Yusuf ZEREN, Mıqdad ISMAILOV, Cemil KARAÇAM

Korovkin-type theorems and their statistical versions in grand Lebesgue spaces

The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace G p −π; π of grand Lebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions is dense in G p −π; π . The analogs of Korovkin theorems are proved in G p −π; π . These results are established in G p −π; π in the sense of statistical convergence. The obtained results are applied to a sequence of operators generated by the Kantorovich polynomials, to Fejer and Abel-Poisson convolution operators.

Keywords:

Grand Lebesgue space, Korovkin theorems, shift operator statistical convergence, positive linear operator, approximation process,

PDF

___

[1] Adams DR. Morrey Spaces. Switzerland: Springer International Publishing, 2015.
[2] Altomare F. Korovkin type theorems and approximation by positive linear operators. Surveys in Approximation Theory 2010; 6: 92-164.
[3] Bilalov BT, Guseynov ZG. Basicity of a system of exponents with a piece-wise linear phase in variable spaces. Mediterranean Journal of Mathematics 2012; 9 (3): 487-498. doi: 10.1007/s00009-011-0135-7
[4] Bilalov BT, Guseynov ZG. Basicity criterion of perturbed system of exponents in Lebesgue spaces with variable summability. Dokladi Akademii Nauk 2011; 436 (5): 586-589. (in Russian)
[5] Bilalov BT, Nazarova TY. On Statistical Convergence in Metric Spaces. Journal of Mathematics Research 2015; 7 (1): 37-43. doi: 10.5539/jmr.v1n1p37
[6] Bilalov BT, Nazarova TY. Statistical convergence on functional sequences. Rocky Mountain Journal of Mathematics 2015; 45 (5): 1413-1423. doi: 10.1216/ RMJ-2015-45-5-1413
[7] Bilalov BT, Nazarova TY. On statistical type convergence in uniform spaces. Bulletin of Iranian Mathematical Society 2016; 42 (4): 975-986.
[8] Bilalov BT, Sadigova SR. On µ-statistical convergence. Proceedings of the American Mathematical Society 2015; 143 (9): 3869-3878.
[9] Bilalov BT . The basis property of a perturbed system of exponentials in Morrey-type spaces. Siberian Mathematical Journal 2019; 60 (2): 249-271. doi: 10-1134/S003744661902006X
[10] Bilalov BT, Gasymov TB, Guliyeva AA. On the solvability of the Riemann boundary value problem in Morrey-Hardy classes. Turkish Journal of Mathematics 2016; 40 (50): 1085-1101. doi: 10.3906/mat-1507-10
[11] Bilalov BT , Huseynli AA, El-Shabrawy SR. Basis Properties of Trigonometric Systems in Weighted Morrey Spaces. Azerbaijan Journal of Mathematics 2019; 9 (2): 166-192
[12] Bilalov BT, Seyidova F. Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces. Turkish Journal of Mathematics 2019; 43: 1850-1866. doi: 10.3906/mat-1901-113
[13] Brown TC, Freedman AR. The uniform density of sets of integers and Fermat’s last Theorem. Comptes Rendus Mathematiques de L’Academie des Sciences 1990; 12 : 1-6.
[14] Capone CC, Fiorenza A. On small Lebesgue spaces. Journal of Function Spaces and Applications 2005; 3 (1): 73-89. doi:10.1155/2005/192538
[15] Castilo RE, Rafeiro H. An Introductory Course in Lebesgue Spaces. Switzerland: Springer International Publishing, 2016.
[16] Connor BJ. The statistical and strong p-Cesaro convergence of sequences. Analysis 1988; 8: 47-63 . doi: 10.1524/anly.1988.8.12.47
[17] Cruz-Uribe DV, Fiorenza A. Variable Lebesgue spaces: Foundations and Harmonic Analysis. New York, USA: Springer Science and Business Media, 2013.
[18] D’onofrio LL, Sbordone C, Schiattarella R. Grand Sobolev spaces and their application in geometric function theory and PDEs, Journal of Fixed Point Theory Applications 2013; 13: 309-340.
[19] Erdos R, Tenenbaum G. Sur les densities de certaines suites dentiers. Proceedings of the London Mathematical Society 1989; 59 (3): 417-438. doi: 10.1112/plms/s3-59.3.417
[20] Fast H. Sur la convergence statistique. Colloquium Mathematicum 1951; 2: 241-244.
[21] Fiorenza A. Duality and reflexivity in grand Lebesgue spaces. Collectanea Mathematica 2000; 51 (2): 131-148.
[22] Fiorenza A, Karadzhov G E. Grand and small Lebesgue spaces and their analogs. Zeitschrift für Analysis und ihre Anwendungen=Journal of analysis and its applications 2004 ; 23 (4): 657-681. doi: 10.4171/ZAA/1215
[23] Fiorenza A, Krbec M, On the domain and range of the maximal operator. Nagoya Mathematical Journal 2000; 158: 43-61. doi: 10.1017 /S0027763000007285
