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In this paper, using the concept of strong summation process, we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from L p;q (loc) into itself. We also study modulus of continuity for L p;q (loc) approximation and give the rate of convergence of these operators.Keywords: A summability, positive linear operators, locally integrable functions, Korovkin type theorem, modulus of continuity, rate of convergence.2010 AMS Classi…cation: 41A25, 41A36. IntroductionThe classical theorem of Korovkin Approximation theory has important applications in the theory of polynomial approximation, in functional analysis, numerical solutions of di¤erential and integral equations [1], [7].The purpose this paper is to study a Korovkin type approximation theorem of a function f by means of sequence of positive linear operators from the space of locally integrable functions into itself with the use of a matrix summability method which includes both convergence and almost convergence. We also obtain rate of convergence in L p;q (loc) approximation with positive linear operators by means of modulus of continuity. Now we recall some information of locally integrable functions given in [6].

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In the present work, using statistical convergence with respect to power series methods, we obtain various Korovkin-type approximation theorems for linear operators defined on derivatives of functions. Then we give an example satisfying our approximation theorem. We study certain rate of convergence related to this method. In the final section we summarize these results to emphasize the importance of the study.

In this paper we mainly deal with I (q) c − convergence. In particular we study bounded multipliers of bounded I (q) c − convergent sequences. We also give some I− core results and characterize the inclusion K−core {Ax} ⊆ I−core {x} for bounded sequences x = (xn) .

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Given a real bounded sequence $x=(x_{j})$ and an infinite matrix $A=(a_{nj})$ Knopp core theorem is equivalent to study the inequality $limsup{Ax} ≤ limsup{x}.$ Recently Fridy and Orhan [6] have considered some variants of this inequality by replacing $limsup{x}$ with statistical limit superior $st - limsup{x}$. In the present paper we examine similar type of inequalities by employing a power series method $P$; a non-matrix sequence-to-function transformation, in place of $A =(a_{nj})$ .

This paper is a continuation of the so far performed studies on the concept of uniform statistical convergence. We first characterize two inequalities concerning the uniform statistical limit superior that lead to two core inclusion results for bounded real sequences. Using the inverse Fourier transformation, we also give a criterion on uniform statistical convergence, and study some factorization results of the space of uniformly statistically convergent sequences. These results are used to give a Korovkin-type approximation theorem.