Pointwise approximation of modified conjugate functions by matrix operators of their Fourier series

We extend the results presented by Xh. Z. Krasniqi (Slight extensions of some theorems on the rate of pointwise approximation of functions from some subclasses of Lp. Acta Comment. Univ. Tartu. Math., 2013, 17, 89–101) and W. Lenski and B. Szal (Approximation of functions belonging to the class Lp(ω) by linear operators. Acta Comment. Univ. Tartu. Math., 2009, 13, 11–24) to the case when a conjugate function depends on r and where in the measures of estimations r-differences of the entries are used.


INTRODUCTION
Let L p (1 ≤ p < ∞) be the class of all 2π-periodic real-valued functions, integrable in the Lebesgue sense with the pth power over Q = [−π, π] with the norm Given a function of class L p let us consider its conjugate trigonometric Fourier series with the partial sums S k f .We know that if f ∈ L 1 , then ψ x (t) 1 2 cot t 2 dt for an odd r, for the A-transformation of S f .In this paper, we will estimate the deviation T n,A f (x) − f r (x, ε) by the function of modulus of continuity type, i.e. nondecreasing continuous function ω having the following properties: We will also consider functions from the following subclass L p ( ω) β of L p : where The above deviation was estimated with r = 1 in [2] and generalized in [1] as follows: and )) .
In our theorems we generalize the above results using f r (x, ε) with r ∈ N instead of f 1 (x, ε) = f (x, ε).In the paper ∑ b k=a = 0 when a > b.

STATEMENT OF THE RESULTS
First we will present the estimates of the quantity T n,A f (x) − f r (x, ε) .Finally, we will formulate some remarks and corollaries.
Theorem 1.Let f ∈ L p , 0 ≤ β < 1 − 1 p and let a function of modulus of continuity type ω satisfy the conditions: for r ∈ N for a natural r ≥ 3 when r is an even natural number, and for r ∈ N ] − 1 } when r is an even natural number.Moreover, let ω satisfy, for a natural r ≥ 2, the conditions: and are true for r ∈ N, then )) .
Theorem 2. Let f ∈ L p , 0 ≤ β < 1 − 1 p and let a function of modulus of continuity type ω satisfy, for r ∈ N, the conditions: and ( 5) with when r is an even natural number.Moreover, let ω satisfy for natural r ≥ 2, the conditions ( 6) and ( 7) with )) .
Remark 2. We note that our extra conditions (8) , (9) , and (11) for a lower triangular infinite matrix A always hold.

Corollary 3. Under the above remarks and the obvious inequality
our results also improve and generalize the mentioned result of Krasniqi [1].
Remark 3. We note that instead of L p ( ω) β one can consider other subclasses of L p generated by any function of modulus continuity type, e.g.ω x such that Remark 4. We note that our condition (12) holds if we take ω (δ ) = δ α with 0 < α < β + 1 p .
It is clear by [5] that Now we present a very useful property of the modulus of continuity.

Lemma 1 ([5]). A function ω of a modulus of continuity type on the interval
Next, we present the known estimates.