A generalization of deferred Cesáro means and some of their applications

Uǧur Deǧer,Mehmet Küçükaslan

doi:10.1186/s13660-014-0532-0

Abstract

The deferred Cesáro transformation, which has useful properties not possessed by the Cesáro transformation, was considered by RP Agnew in 1932. The aim of this paper is to give a generalization of deferred Cesáro transformations by taking account of some well-known transformations and to handle some of their properties as well. On the other hand, we shall consider the approximation by the generalized deferred Cesáro means in a generalized Hölder metric and present some applications of the approach concerning some sequence classes.

Highlights

The deferred Cesáro transformation, which has useful properties not possessed by the Cesáro transformation, was considered by RP Agnew in 1932
One of the basic problems in the theory of approximations of functions and the theory of Fourier series is to examine the degree of approximation in given function spaces by certain methods
Prösdorff has studied the degree of approximation in the Hölder metric and proved the following theorem

Summary

Let f

Be the Fourier series of a function f ∈ L. The summability methods used in approximations belong to these methods By considering the deferred Cesáro means, we write the following notations with conditions ( ) and ( ). In the case bn = n and an = , the methods DbaNn(f ; x) and DbaRn(f ; x) give us the classically known Woronoi-Nörlund and Riesz means, respectively.

Cesáro means

Proof A standard computation shows that

Results on generalized deferred Cesáro means

The matrix

Rbn pan

We write