# Pointwise approximation of modified conjugate functions by matrix operators of conjugate Fourier series of $$2\pi /r$$-periodic functions

## Abstract

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 $$2\pi/r$$-periodic and the measure of approximation depends on r-differences of the entries of the considered matrices.

## 1 Introduction

Let $$L_{2\pi/r}^{p}\ (1\leq p<\infty)$$ be the class of all $$2\pi /r$$-periodic real-valued functions, integrable in the Lebesgue sense with the pth power over $$Q_{r}=$$ $$[-\pi/r,\pi/r]$$ with the norm

$$\Vert f \Vert _{L_{2\pi/r}^{p}}= \bigl\Vert f ( \cdot) \bigr\Vert _{L_{2\pi /r}^{p}}:= \biggl( \int_{Q_{r}}\bigl\vert f(t)\bigr\vert ^{p}\,dt \biggr) ^{1/p},$$

where $$r\in \mathbb{N}$$. It is clear that $$L_{2\pi/r}^{p}\subset L_{2\pi/1}^{p}=L_{2\pi}^{p}$$ and for $$f\in L_{2\pi/r}^{p}$$

$$\Vert f \Vert _{L_{2\pi}^{p}}=r^{1/p} \Vert f \Vert _{L_{2\pi/r}^{p}}.$$

Taking into account the above relations, we will consider, for $$f\in L_{2\pi /r}^{1}$$, the trigonometric Fourier series as such a series of $$f\in L_{2\pi }^{1}$$ in the following form:

$$Sf(x):=\frac{a_{0}(f)}{2}+\sum_{\nu=1}^{\infty} \bigl(a_{\nu}(f)\cos\nu x+b_{\nu}(f)\sin\nu x \bigr)$$

with the partial sums $$S_{k}f$$ and the conjugate one

$$\widetilde{S}f(x):=\sum_{\nu=1}^{\infty} \bigl(a_{\nu}(f)\sin\nu x-b_{\nu }(f)\cos\nu x \bigr)$$

with the partial sums $$\widetilde{S}_{k}f$$. We also know that if $$f\in L_{2\pi}^{1}$$, then

$$\widetilde{f} ( x ) :=-\frac{1}{\pi} \int_{0}^{\pi}\psi_{x} ( t ) \frac{1}{2}\cot\frac{t}{2}\,dt=\lim_{\epsilon\rightarrow0^{+}}\widetilde{f} ( x,\epsilon) =\lim_{\epsilon\rightarrow0^{+}}\widetilde{f}_{r} ( x,\epsilon) ,$$

where, for $$r\in \mathbb{N}$$,

$$\widetilde{f}_{r} ( x,\epsilon) := \textstyle\begin{cases} -\frac{1}{\pi} ( \sum_{m=0}^{ [ r/2 ] -1}\int_{\frac{2m\pi}{r}+\epsilon}^{\frac{2 ( m+1 ) \pi}{r}-\epsilon}+\int_{\frac{2 [ r/2 ] \pi}{r}+\epsilon}^{\frac{ ( 2 [ r/2 ] +1 ) \pi}{r}} ) \psi_{x} ( t ) \frac{1}{2}\cot\frac{t}{2}\,dt&\text{for an odd }r, \\ -\frac{1}{\pi}\sum_{m=0}^{ [ r/2 ] -1}\int_{\frac{2m\pi}{r}+\epsilon}^{\frac{2 ( m+1 ) \pi}{r}-\epsilon}\psi_{x} ( t ) \frac{1}{2}\cot\frac{t}{2}\,dt&\text{for an even }r,\end{cases}$$

and

$$\widetilde{f} ( x,\epsilon) =\widetilde{f}_{1} ( x,\epsilon) :=- \frac{1}{\pi} \int_{\epsilon}^{\pi}\psi_{x} ( t ) \frac{1}{2}\cot\frac{t}{2}\,dt,$$

with

$$\psi_{x} ( t ) :=f ( x+t ) -f ( x-t ) ,$$

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

$$a_{n,k}\geq0\quad\text{when }k,n=0,1,2,\ldots, \qquad\lim _{n\rightarrow \infty}a_{n,k}=0\quad\text{and}\quad\sum _{k=0}^{\infty}a_{n,k}=1,$$

but $$A^{\circ}:= ( a_{n,k} ) _{k=0}^{n}$$, where

$$a_{n,k}=0\quad\text{when }k>n.$$

We will use the notations

$$A_{n,r}=\sum_{k=0}^{\infty} \vert a_{n,k}-a_{n,k+r} \vert ,\qquad A_{n,r}^{\circ}= \sum_{k=0}^{n} \vert a_{n,k}-a_{n,k+r} \vert \text{ }$$

for $$r\in \mathbb{N}$$ and

$$\widetilde{T}_{n,A} f ( x ) :=\sum _{k=0}^{\infty}a_{n,k}\widetilde{S}_{k}f ( x ) \quad ( n=0,1,2,\ldots )$$

for the A-transformation of S̃f.

In this paper, we will study the estimate of $$\vert \widetilde{T}_{n,A} f ( x ) -\widetilde{f}_{r} ( x,\epsilon ) \vert$$ by the function of modulus of continuity type, i.e. a nondecreasing continuous function ω̃ having the following properties: $$\widetilde{\omega} ( 0 ) =0$$, $$\widetilde{\omega}( \delta_{1}+\delta_{2} ) \leq\widetilde{\omega} ( \delta _{1} ) +\widetilde{\omega} ( \delta_{2} )$$ for any $$0\leq \delta_{1}\leq\delta_{2}\leq\delta_{1}+\delta_{2}\leq2\pi$$. We will also consider functions from the subclass $$L_{2\pi/r}^{p} ( \widetilde{\omega} ) _{\beta}$$ of $$L_{2\pi/r}^{p}$$ for $$r\in\mathbb{ N}$$:

$$L_{2\pi/r}^{p} ( \widetilde{\omega} ) _{\beta}= \bigl\{ f \in L_{2\pi/r}^{p}:\widetilde{\omega}_{\beta} ( f,\delta ) _{L_{2\pi/r}^{p}}=O \bigl( \widetilde{\omega} ( \delta) \bigr) \text{ when } \delta\in[ 0,2\pi] \text{ and }\beta\geq0 \bigr\} ,$$

where

$$\widetilde{\omega}_{\beta}f ( \delta) _{L_{2\pi /r}^{p}}=\sup _{0\leq \vert t \vert \leq\delta} \biggl\{ \biggl\vert \sin \frac{rt}{2} \biggr\vert ^{\beta} \bigl\Vert \psi_{\cdot} ( t ) \bigr\Vert _{L_{2\pi/r}^{p}} \biggr\} .$$

It is easy to see that $$\widetilde{\omega}_{0}f ( \cdot ) _{L_{2\pi/r}^{p}}=\widetilde{\omega}f ( \cdot ) _{L_{2\pi /r}^{p}}$$ is the classical modulus of continuity. Moreover, it is clear that for $$\beta\geq\alpha\geq0$$

$$\widetilde{\omega}_{\beta}f ( \delta) _{L_{2\pi/r}^{p}}\leq \widetilde{ \omega}_{\alpha}f ( \delta) _{L_{2\pi/r}^{p}}$$

and consequently

$$L_{2\pi/r}^{p} ( \widetilde{\omega} ) _{\alpha}\subseteq L_{2\pi /r}^{p} ( \widetilde{\omega} ) _{\beta}.$$

The deviation $$\widetilde{T}_{n,A} f ( x ) -\widetilde{f}_{r} ( x,\epsilon )$$ was estimated with $$r=1$$ in [2] and generalized in [1] as follows:

### Theorem A

([1, Theorem 8, p. 95])

If $$f\in L_{2\pi }^{p} ( \widetilde{\omega} ) _{\beta}$$ with $$1< p<\infty$$ and $$0\leq\beta<1-\frac{1}{p}$$, where ω̃ satisfies the conditions:

$$\biggl\{ \int_{\frac{\pi}{n+1} }^{\pi} \biggl( \frac{t^{-\gamma} \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( t ) } \biggr) ^{p}\sin^{\beta p}\frac{t}{2}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma} \bigr)$$
(1)

with $$0<\gamma<\beta+\frac{1}{p}$$ and

$$\biggl\{ \int_{0}^{\frac{\pi}{n+1}} \biggl( \frac{t \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( t ) } \biggr) ^{p}\sin^{\beta p}\frac{t}{2}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{-1} \bigr) ,$$
(2)

then

$$\biggl\vert \widetilde{T}_{n,A^{\circ}} f ( x ) -\widetilde{f} \biggl( x,\frac{\pi}{n+1} \biggr) \biggr\vert =O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}+1}A_{n,1}^{\circ}\widetilde{\omega } \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

The next essential generalizations and improvements in [3, Theorem 1] were given. In these results $$\widetilde{f}_{r} ( x,\epsilon )$$ and $$A_{n,r}$$ (with $$r\in \mathbb{N}$$) instead of $$\widetilde{f}_{1} ( x,\epsilon ) =\widetilde{f}( x,\epsilon )$$ and $$A_{n,1}^{\circ}$$, respectively, were taken. We can formulate them as follows.

### Theorem B

([3, Theorem 1])

If $$f\in L_{2\pi }^{p}$$, $$1< p<\infty$$, $$0\leq\beta<1-\frac{1}{p}$$ and a function ω̃ of modulus of continuity type satisfies the conditions:

$$\biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{t \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{-1} \bigr)$$
(3)

for $$r\in \mathbb{N}$$,

$$\biggl\{ \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t-\frac{2m\pi}{r} ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} ( 1 )$$
(4)

for a natural $$r\geq3$$, where $$m\in \{ 1,\ldots [ \frac{r}{2} ] \}$$ when r is an odd or $$m\in \{ 1,\ldots [ \frac{r}{2} ] -1 \}$$ when r is an even natural number, and

$$\biggl\{ \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2m\pi}{r}+\frac{\pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) ( t-\frac{2m\pi}{r} ) ^{\gamma }} \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma } \bigr) ,$$
(5)

for $$r\in \mathbb{N}$$ with $$0<\gamma<\beta+\frac{1}{p}$$, where $$m\in \{ 0,\ldots [ \frac{r}{2} ] \}$$ when r is an odd or $$m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \}$$ when r is an even natural number. Moreover, let ω̃ satisfy, for a natural $$r\geq 2$$, the conditions:

\begin{aligned} &\biggl\{ \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( \frac{2 ( m+1 ) \pi}{r}-t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} ( 1 ) , \end{aligned}
(6)
\begin{aligned} &\biggl\{ \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) ( \frac{2 ( m+1 ) \pi}{r}-t ) ^{\gamma}} \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma} \bigr) , \end{aligned}
(7)

with $$0<\gamma<\beta+\frac{1}{p}$$, where $$m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \}$$. If a matrix A is such that

$$\sum_{k=0}^{\infty} ( k+1 ) ^{2}a_{n,k}=O \bigl( ( n+1 ) ^{2} \bigr)$$
(8)

and

$$\Biggl[ \sum_{l=0}^{n}\sum _{k=l}^{r+l-1}a_{n,k} \Biggr] ^{-1}=O ( 1 )$$
(9)

with $$r\in \mathbb{N}$$ are true, then

$$\biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert =O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}+1}A_{n,r} \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

### Theorem C

([3, Theorem 2])

Let $$f\in L_{2\pi }^{p}$$, $$1< p<\infty$$, $$0\leq\beta<1-\frac{1}{p}$$ and a function ω̃ of modulus of continuity type satisfy, for $$r\in \mathbb{N}$$, the conditions: (4) and (5) with $$0<\gamma<\beta+\frac{1}{p}$$, where $$m\in \{ 0,\ldots [ \frac{r}{2} ] \}$$ when r is an odd or $$m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \}$$ when r is an even natural number. Moreover, let ω̃ satisfy, for a natural $$r\geq 2$$, the conditions (6) and (7) with $$0<\gamma<\beta+\frac{1}{p}$$, where $$m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \}$$. If a matrix A is such that

$$\sum_{k=0}^{\infty} ( k+1 ) a_{n,k}=O ( n+1 ) ,$$
(10)

and (9) with $$r\in \mathbb{N}$$ are true, then

$$\biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert =O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}+1}A_{n,r} \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

In our theorems we generalize the above results considering $$2\pi/r$$-periodic functions and using simpler assumptions.

In the paper $$\sum_{k=a}^{b}=0$$ when $$a>b$$.

## 2 Statement of the results

To begin with, we will present the estimates of the quantities

$$\biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert \quad\text{and}\quad \biggl\Vert \widetilde{T}_{n,A} f ( \cdot) -\widetilde{f}_{r} \biggl( \cdot, \frac{\pi}{r ( n+1 ) } \biggr) \biggr\Vert _{L_{2\pi/r}^{p}}.$$

Finally, we will formulate some remarks and corollaries.

### Theorem 1

Suppose that $$f\in L_{2\pi/r}^{p}$$, $$1< p<\infty$$, $$r\in\mathbb{N}$$, $$0\leq\beta<1-\frac{1}{p}$$ and a function ω̃ of the modulus of continuity type satisfies the conditions:

$$\biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{t \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2}\vert ^{\beta}}{\widetilde{\omega} ( t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{-1} \bigr) ,$$
(11)

when $$r=1$$ or

$$\biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2}\vert ^{\beta}}{\widetilde{\omega} ( t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} ( 1 ) ,$$
(12)

when $$r\geq2$$, and

$$\biggl\{ \int_{\frac{\pi}{r ( n+1 ) }}^{\frac{\pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) t^{\gamma}} \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma} \bigr) ,$$
(13)

for $$r\in \mathbb{N}$$ with $$0<\gamma<\beta+\frac{1}{p}$$. If a matrix A is such that (8) and (9) are true, then

$$\biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert =O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}+1}A_{n,r} \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

### Theorem 2

Suppose that $$f\in L_{2\pi/r}^{p}$$, $$1< p<\infty$$, $$r\in\mathbb{N}$$, $$0\leq\beta<1-\frac{1}{p}$$ and a function ω̃ of the modulus of continuity type satisfies the conditions (12) and (13) for $$r\in \mathbb{N}$$ with $$0<\gamma<\beta+\frac{1}{p}$$. If a matrix A is such that (10) and (9) are true, then

$$\biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert =O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}+1}A_{n,r} \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

### Remark 1

The Hölder inequality gives

\begin{aligned} \sum_{k=0}^{\infty} ( k+1 ) a_{n,k} &=\sum_{k=0}^{\infty} ( k+1 ) a_{n,k}^{1/2}a_{n,k}^{1/2}\leq \Biggl[ \sum _{k=0}^{\infty} ( k+1 ) ^{2}a_{n,k} \Biggr] ^{1/2} \Biggl[ \sum_{k=0}^{\infty}a_{n,k} \Biggr] ^{1/2} \\ &= \Biggl[ \sum_{k=0}^{\infty} ( k+1 ) ^{2}a_{n,k} \Biggr] ^{1/2} \end{aligned}

and thus the condition (8) implies (10), but the condition (12) implies (11). Therefore Theorems 1 and 2 are not comparable.

### Theorem 3

Let $$f\in L_{2\pi/r}^{p} ( \widetilde{\omega} ) _{\beta}$$, $$1< p<\infty$$, $$r\in\mathbb{N}$$ and $$0\leq\beta<1-\frac{1}{p}$$. If a matrix A is such that (9) and (8) or (10) are true, then

$$\biggl\Vert \widetilde{T}_{n,A} f ( \cdot) - \widetilde{f}_{r} \biggl( \cdot,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\Vert _{L_{2\pi/r}^{p}}=O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}+1}A_{n,r}\widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

### Corollary 1

Taking $$r=1$$ the conditions (11) and (13) in Theorem 1 reduce to (1) and (2). 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 $$\beta>0$$ we can formulate, without proofs, the following theorems.

### Theorem 4

Suppose that $$f\in L_{2\pi/r}^{p}$$, $$1< p<\infty$$, $$r\in\mathbb{N}$$, $$0<\beta<1-\frac{1}{p}$$. Let a function ω̃ of the modulus of continuity type satisfy the conditions:

$$\biggl\{ \int_{\frac{\pi}{r ( n+1 ) }}^{\frac{\pi}{r}} \biggl( \frac{t^{-\gamma} \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma-\frac{1}{p}} \bigr)$$
(14)

for $$\gamma\in ( \frac{1}{p},\frac{1}{p}+\beta )$$ and $$r\in \mathbb{N}$$ (instead of (13)), and

$$\biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{t \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2}\vert ^{\beta}}{\widetilde{\omega} ( t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{-1-\frac{1}{p}} \bigr)$$

when $$r=1$$ or

$$\biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2}\vert ^{\beta}}{\widetilde{\omega} ( t ) } \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{-\frac{1}{p}} \bigr)$$
(15)

when $$r\geq2$$ (instead of (11) and (12), respectively). If a matrix A is such that (9) and (8) are true, then

$$\biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f_{r}} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert =O_{x} \biggl( ( n+1 ) ^{\beta+1}A_{n,r} \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$
(16)

Moreover, if a function ω̃ of the modulus of continuity type and a matrix A satisfy the following conditions: (14) with $$r\in \mathbb{N}$$ and $$\gamma\in ( \frac{1}{p},\frac{1}{p}+\beta )$$, (15) with $$r\in \mathbb{N}$$, (9) and (10), then the estimate (16) is also true.

### Theorem 5

Let $$f\in L_{2\pi/r}^{p} ( \widetilde{\omega} ) _{\beta}$$ with $$1< p<\infty$$, $$r\in\mathbb{N}$$ and $$0<\beta<1-\frac{1}{p}$$. If a matrix A is such that (9) and (8) or (10) are true, then

$$\biggl\Vert \widetilde{T}_{n,A} f ( \cdot) - \widetilde{f}_{r} \biggl( \cdot,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\Vert _{L_{2\pi/r}^{p}}=O_{x} \biggl( ( n+1 ) ^{\beta+1}A_{n,r}\widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

### Remark 2

We note that our extra conditions (9), (8) and (10) for a lower triangular infinite matrix $$A^{\circ}$$ always hold.

### Corollary 2

Considering the above remarks and the obvious inequality

$$A_{n,r}\leq rA_{n,1}\quad\textit{for }r\in \mathbb{N}$$
(17)

our results also improve and generalize the mentioned result of Krasniqi [1].

### Remark 3

We note that instead of $$L_{2\pi/r}^{p} ( \widetilde{\omega} ) _{\beta}$$ one can consider another subclass of $$L_{2\pi/r}^{p}$$ generated by any function of the modulus of continuity type e.g. $$\widetilde{\omega}_{x}$$ such that

$$\widetilde{\omega}_{x}(f,\delta)=\sup_{ \vert t \vert \leq \delta } \bigl\vert \psi_{x} ( t ) \bigr\vert \leq\widetilde{\omega }_{x} ( \delta)$$

or

$$\widetilde{\omega}_{x}(f,\delta)=\frac{1}{\delta} \int_{0}^{\delta } \bigl\vert \psi_{x} ( t ) \bigr\vert \,dt\leq\widetilde{\omega}_{x} ( \delta) .$$

## 3 Auxiliary results

We begin this section by some notations from [5] and [4, Sect. 5 of Chapter II]. Let for $$r=1,2,\ldots$$

$$D_{k,r}^{\circ} ( t ) =\frac{\sin\frac{ ( 2k+r ) t}{2}}{2\sin\frac{rt}{2}},\qquad \widetilde{D^{\circ}}_{k,r} ( t ) =\frac{\cos\frac{ ( 2k+r ) t}{2}}{2\sin\frac{rt}{2}}$$

and

$$\widetilde{D}_{k,r} ( t ) =\frac{\cos\frac{rt}{2}-\cos \frac{ ( 2k+r ) t}{2}}{2\sin\frac{rt}{2}}= \frac{\cos\frac{rt}{2}}{2\sin\frac{rt}{2}}-\widetilde{D^{\circ}}_{k,r} ( t ) .$$

It is clear by [4] that

$$\widetilde{S}_{k}f ( x ) =-\frac{1}{\pi} \int_{-\pi}^{\pi}f ( x+t ) \widetilde{D}_{k,1} ( t ) \,dt$$

and

$$\widetilde{T}_{n,A} f ( x ) =-\frac{1}{\pi} \int_{-\pi }^{\pi}f ( x+t ) \sum _{k=0}^{\infty}a_{n,k}\widetilde{D}_{k,1} ( t ) \,dt.$$

Now, we present a very useful property of the modulus of continuity.

### Lemma 1

([4])

A function ω̃ of modulus of continuity type on the interval $$[0,2\pi]$$ satisfies the following condition:

$$\delta_{2}^{-1}\widetilde{\omega} ( \delta _{2} ) \leq2\delta_{1}^{-1}\widetilde{\omega} ( \delta_{1} ) \quad\textit{for } \delta_{2}\geq\delta _{1}>0.$$

Next, we present the following well-known estimates.

### Lemma 2

([4])

If $$0< \vert t \vert \leq\pi$$ then

$$\bigl\vert \widetilde{D^{\circ}}_{k,1} ( t ) \bigr\vert \leq \frac{\pi}{2 \vert t \vert }, \qquad\bigl\vert \widetilde{D}_{k,1} ( t ) \bigr\vert \leq\frac{\pi}{ \vert t \vert }$$

and, for any real t, we have

$$\bigl\vert D_{k,1}^{\circ} ( t ) \bigr\vert \leq k+ \frac{1}{2},\qquad \bigl\vert \widetilde{D}_{k,1} ( t ) \bigr\vert \leq\frac{1}{2}k ( k+1 ) \vert t \vert , \qquad \bigl\vert \widetilde{D}_{k,1} ( t ) \bigr\vert \leq k+1.$$

### Lemma 3

([5, 6])

Let $$r\in N$$, $$l\in Z$$ and $$(a_{n})\subset \mathbb{C}$$. If $$t \neq \frac{2l\pi}{r}$$, then for every $$m\geq n$$

\begin{aligned} &\sum_{k=n}^{m}a_{k}\sin kt =-\sum_{k=n}^{m} ( a_{k}-a_{k+r} ) \widetilde{D^{\circ}}_{k,r} ( t ) +\sum _{k=m+1}^{m+r}a_{k}\widetilde{D^{\circ}}_{k,-r} ( t ) -\sum _{k=n}^{n+r-1}a_{k}\widetilde{D^{\circ}}_{k,-r} ( t ) , \\ &\sum_{k=n}^{m}a_{k}\cos kt =\sum_{k=n}^{m} ( a_{k}-a_{k+r} ) D_{k,r}^{\circ} ( t ) -\sum_{k=m+1}^{m+r}a_{k}D_{k,-r}^{\circ } ( t ) +\sum_{k=n}^{n+r-1}a_{k}D_{k,-r}^{\circ} ( t ) . \end{aligned}

We additionally need the following estimate as a consequence of Lemma 3.

### Lemma 4

Let $$r\in \mathbb{N}$$, $$l\in \mathbb{Z}$$ and $$(a_{n,k})\subset \mathbb{R} _{0}^{+}$$ for $$n.k\in \mathbb{N} _{0}$$. If $$t \neq \frac{2l\pi }{r}$$, then

$$\Biggl\vert \frac{1}{2}\sum_{k=0}^{\infty}a_{n,k} \cos\frac{ ( 2k+1 ) t}{2} \Biggr\vert \leq\frac{1}{2 \vert \sin\frac{rt}{2}\vert } \Biggl( A_{n,r}+\sum_{k=0}^{r-1}a_{n,k} \Biggr) \leq\frac{1}{\vert \sin\frac{rt}{2} \vert }A_{n,r}.$$

### Proof

By Lemma 3,

\begin{aligned} &\frac{1}{2}\sum_{k=0}^{\infty}a_{n,k} \cos\frac{ ( 2k+1 ) t}{2} \\ &\quad =\frac{1}{2} \Biggl( \sum_{k=0}^{\infty}a_{n,k} \cos kt\cos\frac{t}{2}-\sum_{k=0}^{\infty}a_{n,k} \sin kt\sin\frac{t}{2} \Biggr) \\ &\quad=\frac{\cos\frac{t}{2}}{2} \Biggl( \sum_{k=0}^{\infty} ( a_{n,k}-a_{n,k+r} ) D_{k,r}^{\circ} ( t ) + \sum_{k=0}^{r-1}a_{n,k}D_{k,-r}^{\circ} ( t ) \Biggr) \\ &\qquad{}-\frac{\sin\frac{t}{2}}{2} \Biggl( -\sum_{k=0}^{\infty} ( a_{n,k}-a_{n,k+r} ) \widetilde{D^{\circ}}_{k,r} ( t ) -\sum_{k=0}^{r-1}a_{n,k} \widetilde{D^{\circ}}_{k,-r} ( t ) \Biggr) \end{aligned}

and our inequalities follow. □

We also need some special conditions which follow from the ones mentioned above.

### Lemma 5

Suppose that $$f\in L_{2\pi/r}^{p}$$, where $$1\leq p<\infty$$ and $$r\in \mathbb{N}$$. If the condition (12) holds with any function ω̃ of the modulus of continuity type and $$\beta \geq0$$, then

$$\biggl\{ \int _{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( \frac{2 ( m+1 ) \pi}{r}-t ) } \biggr) ^{p} \biggl\vert \sin\frac{rt}{2} \biggr\vert ^{\beta p}\,dt \biggr\} ^{\frac{1}{p}}=O_{x} ( 1 ) ,$$

where $$m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \}$$.

### Proof

By the substitution $$t=\frac{2 ( m+1 ) \pi}{r}-u$$, we obtain

\begin{aligned} & \biggl\{ \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( \frac{2 ( m+1 ) \pi}{r}-t ) } \biggr) ^{p} \biggl\vert \sin\frac{rt}{2} \biggr\vert ^{\beta p}\,dt \biggr\} ^{1/p} \\ &\quad= \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\vert \psi_{x} ( \frac{2 ( m+1 ) \pi}{r}-u ) \vert }{\widetilde{\omega} ( u ) } \biggl\vert \sin\frac{r}{2} \biggl( \frac{2 ( m+1 ) \pi}{r}-u \biggr) \biggr\vert ^{\beta } \biggr) ^{p}\,du \biggr\} ^{1/p} \\ &\quad= \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\vert \psi_{x} ( u ) \vert }{\widetilde{\omega} ( u ) } \biggl\vert \sin\frac{ru}{2} \biggr\vert ^{\beta} \biggr) ^{p}\,du \biggr\} ^{1/p}. \end{aligned}

Hence, by (12) our estimate follows. □

### Lemma 6

Suppose that $$f\in L_{2\pi/r}^{p}$$, where $$1\leq p<\infty$$ and $$r\in \mathbb{N}$$. If the condition (12) holds with any function ω̃ of the modulus of continuity type and $$\beta \geq0$$, then

$$\biggl\{ \int _{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( t-\frac{2m\pi}{r} ) } \biggr) ^{p} \biggl\vert \sin\frac{rt}{2} \biggr\vert ^{\beta p}\,dt \biggr\} ^{\frac{1}{p}}=O_{x} ( 1 ) ,$$

where $$m\in \{ 0,\ldots [ \frac{r}{2} ] \}$$.

### Proof

By the substitution $$t=\frac{2m\pi}{r}+u$$, analogously to the above proof, we obtain

\begin{aligned} & \biggl\{ \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( t-\frac{2m\pi}{r} ) } \biggr) ^{p} \biggl\vert \sin\frac{rt}{2} \biggr\vert ^{\beta p}\,dt \biggr\} ^{1/p} \\ &\quad= \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\vert \psi_{x} ( \frac{2m\pi}{r}+u ) \vert }{\widetilde{\omega} ( u ) } \biggl\vert \sin\frac{r}{2} \biggl( \frac{2m\pi}{r}+u \biggr) \biggr\vert ^{\beta} \biggr) ^{p}\,du \biggr\} ^{1/p} \\ &\quad\leq \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\vert \psi_{x} ( u ) \vert }{\widetilde{\omega} ( u ) } \biggl\vert \sin\frac{ru}{2} \biggr\vert ^{\beta} \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} ( 1 ) \end{aligned}

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 $$f\in L_{2\pi/r}^{p}$$, where $$1\leq p<\infty$$ and $$r\in \mathbb{N}$$. If the condition (13) holds with any function ω̃ of the modulus of continuity type and $$\gamma ,\beta\geq0$$, then

$$\biggl\{ \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) ( \frac{2 ( m+1 ) \pi}{r}-t ) ^{\gamma}} \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma} \bigr) ,$$

where $$m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \}$$.

### Lemma 8

Suppose that $$f\in L_{2\pi/r}^{p}$$, where $$1\leq p<\infty$$ and $$r\in \mathbb{N}$$. If the condition (13) holds with any function ω̃ of the modulus of continuity type and $$\gamma ,\beta\geq0$$, then

$$\biggl\{ \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2m\pi}{r}+\frac{\pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \vert \sin\frac{rt}{2} \vert ^{\beta}}{\widetilde{\omega} ( t ) ( t-\frac{2m\pi}{r} ) ^{\gamma}} \biggr) ^{p}\,dt \biggr\} ^{1/p}=O_{x} \bigl( ( n+1 ) ^{\gamma} \bigr) ,$$

where $$m\in \{ 0,\ldots [ \frac{r}{2} ] \}$$.

## 4 Proofs of theorems

### 4.1 Proof of Theorem 1

It is clear that for odd r

\begin{aligned} &\widetilde{T}_{n,A} f ( x ) -\widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \\ &\quad=-\frac{1}{\pi} \int_{0}^{\pi}\psi_{x} ( t ) \sum _{k=0}^{\infty }a_{n,k} \widetilde{D}_{k,1} ( t ) \,dt \\ &\qquad{}+\frac{1}{\pi} \Biggl( \sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}+ \int_{\frac{2 [ r/2 ] \pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{ ( 2 [ r/2 ] +1 ) \pi}{r}} \Biggr) \psi_{x} ( t ) \frac{1}{2}\cot\frac{t}{2}\,dt \\ &\quad=-\frac{1}{\pi} \Biggl( \int_{0}^{\frac{\pi}{r ( n+1 ) }}+\sum_{m=1}^{ [ r/2 ] } \int_{\frac{2m\pi}{r}}^{\frac{2m\pi }{r}+\frac{\pi}{r ( n+1 ) }}+\sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \Biggr) \\ &\qquad{}\times\psi_{x} ( t ) \sum_{k=0}^{\infty}a_{n,k} \widetilde{D}_{k,1} ( t ) \,dt \\ &\qquad{}+\frac{1}{\pi} \Biggl( \sum_{m=0}^{ [ r/2 ] } \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}}+\sum_{m=0}^{[r/2]-1} \int_{\frac{ ( 2m+1 ) \pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \Biggr) \psi_{x} ( t ) \sum_{k=0}^{\infty}a_{n,k} \widetilde{D^{\circ}}_{k,1} ( t ) \,dt \\ &\quad=I_{0} ( x ) +I_{1} ( x ) +I_{2} ( x ) +I_{3} ( x ) +I_{4} ( x ) \end{aligned}

and for even r

\begin{aligned} &\widetilde{T}_{n,A} f ( x ) -\widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \\ &\quad=-\frac{1}{\pi} \int_{0}^{\pi}\psi_{x} ( t ) \sum _{k=0}^{\infty }a_{n,k} \widetilde{D}_{k,1} ( t ) \,dt+\frac{1}{\pi}\sum _{m=0}^{ [ r/2 ] -1} \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}\psi_{x} ( t ) \frac{1}{2}\cot\frac{t}{2}\,dt \\ &\quad=-\frac{1}{\pi} \Biggl( \int_{0}^{\frac{\pi}{r ( n+1 ) }}+\sum_{m=1}^{ [ r/2 ] -1} \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}+\sum_{m=0}^{ [ r/2] -1} \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \Biggr) \\ &\qquad{}\times\psi_{x} ( t ) \sum_{k=0}^{\infty}a_{n,k} \widetilde{D}_{k,1} ( t ) \,dt \\ &\qquad{}+\frac{1}{\pi} \Biggl( \sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{ ( 2m+1 ) \pi}{r}}+\sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{ ( 2m+1 ) \pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \Biggr) \\ &\qquad{}\times\psi_{x} ( t ) \sum_{k=0}^{\infty}a_{n,k} \widetilde{D^{\circ}}_{k,1} ( t ) \,dt \\ &\quad=I_{0} ( x ) +I_{1}^{\prime} ( x ) +I_{2} ( x ) +I_{3}^{\prime} ( x ) +I_{4} ( x ) , \end{aligned}

whence

\begin{aligned} & \biggl\vert \widetilde{T}_{n,A} f ( x ) - \widetilde{f}_{r} \biggl( x,\frac{\pi}{r ( n+1 ) } \biggr) \biggr\vert \\ &\quad \leq \bigl\vert I_{0} ( x ) \bigr\vert + \bigl\vert I_{1} ( x ) \bigr\vert + \bigl\vert I_{1}^{\prime} ( x ) \bigr\vert + \bigl\vert I_{2} ( x ) \bigr\vert + \bigl\vert I_{3} ( x ) \bigr\vert + \bigl\vert I_{3}^{\prime} ( x ) \bigr\vert + \bigl\vert I_{4} ( x ) \bigr\vert . \end{aligned}

Next, using Lemma 2, (8), the Hölder inequality with $$p>1$$ and $$q=\frac{p}{p-1}$$ and (11) when $$r=1$$ or (12) when $$r\geq2$$ we get

\begin{aligned} & \bigl\vert I_{0} ( x ) \bigr\vert \\ &\quad=O \bigl( ( n+1 ) ^{2} \bigr) \int_{0}^{\frac{\pi}{r ( n+1 ) }}t \bigl\vert \psi_{x} ( t ) \bigr\vert \,dt \\ &\quad\leq O \bigl( ( n+1 ) ^{2} \bigr) \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{t \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( t ) } \biggr) ^{p}\sin^{\beta p}\frac{rt}{2}\,dt \biggr\} ^{1/p} \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\widetilde{\omega} ( t ) }{\sin^{\beta}\frac{rt}{2}} \biggr) ^{q}\,dt \biggr\} ^{\frac{1}{q}} \\ &\quad\leq O \bigl( ( n+1 ) ^{2} \bigr) O_{x} \bigl( ( n+1 ) ^{-1} \bigr) \widetilde{\omega} \biggl( \frac{\pi}{r ( n+1 ) } \biggr) \biggl\{ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\pi }{rt} \biggr) ^{\beta q}\,dt \biggr\} ^{\frac{1}{q}} \\ &\quad=O_{x} \bigl( ( n+1 ) \bigr) \widetilde{\omega} \biggl( \frac{\pi }{r ( n+1 ) } \biggr) \biggl( \frac{\pi}{r ( n+1 ) } \biggr) ^{\frac {1}{q}-\beta}=O_{x} \bigl( ( n+1 ) ^{\beta+\frac{1}{p}} \bigr) \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) , \end{aligned}

for $$0\leq\beta<1-\frac{1}{p}$$. We note that applying the condition (9) we have

\begin{aligned} \bigl[ ( n+1 ) A_{n,r} \bigr] ^{-1}&= \Biggl[ \sum _{l=0}^{n}A_{n,r} \Biggr] ^{-1}\leq \Biggl[ \sum_{l=0}^{n} \sum_{k=l}^{\infty} \vert a_{n,k}-a_{n,k+r} \vert \Biggr] ^{-1} \\ &\leq \Biggl[ \sum_{l=0}^{n} \Biggl\vert \sum_{k=l}^{\infty} ( a_{n,k}-a_{n,k+r} ) \Biggr\vert \Biggr] ^{-1}= \Biggl[ \sum _{l=0}^{n}\sum _{k=l}^{r+l-1}a_{n,k} \Biggr] ^{-1}=O ( 1 ) , \end{aligned}

whence

$$\bigl\vert I_{0} ( x ) \bigr\vert =O_{x} \biggl( ( n+1 ) ^{1+\beta+\frac{1}{p}}A_{n,r}\widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) .$$

By Lemma 2

\begin{aligned} & \bigl\vert I_{1} ( x ) \bigr\vert + \bigl\vert I_{1}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{2} ( x ) \bigr\vert \\ &\quad\leq \frac{1}{\pi} \Biggl( \sum_{m=1}^{ [ r/2 ] } \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}+\sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{2 ( m+1 ) \pi }{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \Biggr) \frac{ \vert \psi_{x} ( t ) \vert }{t}\,dt \\ &\quad \leq\frac{1}{\pi} \Biggl( \sum_{m=1}^{ [ r/2 ] } \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}+\sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{2 ( m+1 ) \pi }{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \Biggr) \frac{ \vert \psi_{x} ( t ) \vert }{\pi/r}\,dt \end{aligned}

and using the Hölder inequality with $$p>1$$ and $$q=\frac{p}{p-1}$$

\begin{aligned} & \bigl\vert I_{1} ( x ) \bigr\vert + \bigl\vert I_{1}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{2} ( x ) \bigr\vert \\ &\quad \leq O_{x} ( 1 ) \sum_{m=1}^{ [ r/2 ] } \biggl[ \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \sin^{\beta}\frac{rt}{2}}{\widetilde{\omega} ( t-\frac{2m\pi}{r} ) } \biggr) ^{p}\,dt \biggr] ^{\frac{1}{p}} \\ &\qquad{}\times \biggl[ \int_{\frac{2m\pi}{r}}^{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\widetilde{\omega} ( t-\frac{2m\pi}{r}) }{\sin^{\beta}\frac{rt}{2}} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}} \\ &\qquad{}+O_{x} ( 1 ) \sum_{m=1}^{ [ r/2 ] -1} \biggl[ \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert \sin^{\beta}\frac{rt}{2}}{\widetilde{\omega} ( \frac{2 ( m+1 ) \pi}{r}-t ) } \biggr) ^{p}\,dt \biggr] ^{\frac{1}{p}} \\ &\qquad{}\times \biggl[ \int_{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}^{\frac{2 ( m+1 ) \pi}{r}} \biggl( \frac{\widetilde{\omega} ( \frac{2 ( m+1 ) \pi}{r}-t ) }{\sin^{\beta}\frac{rt}{2}} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}}. \end{aligned}

Hence, by Lemmas 5 and 6 with (12) and (9),

\begin{aligned} &\bigl\vert I_{1} ( x ) \bigr\vert + \bigl\vert I_{1}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{2} ( x ) \bigr\vert \\ &\quad=O_{x} ( 1 ) \widetilde{\omega} \biggl( \frac{\pi}{r ( n+1 ) } \biggr) \biggl[ \int_{0}^{\frac{\pi}{r ( n+1 ) }} \biggl( \frac{1}{\sin^{\beta}\frac {rt}{2}} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}} \\ &\quad =O_{x} \bigl( ( n+1 ) ^{\beta-\frac{1}{q}} \bigr) \widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) =O_{x} \biggl( ( n+1 ) ^{\beta+\frac{1}{p}}A_{n,r}\widetilde{\omega} \biggl( \frac{\pi}{n+1} \biggr) \biggr) , \end{aligned}

for $$0\leq\beta<1-\frac{1}{p}$$.

In the case of the last integrals, applying Lemma 4 we obtain

\begin{aligned} & \bigl\vert I_{3} ( x ) \bigr\vert + \bigl\vert I_{3}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{4} ( x ) \bigr\vert \\ &\quad\leq\frac{1}{\pi} \Biggl( \sum_{m=0}^{ [ r/2 ] } \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{ ( 2m+1 ) \pi}{r}}+\sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{ ( 2m+1 ) \pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \Biggr) \frac{ \vert \psi_{x} ( t ) \vert }{\vert \sin\frac{t}{2}\sin\frac{rt}{2} \vert }A_{n,r}\,dt. \end{aligned}

Using the estimates $$\vert \sin\frac{t}{2} \vert \geq\frac{\vert t \vert }{\pi}$$ for $$t\in [ 0,\pi ]$$, $$\vert \sin\frac{rt}{2} \vert \geq\frac{rt}{\pi}-2m$$ for $$t\in [ \frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) },\frac{ ( 2m+1 ) \pi}{r} ]$$, where $$m\in \{ 0,\ldots, [ r/2 ] \}$$ and $$\vert \sin\frac{rt}{2} \vert \geq2 ( m+1 ) -\frac{rt}{\pi}$$ for $$t\in [ \frac{ ( 2m+1 ) \pi}{r},\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) } ]$$, where $$m\in \{ 0,\ldots, [ r/2 ] -1 \}$$, we obtain

\begin{aligned} & \bigl\vert I_{3} ( x ) \bigr\vert + \bigl\vert I_{3}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{4} ( x ) \bigr\vert \\ &\quad\leq A_{n,r}\sum_{m=0}^{ [ r/2 ] } \int_{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{ ( 2m+1 ) \pi}{r}}\frac{\vert \psi_{x} ( t ) \vert }{\frac{rt}{\pi} ( t-\frac{2m\pi}{r} ) }\,dt \\ &\qquad{}+A_{n,r}\sum_{m=0}^{ [ r/2 ] -1} \int_{\frac{ ( 2m+1 ) \pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }}\frac{ \vert \psi_{x} ( t ) \vert }{\frac{rt}{\pi} [ \frac{2 ( m+1 ) \pi}{r}-t ] }\,dt. \end{aligned}

By the Hölder inequality with $$p>1$$ and $$q=\frac{p}{p-1}$$ we have

\begin{aligned} & \bigl\vert I_{3} ( x ) \bigr\vert + \bigl\vert I_{3}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{4} ( x ) \bigr\vert \\ &\quad\leq\frac{\pi}{r}A_{n,r}\sum_{m=0}^{ [ r/2 ] } \biggl[ \int _{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2m\pi }{r}+\frac{\pi}{r}} \biggl( \frac{ \vert \psi_{x} ( t ) \vert }{\widetilde{\omega} ( t ) ( t-\frac{2m\pi}{r}) ^{\gamma}} \biggl\vert \sin\frac{rt}{2} \biggr\vert ^{\beta} \biggr) ^{p}\,dt \biggr] ^{\frac{1}{p}} \\ &\qquad{}\times \biggl[ \int _{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2m\pi}{r}+\frac{\pi}{r}} \biggl( \frac{\widetilde{\omega} ( t ) ( t-\frac{2m\pi}{r} ) ^{\gamma}}{t ( t-\frac{2m\pi }{r} ) \vert \sin\frac{rt}{2} \vert ^{\beta}} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}} \\ &\qquad{}+\frac{\pi}{r}A_{n,r}\sum_{m=0}^{ [ r/2 ] -1} \biggl[ \int _{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \biggl( \frac{ \vert \psi _{x} ( t ) \vert }{\widetilde{\omega} ( t ) ( \frac{2 ( m+1 ) \pi}{r}-t ) ^{\gamma}} \biggl\vert \sin\frac{rt}{2} \biggr\vert ^{\beta} \biggr) ^{p}\,dt \biggr] ^{\frac{1}{p}} \\ &\qquad{}\times\biggl[ \int _{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \biggl( \frac {\widetilde{\omega} ( t ) ( \frac{2 ( m+1 ) \pi }{r}-t ) ^{\gamma}}{t ( \frac{2 ( m+1 ) \pi}{r}-t ) \vert \sin\frac{rt}{2} \vert ^{\beta}} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}}. \end{aligned}

Further, using Lemmas 7 and 8 with (13) and Lemma 1 we get

\begin{aligned} & \bigl\vert I_{3} ( x ) \bigr\vert + \bigl\vert I_{3}^{\prime } ( x ) \bigr\vert + \bigl\vert I_{4} ( x ) \bigr\vert \\ &\quad\leq O_{x} ( 1 ) A_{n,r}\sum _{m=0}^{ [ r/2 ] } ( n+1 ) ^{\gamma} \biggl[ \int _{\frac{2m\pi}{r}+\frac{\pi}{r ( n+1 ) }}^{\frac{2m\pi}{r}+\frac{\pi}{r}} \biggl( \frac{\widetilde{\omega} ( t ) ( t-\frac{2m\pi}{r} ) ^{\gamma }}{t ( t-\frac{2m\pi}{r} ) \vert \sin\frac{rt}{2} \vert ^{\beta}} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}} \\ &\qquad{}+O_{x} ( 1 ) A_{n,r}\sum_{m=0}^{ [ r/2 ] -1} ( n+1 ) ^{\gamma} \biggl[ \int _{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r}}^{\frac{2 ( m+1 ) \pi}{r}-\frac{\pi}{r ( n+1 ) }} \biggl( \frac{\widetilde{\omega} ( t ) ( \frac{2 ( m+1 ) \pi}{r}-t ) ^{\gamma}}{t ( \frac{2 ( m+1 ) \pi}{r}-t ) \vert \sin\frac{rt}{2} \vert ^{\beta }} \biggr) ^{q}\,dt \biggr] ^{\frac{1}{q}} \\ &\quad= O_{x} ( 1 ) A_{n,r} \Biggl[ \sum _{m=0}^{ [ r/2 ] } ( n+1 ) ^{\gamma} \biggl\{ \int _{\frac{\pi}{r(n+1)}}^{\frac{\pi}{r}} \biggl( \frac{\widetilde{\omega} ( t+\frac{2m\pi}{r} ) t^{\gamma -1}}{ ( t+\frac{2m\pi}{r} ) \vert \sin\frac{rt}{2}\vert ^{\beta}} \biggr) ^{q}\,dt \biggr\} ^{\frac{1}{q}} \\ &\qquad{}+ \sum_{m=0}^{ [ r/2 ] -1} ( n+1 ) ^{\gamma } \biggl\{ \int _{\frac{\pi}{r(n+1)}}^{\frac{\pi}{r}} \biggl( \frac{\widetilde{\omega} ( \frac{2(m+1)\pi}{r}-t ) t^{\gamma-1}}{( \frac{2(m+1)\pi}{r}-t ) \vert \sin\frac{rt}{2} \vert ^{\beta}} \biggr) ^{q}\,dt \biggr\} ^{\frac{1}{q}} \Biggr] \\ &\quad=O_{x} ( 1 ) A_{n,r} ( n+1 ) ^{\gamma} \biggl\{ \int _{\frac{\pi}{r(n+1)}}^{\frac{\pi}{r}} \biggl( \frac{\widetilde{\omega} ( t ) t^{\gamma-1}}{t \vert \sin\frac{rt}{2}\vert ^{\beta}} \biggr) ^{q}\,dt \biggr\} ^{\frac{1}{q}} \\ &\quad=O_{x} ( 1 ) A_{n,r} ( n+1 ) ^{1+\gamma} \widetilde{\omega} \biggl( \frac{\pi}{r ( n+1 ) } \biggr) \biggl( \int _{\frac{\pi}{r(n+1)}}^{\frac{\pi}{r}}t^{ ( \gamma-1-\beta ) q}\,dt \biggr) ^{\frac{1}{q}} \\ &\quad =O_{x} ( 1 ) A_{n,r} ( n+1 ) ^{1+\gamma} \widetilde{\omega} \biggl( \frac{\pi}{r ( n+1 ) } \biggr) ( n+1 ) ^{1+\beta-\gamma-\frac{1}{q}} \\ &\quad=O_{x} \biggl( ( n+1 ) ^{1+\beta+\frac{1}{p}}A_{n,r} \widetilde{\omega} \biggl( \frac{\pi}{ ( n+1 ) } \biggr) \biggr) \end{aligned}

for $$0<\gamma<\beta+\frac{1}{p}$$.

Collecting the partial estimates our statement follows.

### 4.2 Proof of Theorem 2

The proof is the same as above, but for estimate of $$\vert I_{0} ( x ) \vert$$ we only used the inequality $$\vert \widetilde{D}_{k,1} ( t ) \vert \leq k+1$$ from Lemma 2, and the condition (10) instead of (8).

### 4.3 Proof of Theorem 3

We note that for the estimate of $$\Vert \widetilde{T}_{n,A} f ( \cdot ) -\widetilde{f}_{r} ( \cdot,\frac{\pi}{(n+1)}) \Vert _{L_{2\pi}^{p}}$$ we need the conditions on ω̃ from the assumptions of Theorems 1 or 2. These conditions always hold with $$\Vert \psi_{\cdot} ( t ) \Vert _{L_{2\pi /r}^{p}}$$ instead of $$\vert \psi_{x} ( t ) \vert$$ and thus the desired result follows.

## References

1. Krasniqi, X.Z.: Slight extensions of some theorems on the rate of pointwise approximation of functions from some subclasses of $$L^{p}$$. Acta Comment. Univ. Tartu Math. 17, 89–101 (2013)

2. Łenski, W., Szal, B.: Approximation of functions belonging to the class $$L^{p}(\omega)$$ by linear operators. Acta Comment. Univ. Tartu Math. 13, 11–24 (2009)

3. Łenski, W., Szal, B.: Pointwise approximation of modified conjugate functions by matrix operators of their Fourier series. Proc. Est. Acad. Sci. 67(1), 50–60 (2018)

4. Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (2002)

5. Szal, B.: On L-convergence of trigonometric series. J. Math. Anal. Appl. 373, 449–463 (2011)

6. Szal, B.: A new class of numerical sequences and its applications to uniform convergence of sine series. Math. Nachr. 284(14–15), 1985–2002 (2011)

## Author information

Authors

### Contributions

MK, WŁ and BS contributed equally in all stages to the writing of the paper. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Bogdan Szal.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and permissions

Kubiak, M., Łenski, W. & Szal, B. Pointwise approximation of modified conjugate functions by matrix operators of conjugate Fourier series of $$2\pi /r$$-periodic functions. J Inequal Appl 2018, 92 (2018). https://doi.org/10.1186/s13660-018-1684-0