#### Abstract

We show the results, corresponding to theorem of Lal (2009), on the rate of pointwise approximation of functions from the pointwise integral Lipschitz classes by matrix summability means of their Fourier series as well as the theorems on norm approximations.

#### 1. Introduction

Let be the class of all -periodic real-valued functions integrable in the Lebesgue sense with th power over with the norm and consider the trigonometric Fourier series and conjugate one with the partial sums and , respectively. We know that if , then where with exists for almost all [1, Th. (3.1)IV].

Let be an infinite lower triangular matrix of real numbers such that and let the transformationsof and be given by respectively. Denote, for ,

We define two classes of sequences (see [2]).

A sequence of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly , if it has the property for all natural numbers , where is a constant depending only on .

A sequence of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly , if it has the property for all natural numbers , or only for all if the sequence has only finite nonzero terms and the last nonzero terms is .

Now, we define another class of sequences.

Followed by Leindler [3], a sequence of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly , if it has the property for all natural numbers , where is a constant depending only on .

Analogously, a sequence of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly , if it has the property for all positive integer , where the sequence has only finite nonzero terms and the last nonzero term is and where is a constant depending only on .

It is clear that (see [4])

Consequently, we assume that the sequence is bounded, that is, that there exists a constant such that holds for all , where denotes the sequence of constants appearing in the inequalities (1.12) or (1.13) for the sequences .

Now, we can give the conditions to be used later on. We assume that, for all and , where is the same as above, hold if belong to or , for , respectively.

As a measure of approximation of functions by the above means, we use the generalized pointwise moduli of continuity of in the space defined for by the formulas where It is clear that, for , It is easily seen that and are the classical pointwise moduli of continuity.

The deviation with special form of matrix was estimated in the norm of by Lal [5, Theoremββ2, page 347] as follows.

Theorem A. *If
**
then
**
where is a nonnegative and nonincreasing sequence, and the function of modulus of continuity type will be defined in the next section.*

In this note we show that the conditions (1.21), (1.22), and (1.23) are superfluous when we use the pointwise modulus of continuity.

In our theorems, we will consider the pointwise deviations , with the matrix whose rows belong to the classes of sequences and . Consequently, we also give some results on norm approximation.

We will write if there exists a positive constant , sometimes depending on some parameters, such that .

#### 2. Statement of the Results

Let us define, for a fixed , a function (or ) of modulus of continuity type on the interval , that is, a nondecreasing continuous function having the following properties: for any . It is easy to conclude that the function nondecreases in . Let where , and are also the functions of modulus of continuity type. It is clear that, for ,

Now, we can formulate our main results on the degrees of pointwise summability.

Theorem 2.1. *Let with . If is such that , where , then
**
for considered .*

Theorem 2.2. *Let with . If is such that , where , then
**
for considered .*

Theorem 2.3. *Let with . If is such that , where , then
**
for considered .*

Theorem 2.4. *Let with . If is such that , where , then
**
for considered .*

Theorem 2.5. *Let , and
**
holds with when or with when . If is such that , where , then
**
for considered such that exists.*

Theorem 2.6. *Let , and (2.8) holds with when or with when . If and , where , then
**
for considered such that exists.*

Consequently, we formulate the results on norm approximation.

Theorem 2.7. *Let with is such that , where , then
*

Theorem 2.8. *Let with is such that , where , then
*

Theorem 2.9. *Let with is such that , where , then
*

Theorem 2.10. *Let with is such that , where , then
*

Theorem 2.11. *Let , and
**
holds with . If and , where , then
*

Theorem 2.12. *Let , and (2.15) holds with . If is such that , where , then
*

*Remark 2.13. *In the case , we can suppose that the expression nondecreases in instead of the assumption .

*Remark 2.14. *Under additional assumptions , and , the degree of approximation in Theorem 2.1 is following . The same degree of approximation we obtain in Theorem 2.2 under the assumption . In case of the remaining theorems, we can use the same remarks.

*Remark 2.15. *If we consider the classical modulus of continuity of the function instead of the modulus , then the condition holds for every function and thus . The same remark for conjugate functions holds too.

*Remark 2.16. *Our theorems will be also true if we consider function from with the following norm:

*Remark 2.17. *We can observe that, taking , we obtain the mean considered by Lal [5], and if is monotonic with respect to , then . Therefore, if is a nonincreasing sequence such that , then, from Theorem 2.1, we obtain the corrected form of the result of Lal [5] (i.e., with condition instead of (1.22)).

#### 3. Auxiliary Results

We begin this section by some notations following Zygmund [1].

It is clear that where Hence, where Now, we formulate some estimates for the conjugate Dirichlet kernel.

Lemma 3.1 (see [1]). *If , then
**
and, for any real , we have
*

Lemma 3.2 (cf. [2, 6]). *If , then
**
and if , then
**
for , where .*

Lemma 3.3 (see [7]). *If , then
**
and if , then
**
for , where .*

#### 4. Proofs of the Results

##### 4.1. Proof of Theorem 2.1

As usual, By the HΓΆlder inequality and Lemma 3.1, for , Using Lemma 3.2 and the HΓΆlder inequality , Since , we have

Collecting these estimates, we obtain the desired result.

##### 4.2. Proof of Theorem 2.2

The proof is the same as the proof of Theorem 2.1, the only difference is that, in place of , we have to write the quantity for which we suppose the same order.

##### 4.3. Proof of Theorem 2.3

We start with the obvious relations By the HΓΆlder inequality and Lemma 3.1, we have for .

Using Lemma 3.3 and the HΓΆlder inequality , we get Since , we have and our proof is complete.

##### 4.4. Proof of Theorem 2.4

The proof is the same as the proof of Theorem 2.3, the only difference is that, in place of , we have to write the quantity for which we suppose the same order.

##### 4.5. Proof of Theorem 2.5

We start with the obvious relations Lemma 3.1 gives If and , then, by the HΓΆlder inequality , or if , then Therefore, using (2.8), we have The same estimation as in the proof of Theorem 2.3 gives Collecting these estimates, we obtain the desired result.

##### 4.6. Proof of Theorem 2.6

For the proof, we use the analogical remark as these in the proofs of Theorems 2.2 and 2.4.

##### 4.7. Proofs of Theorems from 2.7 to 2.12

The proofs are similar to these above. We have only to use the generalized Minkowski inequality.

For example, in case of Theorem 2.7, we get

This completes the proof of Theorem 2.7.