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In this paper, we use summability methods on the approximation to derivatives of functions by a family of linear operators acting on weighted spaces. This point of view enables us to overcome the lack of ordinary convergence in the approximation. To support this idea, at the end of the paper, we will give a sequence of positive linear operators obeying the arithmetic mean approximation (or, approximation with respect to the Cesàro method) although it is impossible in the usual sense. Some graphical...
2000 Mathematics Subject Classification: 41A25, 41A36, 40G15.
In this paper, we obtain some statistical Korovkin-type approximation theorems including fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
2000 Mathematics Subject Classification: 41A25, 41A36.
In the present paper, we improve the classical trigonometric Korovkin theory by using the concept of statistical convergence from the summability theory and also by considering the fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.
In the present paper we are concerned with convergence in -density and -statistical convergence of sequences of functions defined on a subset of real numbers, where is a finitely additive measure. Particularly, we introduce the concepts of -statistical uniform convergence and -statistical pointwise convergence, and observe that -statistical uniform convergence inherits the basic properties of uniform convergence.
We obtain solutions to some conjectures about the nonlinear difference equation
More precisely, we get not only a condition under which the equilibrium point of the above equation is globally asymptotically stable but also a condition under which the above equation has a unique positive cycle of prime period two. We also prove some further results.
In this paper, considering A-statistical convergence instead of Pringsheim’s sense for double sequences, we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued Bögel-type continuous and periodic functions on the whole real two-dimensional space. A strong application is also presented. Furthermore, we obtain some rates of A-statistical convergence in our approximation.
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