Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimates through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix does not have monotone rows. In this paper, the first author continues the work in the direction for to be a -matrix. We extend two theorems on summability matrix of Deger et al. (2012) where they have extended two theorems of Chandra (2002) using -method obtained by deleting a set of rows from Cesàro matrix . Our theorems also generalize two theorems of Leindler (2005) to -matrix which in turn generalize the result of Chandra (2002) and Quade (1937).

“In memory of Professor K. V. Mital, 1918 - 2010.”

1. Introduction

Let be a periodic signal (function) and let . Let denote the partial sums, called trigonometric polynomials of degree (or order) , of the first terms of the Fourier series of at a point .

The integral modulus of continuity of is defined by

If, for , then . Throughout will denote the -norm, defined by

A positive sequence is called almost monotone decreasing (increasing) if there exists a constant , depending on the sequence only, such that, for all ,

Such sequences will be denoted by and , respectively. A sequence which is either or is called almost monotone sequence and will be denoted by . Let be an infinite subset of and as the range of strictly increasing sequence of positive integers; say . The Cesàro submethod is defined as where is a sequence of real or complex numbers. Therefore, the -method yields a subsequence of the Cesàro method , and hence it is regular for any . is obtained by deleting a set of rows from Cesàro matrix. The basic properties of -method can be found in [1, 2]. In the present paper, we will consider approximation of by trigonometric polynomials and of degree (or order) , where and by convention .

The case for all of either or yields

We also use

Mittal and Rhoades [3, 4] have initiated the study of error estimates through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix does not have monotone rows. In this paper, the first author continues the work in the direction for to be a -matrix. Recently, Chandra [5] has proved three theorems on the trigonometric approximation using -matrix. Some of them give sharper estimates than the results proved by Quade [6], Mohapatra and Russell [7], and himself earlier [8]. These results of Chandra [5] are improved in different directions by different investigators such as Leindler [9] who dropped the monotonicity on generating sequence and Mittal et al. [10, 11] who used more general matrix while very recently Deger et al. [12] used more general -method in view of Armitage and Maddox [1].

2. Known Results

Leindler [9] proved the following.

Theorem 1 (see [9]). If and be positive. If one of the conditions(i) , and ,(ii) , and , (iii) , and ,(iv) , and (10) holds,(v) , and
maintains, then

Theorem 2 (see [9]). Let . If the positive satisfies conditions (10) and hold, then

Deger et al. [12] proved.

Theorem 3 (see [12]). Let and let be positive such that
If either (i)   , and is monotonic or (ii)   , and is nondecreasing, then

Theorem 4 (see [12]). Let . If the positive satisfies condition (13) and is nondecreasing, then

3. Main Results

In this paper we generalize Theorems 3 and 4 of Deger et al. [12], by dropping monotonicity on the elements of the matrix rows which in turn generalize Theorems 1 and 2, respectively, of Leindler [9] to a more general -method. We prove the following.

Theorem 5. If and is positive and if one of the following conditions(i) , and ,(ii) , , and (13) holds,(iii) , and ,(iv) , and (13) holds,(v) , and maintains, then

Theorem 6. Let . If the positive satisfies (13) and the condition holds, then

Remarks. If , then our Theorems 5 and 6 reduce to Theorems 1 and 2, respectively.

Deger et al. [12] have used monotone sequences in Theorems 3 and 4, while our Theorems 5 and 6 claim less than the requirements of their theorems. For example, the condition of the sum in (iii) of Theorem 5 is always satisfied if the sequence is nonincreasing; that is, while if sequence is nondecreasing and condition (13) holds, then the condition in (iv) of Theorem 5 is also satisfied; that is,

Thus our theorems generalize the two theorems of Deger et al. [12] under weaker assumptions and give sharper estimates because all the estimates of Deger et al. [12] are in terms of , while our estimates are in terms of and for .

4. Lemmas

We will use the following lemmas in the proof of our theorems.

Lemma 1 (see [6]). If , for and , then

Lemma 2 (see [6]). If , for , then

Lemma 3 (see [6]). If , then

Lemma 4. Let or let and satisfy (13). Then, for , holds.

Proof. Let denote the integral part of . Then, if ,
If and (13) holds, then
This completes the proof of Lemma 4.

5. Proof of the Main Results

Proof of Theorem 5. We prove cases (i) and (ii) together. Since thus in view of Lemmas 1 and 4 and condition (13), we have
Next we consider case (iv).
Let and . By Abel’s transformation, we get and thus
Hence again by Abel’s transformation, we get
Since thus by Lemma 2
In view of (31) and (33), we obtain
Next we will verify by the induction that
If ,
Thus (37) holds for . Now let us assume that (37) is true for and we verify . Since thus (37) is proved for ; that is, (37) is true for any . Using (36) and (37) and interchanging the order of summation, we get
Now combining this with the assumption , we get from (34)
This and Lemma 1 with yield
In the proof of case (iii), we first verify that the condition implies that
In view of (36) and (37)
Denoting again by the integral part of , then, by Abel’s transformation, we have at the last step; we have used the condition . Consider the following:
Furthermore, using again our assumption, we get
Summing up our partial results, we verified (43). Thus (34) and Lemma 1 again yield
Now, we prove case (v), by using (26), , and Abel’s transformation
Hence in view of Lemma 3
Herewith case (v) is also verified and thus the proof of Theorem 5 is complete.

Proof of Theorem 6. Since , so, in view of the assumptions of Theorem 6, we get
This proves Theorem 6.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors are thankful to the learned referee Dr. A. I. Zayed for his valuable comments and suggestions to improve the presentation of the paper.