[24] Fratta GDi, Fiorenza A. A direct approach to the duality of grand and small Lebesgue spaces. Nonlinear Anaysis 2009; 70 (7): 2582-2592. doi: 10.1016/j.na.2008.03.044
[25] Freedman AR, Sember JJ. Densities and summability. Pacific Journal of Mathematics 1981; 95 (2): 293-305. doi: 10.2140/pjm.1981.95.293
[26] Fridy JA. On statistical convergence. Analysis 1985; 5: 301-313. doi: 10.1524/anly.1985.5.4.301
[27] Fridy JA, Khan MK. Tauberian theorems via statistical convergence Journal of Mathematical Analysis and Applications 1998; 228 (1): 73-95. doi: 10.1006/jmaa.1998.6118
[28] Gupta B, Fiorenza A, Jain P. The maximal theorem in weighted grand Lebesgue spaces. Studia Mathematica 2008; 188 (2): 123-133. doi: 10.4064/sm188-2-2
[29] Iwaniec T, Sbordone C. Riesz transforms and elliptic PDEs with VMO coefcients, Journal d’Analyse Mathematique 1998; 74: 183-212. doi: 10.1007/BF02819450
[30] Kokilashvili VM, Meskhi A, Rafeiro H, Samko S. Integral Operators in Non-standard Function Spaces: Volume1: Variable Exponent Lebesgue and Amalgam Spaces . Birkhauser, 2016.
[31] Kokilashvili V. Boundedness criteria for singular integrals in weighted grand lebesque spaces. Journal of Mathematical Sciences 2010; 170: 20-33. doi. 10.1007/s10958-010-0076-X
[32] Kokilashvili V. Boundedness criterion for the Cauchy singular integral operator in weighted grand Lebesgue spaces and application to the Riemann problem. Proceedengs.of A.Razmadze Mathematical Institute. 2009; 151: 129-133.
[33] Kuchukaslan M, Deger U, Dovgoshey O. On the statistical convergence of metric valued sequences. Ukrainian Mathematical Journal 2014; 66 (5): 796-805. doi: 10.1007/s11253-014-0974-z
[34] Kuchukaslan M, Deger U. On statistical boundedness of metric valued sequences. European Journal of Pure and Applied Mathematics 2012; 5 (2): 174-186.
[35] Maddox IJ, Statistical convergence in a locally convex space. Mathematical Proceedings of the Cambridge Philosophical Society 1998; 104: 141-145. doi: 10.1017/s0305004100065312[13] Brown TC, Freedman AR. The uniform density of sets of integers and Fermat’s last Theorem. Comptes Rendus Mathematiques de L’Academie des Sciences 1990; 12 : 1-6.
[36] Maharam D. Finitely additive measures on the integers. Sankhya:The Indian Jounal of Statistics 1976; 38: 44-59.
[37] Maio GD, Kocinac LDR. Statistical convergence in topology. Topology and its Applications 2008; 156 (1): 28-45. doi: 10.1016/j.topol.2008.01.015
[38] Morrey CB.On the solutions of quasi-linear elliptic partial differential equations. Transactions of the American Mathematical Society 1938; 43 : 126-166. doi: 10.1090/S0002-9947-1938-1501936-8
[39] Peterson GM. Regular Matrix Tramformations. London, UK: McGraw-Hill, 1966.
[40] Rafeiro H, Vargas A. On the compactness in grand spaces. Georgian Mathematical Journal 2015; 22 (1) : 141-152. doi: 10-1515/gmj-2014-0060
[41] Sadigova SR, Hasanli RR, Karacam C. On a space of µ- statistical continous functions. Proceeding of the Institute of Mathematics and Mechanics 2018; 44 (1): 70-80.
[42] Samko N. Weight Hardy and singular operators in Morrey spaces. Journal of Mathematical Analysis and Applications 2009; 350 (1): 56-72 . doi: 10.1016/j.jmaa.2008.09.021
[43] Samko SG, Umarkhadzhiev SM. On Iwaniec-Sbordone spaces on sets which may have infinite measure. Azerbaijan Journal of Mathematics 2011; 1 (1): 67-84.
[44] Schoenberg IJ. The integrability of certain functions and related summability methods. The American Mathematical Monthly 1959; 66 (5): 361-375. doi: 10.1080/00029890.1959.11989303
[45] Sharapudinov II. On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces. Azerbaijan Journal of Mathematics 2014; 4 (2): 55-72.
[46] Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicum 1951; 2: 73-74.
[47] Xianling F, Dun Z. On the spaces L p(x) (Ω) and W m,p(x) (Ω). Journal of Mathematical Analysis and Appications 2001; 263 (1): 424-446. doi: 10.1006/jmaa.2000.7617
[48] Zeren Y. Some questions of harmonic analysis in weighted Morrey type spaces. Afrika Mathematica 2019; 30 (4): 129-150. doi:10.1007/s13370-018-0622-0
[49] Zorko CT. Morrey spaces. Proceedings of the American Mathematical Society.1986; 98: 586-592.
[50] Zygmund AA. Trigonometric Series: Volume 1. 2nd ed. Cambridge, UK: Cambridge University Press, 1959.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv