On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

The results generalizing some theorems on summability are shown. The same degrees of pointwise approximation as in earlier papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results, the estimation on norm approximation is derived. Some special cases as corollaries are also formulated.

Sequence  := (  ) of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly  ∈ HBVS, if it has the property ≤  ()   , (10) for all positive integer , or only for all  ≤  if the sequence  has only finite nonzero terms and the last nonzero term is   .Now, we define the other classes of sequences.
Following Leindler [3], sequence  := (  ) of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly  ∈ MRBVS, if it has the property for all positive integer .Analogously as in [4], sequence  := (  ) of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly  ∈ MHBVS, if it has the property for all positive integers  < , where the sequence  has only finite nonzero terms and the last nonzero term is   .It is clear that (see [5]) Consequently, we assume that the sequence ((  )) ∞ =0 is bounded, that is, that there exists a constant  such that 0 ≤  (  ) ≤  (14) holds for all , where (  ) denote the constants for the sequences   = ( , )  =0 ,  = 0, 1, 2, . . .appearing in the inequalities (11) and (12) as ().
Now we can give the conditions to be used later on.We assume that for all  and 0 ≤  < hold if ( , )  =0 belongs to MRBVS and MHBVS, for  = 1, 2, . .., respectively.
We also define two hump matrices in the following way: a lower triangular matrix  = ( , ) is called a maximal hump matrix if, for each , there exists integer  0 =  0 (), such that ( , )  0 −1 =0 is nondecreasing for 0 ≤  <  0 and ( , )  = 0 is nonincreasing for  0 ≤  ≤ , but otherwise we will have a minimal hump matrix.The hump matrices were defined and considered in [6,7].
As a measure of approximation of f by T,, , we use the pointwise modulus of continuity of  in the space   defined by the formula The deviation T,  − f = T,,  − f, with  , = 1 and 0 otherwise, was estimated at the point as well as in the norm of   by Qureshi [8] and Lal and Nigam [9].These results were generalized by Qureshi [10].The next generalization was obtained by Lal [11].In the case where  is an arbitrary positive number with ( + ) + 1 < 0 and  −1 +  −1 = 1,  > 1.
In this paper we will consider the deviations T,, (⋅) − f(⋅) and T,, (⋅) − f(⋅, /( + 1)) in general form.In the theorems we formulate the general conditions for the functions and the modulus of continuity obtaining the same degrees of approximation as above and sometimes essentially better one.Finally, we also give some results on norm approximation with essentially better degrees of approximation.The obtained results generalize the results from [4,9].
We will write  1 ≪  2 if there exists positive constant , sometimes depending on some parameters, such that  1 ≤  2 .

Let
where w is a positive, with w (0) = 0, and nondecreasing continuous function.
We can now formulate our main results.At the beginning, we formulate the results on the degrees of pointwise summability of conjugate series.
Theorem 3. Let  ∈   (w  )  with 1 <  < ∞, and let w satisfy with some  ≥ 0. If the entries of matrix  satisfy condition ( 22) for 0 ≤  ≤ ] and if matrix  is a maximal or minimal hump matrix with for almost all considered  such that f() exists.
Next, we formulate the results on estimates of   norm of the deviation considered above.In case of the deviation T,, (⋅) − f(⋅), let where ω is positive, with ω(0) = 0, and almost nondecreasing continuous function.
If the entries of matrix  satisfy condition ( 22) for 0 ≤  ≤ ] and if matrix  is a maximal or minimal hump matrix with where 0 ≤  ≤ 1.
Theorem 5. Let  ∈   ( ω)  with 1 <  < ∞, where ω instead of w satisfies (24) with some  ≥ 0. If the entries of matrix  satisfy condition (22) for 0 ≤  ≤ ] and if matrix  is a maximal or minimal hump matrix with Finally, we give corollary and remarks as an application of our results.
Remark 8. Taking  = 0, we have, by Theorem 3 with w () =   , for 0 <  < 1 and  > 1, the estimate like in [12] with the better order of approximation without any additional assumptions.

Auxiliary Results
We begin this section by some notations following A. Zygmund [1, Section 5 of Chapter II].
It is clear that where where Now, we formulate some estimates for the conjugate Dirichlet kernels.
The relation ( Now, our proof is complete.
Proof.Since the above formulas are similar, we prove the first one only.If 0 <  ≤  0 , then whence (1/( + 1)) ∑  =0  , nondecreases with respect to  and therefore But if  ≥  0 , then and our proof is complete.
holds for every natural  and all real .
Proof.The proof follows by the easy account Now, our proof is complete.) .

Proofs of the Results
Collecting these estimates, we obtain the desired result.(

Proofs of
Thus proof is complete.