Abstract
We first present a notion of a double lacunary sequence on product time scales. Using this notion, we define the notions of the double lacunary statistical convergence and double lacunary strongly p-Cesàro summability of 2-multiple functions on product time scales and we study some fundamental properties of both notions. We also present a theorem that connects the above-mentioned two concepts. Furthermore, we define a refinement of a double lacunary sequence on product time scales and provide some fundamental properties as well as inclusion theorems for a refined and a non-refined double lacunary sequence on product time scales.
Acknowledgements
We are thankful to the referees for their valuable comments and suggestions.
References
[1] R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999), no. 1–2, 3–22. 10.1007/BF03322019Search in Google Scholar
[2] B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, Qualitative Theory of Differential Equations (Szeged 1988), Colloq. Math. Soc. János Bolyai 53, North-Holland, Amsterdam (1990), 37–56. Search in Google Scholar
[3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0201-1Search in Google Scholar
[4] R. C. Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335–346. 10.2307/2372456Search in Google Scholar
[5] A. Cabada and D. R. Vivero, Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: Application to the calculus of Δ-antiderivatives, Math. Comput. Modelling 43 (2006), no. 1–2, 194–207. 10.1016/j.mcm.2005.09.028Search in Google Scholar
[6] C. Çakan, B. Altay and H. Çoşkun, Double lacunary density and lacunary statistical convergence of double sequences, Studia Sci. Math. Hungar. 47 (2010), no. 1, 35–45. 10.1556/sscmath.2009.1110Search in Google Scholar
[7] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. 10.4064/cm-2-3-4-241-244Search in Google Scholar
[8] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), no. 1, 43–51. 10.2140/pjm.1993.160.43Search in Google Scholar
[9] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003), no. 1, 107–127. 10.1016/S0022-247X(03)00361-5Search in Google Scholar
[10] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1–2, 18–56. 10.1007/BF03323153Search in Google Scholar
[11] S. Hilger, Special functions, Laplace and Fourier transform on measure chains, Dynam. Systems Appl. 8 (1999), no. 3–4, 471–488. Search in Google Scholar
[12] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2006), no. 2, 391–415. 10.1016/j.cam.2005.02.011Search in Google Scholar
[13] F. Móricz, Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2003), no. 1, 82–89. 10.1007/s00013-003-0506-9Search in Google Scholar
[14] R. F. Patterson and E. Savaş, Lacunary statistical convergence of double sequences, Math. Commun. 10 (2005), no. 1, 55–61. Search in Google Scholar
[15] E. Savaş, Lacunary statistical convergence of double sequences in topological groups, J. Inequal. Appl. 2014 (2014), Paper No. 480. 10.1186/1029-242X-2014-480Search in Google Scholar
[16] E. Savaş and R. F. Patterson, Lacunary statistical convergence of multiple sequences, Appl. Math. Lett. 19 (2006), no. 6, 527–534. 10.1016/j.aml.2005.06.018Search in Google Scholar
[17] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. 10.2307/2308747Search in Google Scholar
[18] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), no. 1, 73–74. Search in Google Scholar
[19] C. Turan and O. Duman, Convergence methods on time scales, AIP Conf. Proc. 1558 (2013), 1120–1123. 10.1063/1.4825704Search in Google Scholar
[20] C. Turan and O. Duman, Statistical convergence on time scales and its characterizations, Advances in Applied Mathematics and Approximation Theory, Springer Proc. Math. Stat. 41, Springer, New York (2013), 57–71. 10.1007/978-1-4614-6393-1_3Search in Google Scholar
[21] E. Yilmaz, Y. Altin and H. Koyunbakan, Statistical convergence of multiple sequences on a product time scale, Georgian Math. J. 27 (2020), no. 3, 485–492. 10.1515/gmj-2018-0051Search in Google Scholar
[22] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge University, Cambridge, 1979. Search in Google Scholar
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Articles in the same Issue
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- The generalized Drazin inverse of an operator matrix with commuting entries
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Articles in the same Issue
- Frontmatter
- On skew derivations and antiautomorphisms in prime rings
- On the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces and inclusion between them
- Solution of generalized fractional kinetic equations with generalized Mathieu series
- The generalized Drazin inverse of an operator matrix with commuting entries
- Asymptotic analysis of fundamental solutions of hypoelliptic operators
- Calculation of Reynolds equation for the generalized non-Newtonian fluids and its asymptotic behavior in a thin domain
- Double lacunary statistical convergence of Δ-measurable functions on product time scales
- On ρ-statistical convergence in neutrosophic normed spaces
- On the comparison of translation invariant convex differentiation bases
- The quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring
- Some classes of topological spaces and the space of G-permutation degree
- BV capacity and perimeter in abstract Wiener spaces and applications
- New approach on the study of operator matrix
- Comparison of several numerical solvers for a discretized nonlinear diffusion model with source terms
- Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform