Abstract and Applied Analysis

Volume 2012 (2012), Article ID 513206, 20 pages

doi:10.1155/2012/513206

Research Article

On the Convergence of Absolute Summability for Functions of Bounded Variation in Two Variables

Ying Mei¹ and Dansheng Yu²

¹Department of Mathematics, Lishui University, Lishui, Zhejiang 323000, China

²Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 25 September 2012; Accepted 12 November 2012

Academic Editor: Jaume Giné

Copyright © 2012 Ying Mei and Dansheng Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By adopting some new ideas, we obtain the estimates of an absolute convergence for the functions of the bounded variation in two variables. Our results generalize the related results of Humphreys and Bojanic (1999) and Wang and Yu (2003) from one dimension to two dimensions and can be applied to several summability methods.

1. Introduction

Let be -periodic functions integrable on . Denote its Fourier series by

where the Fourier coefficients and are defined as follows:

Denoted by the th partial sums of the Fourier series (1.1), that is,

When is of bounded variation, Bojanić [1] obtained the following result on the rate of the convergence of approximation by .

Theorem A. If a periodic function is of bounded variation on the interval , then the following estimate holds for every and :

where and is the total variation of on .

There are a lot of interesting generalizations that have been achieved by many authors (see [1–9]). Among them, Jenei [4] and Móricz [5] generalized Theorem A to the double Fourier series of functions of bounded variation; Humphreys and Bojanic [3], Wang and Yu [7] investigated the absolute convergence of Cesàro means of Fourier series.

Let be the Cesàro matrix of order , that is,

where

Then, the so-called Cesàro means of order of (1.1) is defined by (denote by ):

Humphreys and Bojanic [3] investigated the rate of the absolute convergence of Cesàro means of the series (1.1). Their result can be read as follows.

Theorem B. Let and be -periodic functions of bounded variation on . Then for and , one has

where

and be the total variation of on .

However, Wang and Yu [7] showed that Theorem A is not correct when . In fact, they proved the following.

Theorem C. Suppose , and are -periodic functions of bounded variation on . Then for , one has

and there exists a -periodic function of bounded variation on and a point such that

Motivated by Theorems B and C, and the results of Jenei [4] and Móricz [5] on the double Fourier series, we will investigate the absolute convergence of a kind of very general summability of the double Fourier series. Our new results not only generalize Theorems B and C to the double case, but also can be applied to many other classical summability methods. We will present our main results on Section 2. Proofs will be given in Section 3. In Section 4, we will apply our results to some classical summability methods.

2. The Main Results

Let be a function periodic in each variable with period and integrable on the two-dimensional torus in Lebesgue’s sense, in symbol, . The double Fourier series of a complex-valued function is defined by

where the are the Fourier coefficients of :

Let be a bounded and closed rectangle on the plane. A function defined on is said to be of bounded variation over in the sense of Hardy-Krause, in symbol, , if

(i) the total variation of over is finite, that is,

 where the supremum is extended for all finite partitions

 of the intervals and ;(ii) the marginal functions and are of bounded variation over the intervals and , respectively.

For any , define

For convenience, write

For any double sequence , define

For any fourfold sequence , write

A doubly infinite matrix is said to be doubly triangular if for or . The th term of the -transform of the double Fourier series (2.1) is defined by

where is the th partial sum of (2.1), that is,

Write

Set

Our main results are the following Theorems 2.1 and 2.2.

Theorem 2.1. Let be a periodic function and . Assume that is a lower doubly triangular matrix satisfying

and there exist constants such that

Then for , , one has

Theorem 2.2. Let and satisfy all the conditions of Theorem 2.1, except (2.15) is replaced by

where or , , , . Then for , , one has

3. Proofs of Results

Lemma 3.1 (see [10]). If is continuous on a rectangle and , then

(i) is integrable with respect to over in the sense of Riemann-Stieltjes integral;(ii) is integrable with respect to over in the sense of Riemann-Stieltjes integral and

Lemma 3.2 (see [10]). If and is absolutely continuous on , that is,

for some function and constant , then

Proof of Theorem 2.1. By the definition of (see (2.9)), we have

Therefore,

Noting that

we have

where in the last equality, Lemma 3.2 is applied.

By (3.7) and Lemma 3.1, we have

By (2.14), we have (Denote by and the characteristic functions of and , respectively.)

For , by the first inequality of (2.15),

Similarly, we also have

By (2.15), we deduce that

We get (2.16) by combining (3.8)–(3.12).

Proof of Theorem 2.2. Set

By Abel’s transformation, we have

Therefore, by the following well-known inequalities: , , and condition (2.17), we see that (2.15) holds with , and thus we get Theorem 2.2 from Theorem 2.1.

