Pointwise Approximation of Functions from Lp w β by Linear Operators of Their Fourier Series

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.


Introduction
Let L p 1 ≤ p < ∞ be the class of all 2π-periodic real-valued functions integrable in the Lebesgue sense with pth power over Q −π, π with the norm for all natural numbers m, where K c is a constant depending only on c.
A sequence c : c n of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly c ∈ HBVS, if it has the property 12 for all natural numbers m, where K c is a constant depending only on c. Analogously, a sequence c : c k of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly c ∈ MHBVS, if it has the property for all positive integer m < n, where the sequence c has only finite nonzero terms and the last nonzero term is c n and where K c is a constant depending only on c.It is clear that see 4 RBVS MRBVS, HBVS MHBVS.

1.14
Consequently, we assume that the sequence K α n ∞ n 0 is bounded, that is, that there exists a constant K such that holds for all n, where K α n denotes the sequence of constants appearing in the inequalities 1.12 or 1.13 for the sequences α n a n, k n k 0 .Now, we can give the conditions to be used later on.We assume that, for all n and 0 ≤ m < n, where K is the same as above, hold if α n a n, k n k 0 belong to MRBVS or MHBVS, for n 0, 1, 2, . . ., respectively.
As a measure of approximation of functions by the above means, we use the generalized pointwise moduli of continuity of f in the space L p defined for β ≥ 0 by the formulas where It is clear that, for β > α ≥ 0, The deviation T n, A f − f with special form of matrix A was estimated in the norm of L p by Lal 5, Theorem 2, page 347 as follows.
where P n n ν 0 p ν , p ν is a nonnegative and nonincreasing sequence, and the function ω 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 / n 2 with the matrix whose rows belong to the classes of sequences MRBVS and MHBVS.Consequently, we also give some results on norm approximation.
We will write I 1 I 2 if there exists a positive constant K, sometimes depending on some parameters, such that I 1 ≤ KI 2 .

Statement of the Results
Let us define, for a fixed x, a function w x or w x of modulus of continuity type on the interval 0, 2π , that is, a nondecreasing continuous function having the following properties: where w x , w x w, and w are also the functions of modulus of continuity type.It is clear that, for β > α ≥ 0,

2.3
Now, we can formulate our main results on the degrees of pointwise summability.
for considered x such that f x exists.
Theorem 2.6.Let f ∈ L p w x β , and 2.8 holds with q p p − 1 −1 when β > 0 or with q 1 when for considered x such that f x exists.

Journal of Function Spaces and Applications 7
Consequently, we formulate the results on norm approximation.

2.16
Theorem 2.12.Let f ∈ L p w x β , and 2.15 holds with q p p −

2.17
Remark 2.13.In the case p ≥ 1 specially if p 1 , we can suppose that the expression t −β w x t nondecreases in t instead of the assumption β < 1 − 1/p .

Journal of Function Spaces and Applications
Remark 2.14.Under additional assumptions a nn O 1/n , β 0, and w x t O t α 0 < α ≤ 1 , the degree of approximation in Theorem 2.1 is following O n 1/p −α .The same degree of approximation we obtain in Theorem 2.2 under the assumption a n0 O 1/n .In case of the remaining theorems, we can use the same remarks.
Remark 2.15.If we consider the classical modulus of continuity ω 0 f δ L p of the function f instead of the modulus w, then the condition w •, β f δ L p L p ≤ ω 0 f δ L p holds for every function f ∈ L p and thus L p w β L p .The same remark for conjugate functions holds too.
Remark 2.16.Our theorems will be also true if we consider function f from L p β ω with the following norm:

2.18
Remark 2.17.We can observe that, taking a nk 1/ n 1 n ν k p ν−k /P ν , we obtain the mean considered by Lal 5 , and if p ν is monotonic with respect to ν, then a n, k n k 0 ∈ MRBVS.Therefore, if p ν is a nonincreasing sequence such that P τ n ν τ P −1 ν O τ , then, from Theorem 2.1, we obtain the corrected form of the result of Lal 5 i.e., with condition

Auxiliary Results
We begin this section by some notations following Zygmund 1 .
It is clear that where

3.2
Journal of Function Spaces and Applications 9 Hence, where

3.4
Now, we formulate some estimates for the conjugate Dirichlet kernel.

Proof of Theorem 2.1
As usual,

4.4
Collecting these estimates, we obtain the desired result.

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 A n, τ , we have to write the quantity A n, n−2τ for which we suppose the same order.

Proof of Theorem 2.3
We start with the obvious relations

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 A n, τ , we have to write the quantity A n, n−2τ for which we suppose the same order.

Proof of Theorem 2.5
We start with the obvious relations 4.9 Therefore, using 2.8 , we have

4.13
The same estimation as in the proof of Theorem 2.3 gives Collecting these estimates, we obtain the desired result.

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.

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

4.15
This completes the proof of Theorem 2.7.

Theorem 2 . 5 .
Let f ∈ L p w x β , and 11 for all natural numbers m, or only for all m ≤ N if the sequence c has only finite nonzero terms and the last nonzero terms is c N .Now, we define another class of sequences.Followed by Leindler 3 , a sequence c : c n of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly c ∈ MRBVS, if it has the property