Summability methods for two variables measurable functions

This dissertation comprises of ten chapters. The first section is devoted to the introduction. Chapter 2 contains preliminary notions, basic definitions, examples and some significant well known results. In Chapter 3, the notion of double statistical boundedness and double lacunary statistical boundedness for double sequences are presented. In Chapter 4, by using two non-negative real valued Lebesgue measurable functions on (1,∞), the notions of asymptotically I_λ-statistically equivalent of order α and strongly I_λ-asymptotically equivalent of order α are introduced. In Chapter 5, Borwein`s [5] results is extended to multidimensional Cesáro type summable function spaces. In Chapter 6, the notion of λμ-double asymptotically statistically equivalent and strongly λμ-double asymptotically equivalent function spaces for two variables measurable functions are presented. In Chapter 7, I_λ-λμ-double statistical convergence of two variables functions and a new approach to the concept of I_λ-λμ-double lacunary statistical convergence are presented, and the relationship between those two concepts are examined. The purpose of Chapter 8 is to obtain some results in Summability Theory by considering integrable functions in the Gauge sense. In Chapter 9, the new methods of Summability Theory are introduced by considering Pringsheim limits and Gauge integrable two variables measurable functions defined on (1,∞)×(1,∞). The last section is devoted to the results and recommendations. Keywords: Summability Theory, Statistical Convergence, Pringsheim Limit, Strongly Summability, Gauge Integral, Two Variables Measurable Functions