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A generalization of deferred Cesáro means and some of their applications
- Uǧur Deǧer^{1}Email author and
- Mehmet Küçükaslan^{1}
https://doi.org/10.1186/s13660-014-0532-0
© Deǧer and Küçükaslan; licensee Springer 2015
- Received: 21 March 2014
- Accepted: 17 December 2014
- Published: 15 January 2015
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.
Keywords
- deferred Cesáro method
- Hölder metric
- trigonometric polynomials
- almost monotone sequences
- degree of approximation
MSC
- 40Gxx
- 41A25
- 42A10
1 Definitions and some notations
In this paper we are interested in the following two statements and will proceed in these directions.
1. 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. Naturally, there arises the question how we can generalize these approximation methods. The summability methods used in approximations belong to these methods. In this sense, we will give a generalization of deferred Cesáro means which includes Woronoi-Nörlund and Riesz methods as a summability method in Section 1. We know that the Nörlund and Riesz methods generalize the well-known Cesáro method which has an important place in this theory. In Section 4, we will establish some of summability properties related to this generalization.
2. As an application of these methods in theory of Fourier series, we will consider the degree of approximation in accordance with generalized deferred Cesáro means in a generalized Hölder metric in Section 2 and present some applications of the approach concerned with some sequence classes in Section 3.
2 Approximation by generalized deferred Cesáro means in generalized Hölder metric
Theorem A
[5]
The case \(\beta=0\) in Theorem A is due to Alexits [6]. Chandra obtained a generalization of Theorem A by considering the Woronoi-Nörlund transform [7]. Later, Mohapatra and Chandra considered the problem by a matrix means of the Fourier series of \(f\in H_{\alpha}\) [8].
In this section, we shall consider the degree of approximation of \(f\in H(\alpha,p)\) with respect to the norm in the space \(H(\alpha,p)\) by the deferred Woronoi-Nörlund means and the deferred Riesz means of the Fourier series of the function f by taking into account the method in [9].
Theorem 2.1
Proof
Corollary 2.2
Corollary 2.3
The next result is related to the deferred Woronoi-Nörlund means in a generalized Hölder metric.
Theorem 2.4
Proof
Analogous results to Corollary 2.2 and Corollary 2.3 can also be given for the deferred Nörlund means. Moreover, since \(D_{0}^{n}R_{n}\) and \(D_{0}^{n}N_{n}\) in the case \(p_{n}=1\) (for all n) coincide with the Cesáro method \(C_{1}\), Theorem 2.1 and Theorem 2.4 are reduced to the result of Prösdorff in \(H(\alpha,\infty)\) space. Furthermore we know that if \(p_{n}=1\) for all n, then these two methods give us the deferred Cesáro means. Therefore our results in Theorem 2.1 and Theorem 2.4 coincide with the results relevant to the deferred Cesáro means of Das et al. (see [9]) in some cases.
3 Applications related to some sequence classes
4 Results on generalized deferred Cesáro means
Theorem 4.1
\((R,p)\subset(D_{a}^{b}R,p)\) if and only if \((D_{a}^{b}R,p)\) is proper.
Proof
The next result is associated with the fact that Riesz transformations contain deferred Riesz transformations in which case we have the following.
Theorem 4.2
\((D_{a}^{n}R,p)\subset(R,p)\).
Proof
Let \(d_{n,k}\) and \(c_{n,k}\) be different from each other for at most a finite set of values of n. We know that since the two transformations \(\sigma_{n}=\sum_{k=1}^{\infty}d_{n,k}x_{k}\) and \(\sigma_{n}^{1}=\sum_{k=1}^{\infty}c_{n,k}x_{k}\) are equivalent, we write the following corollary as a result of Theorem 4.2.
Corollary 4.3
\((D_{a}^{b}R,p)\subset(R,p)\) whenever \(b_{n}=n\) for almost all n.
Taking into account Theorem 4.1 and Theorem 4.2, we can write the next result.
Theorem 4.4
\((D_{a}^{n}R, p)\sim(R,p)\) if and only if \((D_{a}^{n}R,p)\) is proper where the symbol ∼ denotes equivalence between transformations.
In the case \(p_{n}=1\) for all n Theorem 4.1, Theorem 4.2, and Theorem 4.4 give us Theorem 4.1, Theorem 6.1, and Theorem 6.2 in [1], respectively.
Theorem 4.5
\((D_{a}^{b}R,p)\subset(R,p)\) whenever \((b_{n})\) includes almost all positive integers.
Proof
Declarations
Acknowledgements
The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advice. The first author was supported by the Council of Higher Education of Turkey under a Postdoctoral grant.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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