1985.
  • [14] J.A. Fridy, and M. K. Khan, “Tauberian theorems via statistical convergence,” J. Math. Anal. Appl., 228, 73-95, 1988.
  • [15] A. D. Gadjiev, and C. Orhan, “Some approximation theorems via statistical convergence,” Rocky Mountain J. Math., 32(1), 508-520, 2002.
  • [16] L. Liendler, “ber die verallgemeinerte de la Vallee-Poussinsche Summierbarkeit allgemeiner Orthogonalreihen,” (German), Acta Math. Hungar., 16, 375-387, 1965.
  • [17] H. I. Miller, “A measure theoretical subsequence characterization of statistical convergence,” Trans. Amer. Math. Soc., 347, 1811-1819, 1995.
  • [18] F. Moricz, “Statistical convergence of multiple sequences,” Arch. Math., 81, 82-89, 2003.
  • [19] M. Mursaleen, “ -statistical convergence,” Math. Slovaca, 50(1), 111-115, 2000.
  • [20] M. Mursaleen, C. Cakan, and S. A. Mohiuddine, E. Savas, “Generalized statistical convergence and statistical core of double sequences,”Acta Math. Sin. (Engl. Ser.), 26(11), 2131-2144, 2010.
  • [21] M. Mursaleen, and O. H. Edely, “Statistical convergence of double sequences,” J. Math. Anal. Appl., 288, 223-231, 2003.
  • [22] M. Mursaleen, and C. Belen, “On statistical Lacunary summability of double sequences,” Filomat, 28(2), 231-239, 2014.
  • [23] M. Mursaleen, and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,” Chaos Solitons Fractals, 41, 2414-2421, 2009.
  • [24] F. Nuray, “ -strongly summable and -statistically convergent functions,” Iran. J. Sci. Technol. Trans. A Sci., A4(34), 335-339, 2010.
  • [25] S. Pehlivan, and M. A. Mamedov, “Statistical cluster points and turnpike,” Optimization, 48, 93-106, 2000.
  • [26] A. Pringsheim, “Zur theorie der zweifach unendlichen zahlen folgen,” Math. Ann., 53, 289-321, 1900.
  • [27] R. F. Patterson, R. Savaş, and E. Savaş, “Multidimensional linear functional connected with double strongly cesaro summability,” Indian J. Pure, App. Math., (accepted-preprint).
  • [28] R. Savas, “ double statistical convergence of function”, Filomat, 33(2), 519-524, 2019.
  • [29] R. Savaş, and R. F. Patterson, “I-lacunary strongly summability for multidimensional measurable,” preprint.
  • [30] I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Amer. Math. Monthly, 66, 361-375, 1959.
  • [31] H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloq. Math, 2, 73-74, 1951.
  • [32] A. Zygmund, Trigonometric Series. Cambridge, Cambridge University Press, 1979.
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Rabia SAVAS 0000-0002-4911-9067

Yayımlanma Tarihi 31 Mayıs 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 15 Sayı: 1

Kaynak Göster

IEEE R. SAVAS, “Strongly Summable Bivariate Measurable Functions of Weight g”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, c. 15, sy. 1, ss. 80–89, 2020, doi: 10.29233/sdufeffd.694376.

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