Pointwise approximation of modified conjugate functions by matrix operators of conjugate Fourier series of 2π/r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\pi /r$\end{document}-periodic functions

We extend the results of Xh. Z. Krasniqi (Acta Comment. Univ. Tartu Math. 17:89–101, 2013) and the authors (Acta Comment. Univ. Tartu Math. 13:11–24, 2009; Proc. Est. Acad. Sci. 67:50–60, 2018) to the case when considered function is 2π/r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\pi/r$\end{document}-periodic and the measure of approximation depends on r-differences of the entries of the considered matrices.


Introduction
Let L p 2π /r (1 ≤ p < ∞) be the class of all 2π/r-periodic real-valued functions, integrable in the Lebesgue sense with the pth power over Q r = [-π/r, π/r] with the norm Taking into account the above relations, we will consider, for f ∈ L 1 2π /r , the trigonometric Fourier series as such a series of f ∈ L 1 2π in the following form: a ν (f ) cos νx + b ν (f ) sin νx with the partial sums S k f and the conjugate one a ν (f ) sin νxb ν (f ) cos νx with the partial sums S k f . We also know that if f ∈ L 1 2π , then where, for r ∈ N, )ψ x (t) 1 2 cot t 2 dt for an odd r, for an even r, exist for almost all x (cf. [4,Th. (3.1) IV]). Let A := (a n,k ) be an infinite matrix of real numbers such that a n,k ≥ 0 when k, n = 0, 1, 2, . . . , lim n→∞ a n,k = 0 and ∞ k=0 a n,k = 1, but A • := (a n,k ) n k=0 , where a n,k = 0 when k > n.
We will use the notations |a n,ka n,k+r |, A • n,r = n k=0 |a n,ka n,k+r | for r ∈ N and for the A-transformation of Sf .
In this paper, we will study the estimate of | T n,A f (x)f r (x, )| by the function of modulus of continuity type, i.e. a nondecreasing continuous function ω having the following properties: ω(0) = 0, ω(δ 1 + δ 2 ) ≤ ω(δ 1 ) + ω(δ 2 ) for any 0 ≤ δ 1 ≤ δ 2 ≤ δ 1 + δ 2 ≤ 2π . We will also consider functions from the subclass L p 2π /r ( ω) β of L p 2π /r for r ∈ N: where is the classical modulus of continuity. Moreover, it is clear that for β ≥ α ≥ 0 The deviation T n,A f (x)f r (x, ) was estimated with r = 1 in [2] and generalized in [1] as follows: where ω satisfies the conditions: The next essential generalizations and improvements in [3, Theorem 1] were given. In these results f r (x, ) and A n,r (with r ∈ N) instead of f 1 (x, ) = f (x, ) and A • n,1 , respectively, were taken. We can formulate them as follows.
p and a function ω of modulus of continuity type satisfies the conditions: for r ∈ N, for r ∈ N with 0 < γ < β + 1 p , where m ∈ {0, . . . [ r 2 ]} when r is an odd or m ∈ {0, . . . [ r 2 ] -1} when r is an even natural number. Moreover, let ω satisfy, for a natural r ≥ 2, the conditions: and n l=0 r+l-1 k=l a n,k with r ∈ N are true, then p and a function ω of modulus of continuity type satisfy, for r ∈ N, the conditions: (4) and (5) when r is an even natural number. Moreover, let ω satisfy, for a natural r ≥ 2, the conditions (6) and and (9) with r ∈ N are true, then = O x (n + 1) β+ 1 p +1 A n,r ω π n + 1 .
In our theorems we generalize the above results considering 2π/r-periodic functions and using simpler assumptions.
In the paper b k=a = 0 when a > b.

Statement of the results
To begin with, we will present the estimates of the quantities and Finally, we will formulate some remarks and corollaries.
p and a function ω of the modulus of continuity type satisfies the conditions: when r ≥ 2, and for r ∈ N with 0 < γ < β + 1 p . If a matrix A is such that (8) and (9) are true, then Theorem 2 Suppose that f ∈ L p 2π /r , 1 < p < ∞, r ∈ N, 0 ≤ β < 1 -1 p and a function ω of the modulus of continuity type satisfies the conditions (12) and (13) for r ∈ N with 0 < γ < β + 1 p . If a matrix A is such that (10) and (9) are true, then
Thus we obtain the results from [2] and Theorem A [1, Theorem 8, p. 95], but in the case of [3] (Theorem B and C) we reduce the assumptions.
Next, using more natural conditions when β > 0 we can formulate, without proofs, the following theorems.
Let a function ω of the modulus of continuity type satisfy the conditions: for γ ∈ ( 1 p , 1 p + β) and r ∈ N (instead of (13)), and when r ≥ 2 (instead of (11) and (12), respectively). If a matrix A is such that (9) and (8) are true, then Moreover, if a function ω of the modulus of continuity type and a matrix A satisfy the following conditions: (14) with r ∈ N and γ ∈ ( 1 p , 1 p + β), (15) with r ∈ N, (9) and (10), then the estimate (16) is also true.
Remark 2 We note that our extra conditions (9), (8) and (10) for a lower triangular infinite matrix A • always hold.

Corollary 2 Considering the above remarks and the obvious inequality
A n,r ≤ rA n,1 for r ∈ N (17) our results also improve and generalize the mentioned result of Krasniqi [1].
Remark 3 We note that instead of L p 2π /r ( ω) β one can consider another subclass of L p 2π /r generated by any function of the modulus of continuity type e.g. ω x such that or

Auxiliary results
We begin this section by some notations from [5] and It is clear by [4] that Now, we present a very useful property of the modulus of continuity.

Lemma 1 ([4])
A function ω of modulus of continuity type on the interval [0, 2π] satisfies the following condition: Next, we present the following well-known estimates.

Lemma 2 ([4])
If 0 < |t| ≤ π then and, for any real t, we have We additionally need the following estimate as a consequence of Lemma 3.
We also need some special conditions which follow from the ones mentioned above.
Lemma 5 Suppose that f ∈ L p 2π /r , where 1 ≤ p < ∞ and r ∈ N. If the condition (12) holds with any function ω of the modulus of continuity type and β ≥ 0, then Proof By the substitution t = 2(m+1)π r u, we obtain Hence, by (12) our estimate follows.
Lemma 6 Suppose that f ∈ L p 2π /r , where 1 ≤ p < ∞ and r ∈ N. If the condition (12) holds with any function ω of the modulus of continuity type and β ≥ 0, then Proof By the substitution t = 2mπ r + u, analogously to the above proof, we obtain 2mπ and we have the desired estimate. Now, we formulate another two lemmas without proofs. We can prove them in the same way as Lemmas 5 and 6, respectively.
Lemma 7 Suppose that f ∈ L p 2π /r , where 1 ≤ p < ∞ and r ∈ N. If the condition (13) holds with any function ω of the modulus of continuity type and γ , β ≥ 0, then Lemma 8 Suppose that f ∈ L p 2π /r , where 1 ≤ p < ∞ and r ∈ N. If the condition (13) holds with any function ω of the modulus of continuity type and γ , β ≥ 0, then