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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

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

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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

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