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 [19]). 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,



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
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