4. Applications

4.1. Cesàro’s Means

Let be the double Cesàro matrix of order with , that is, has entries

Then for , ,

In particular,

In other words, satisfies condition (2.13) of Theorem 2.1.

By the well-known inequality (e.g., see [11])

we deduce that

which means condition (2.14) of Theorem 2.1.

For , we have

where

By (4.4), we observe that

Since decreases monotonically to , converges for , and

for .

By (4.4)–(4.9), we obtain that

On the other hand, we have

For , we have that

Now, by (4.10) and (4.11), we deduce that

for , .

When , , by (4.10) and (4.11) again,

for . By (4.10)–(4.12), we have

Similarly, we have

for and

for .

Setting

then we have

by (4.13)–(4.17).

From (4.19), we have the following corollary of Theorem 2.1, which generalizes Theorem B from one dimension to two dimensions.

Theorem 4.1. Let be a periodic function and and be the double Cesàro matrix of order with . Then for , , one has

4.2. Bernstein-Rogosinski’s Means

The so-called Bernstein-Rogosinski means of Fourier series (1.1) are defined by

Now, we introduce the following Bernstein-Rogosinski means of double Fourier series (2.1):

Direct calculations yield that

Thus, can be regarded as a -transformation of Fourier series (1.1) with of entries defined as

By direct calculations, we have

Therefore, (4.26) and (4.27) imply (2.13), while from (4.25)–(4.28), we get

which implies (2.14).

Finally, we verify that satisfies condition (2.17); thus satisfies all the conditions of Theorem 2.2. By direct calculations, we deduce that

Analogously,

Now, (4.30) and (4.31) imply (2.17).

4.3. Riesz’s Means

Let and be positive sequences such that ,     , and let be a lower triangular matrix with entries   , , ; . Then the -transformation of Fourier series (2.1) is known as Riesz’s mean.

Proposition 4.2. Let and be positive nondecreasing sequences such that (one says , if there are two positive constants and such that .)

Then satisfies all the conditions of Theorem 2.2.

Proof. Direct calculations yield that (set )

It follows from (4.35) that satisfies condition (2.13).

Since and are nondecreasing, by (4.32), (4.34)–(4.36), we have

Hence, satisfies condition (2.14).

Since and are nondecreasing, by (4.32)–(4.36), we have

Similarly,

Now, by (4.38) and (4.39), we show that also satisfies condition (2.17).

Acknowledgments

A Project Supported by Scientific Research Fund of Zhejiang Provincial Education Department (Y201223607). Research of the second author is supported by NSF of China (10901044), Qianjiang Rencai Program of Zhejiang Province (2010R10101), SRF for ROCS, SEM, and Program for excellent Young Teachers in HZNU.

References

R. Bojanić, “An estimate of the rate of convergence for Fourier series of functions of bounded variation,” Institut Mathématique, vol. 26, pp. 57–60, 1979.

A. S. Belov and S. A. Telyakovskii, “Refinement of the Dirichlet-Jordan and Young's theorems on Fourier series of functions of bounded variation,” Sbornik, vol. 198, pp. 777–791, 2007.

N. Humphreys and R. Bojanic, “Rate of convergence for the absolutely (C, α) summable Fourier series of functions of bounded variation,” Journal of Approximation Theory, vol. 101, no. 2, pp. 212–220, 1999.

A. Jenei, “On the rate of convergence of Fourier series of two-dimensional functions of bounded variation,” Analysis Mathematica, vol. 35, no. 2, pp. 99–118, 2009.

F. Móricz, “A quantitative version of the Dirichlet-Jordan test for double Fourier series,” Journal of Approximation Theory, vol. 71, no. 3, pp. 344–358, 1992.

S. A. Telyakovskiĭ, “On the rate of convergence of Fourier series of functions of bounded variation,” Mathematical Notes, vol. 72, pp. 872–876, 2002.

K. Y. Wang and D. Yu, “On the absolute (C, α) convergence for functions of bounded variation,” Journal of Approximation Theory, vol. 123, no. 2, pp. 300–304, 2003.

C. B. Lu and C. Q. Gu, “The computation of the inverse of block-wise centrosymmetric matrices,” Publicationes Mathematicae Debrecen, vol. 1-2, no. 82, pp. 1–17, 2013.

C. B. Lu and C. Q. Gu, “The computation of the square roots of circulant matrices,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6819–6829, 2011.

F. Móricz, “Pointwise behavior of double Fourier series of functions of bounded variation,” Monatshefte für Mathematik, vol. 148, no. 1, pp. 51–59, 2006.

A. Zygmund, Trigonometric Series, vol. 1, Cambridge University, New York, NY, USA, 1959.