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Pointwise approximation of modified conjugate functions by matrix operators of conjugate Fourier series of \(2\pi /r\)-periodic functions
Journal of Inequalities and Applications volume 2018, Article number: 92 (2018)
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
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 \(f\in L_{2\pi/r}^{p}\)
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:
with the partial sums \(S_{k}f\) and the conjugate one
with the partial sums \(\widetilde{S}_{k}f\). We also know that if \(f\in L_{2\pi}^{1}\), then
where, for \(r\in \mathbb{N} \),
and
with
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
but \(A^{\circ}:= ( a_{n,k} ) _{k=0}^{n}\), where
We will use the notations
for \(r\in \mathbb{N} \) and
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 \(r\in\mathbb{ N}\):
where
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 \(\beta\geq\alpha\geq0\)
and consequently
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:
with \(0<\gamma<\beta+\frac{1}{p}\) and
then
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:
for \(r\in \mathbb{N} \),
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
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:
with \(0<\gamma<\beta+\frac{1}{p}\), where \(m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \} \). If a matrix A is such that
and
with \(r\in \mathbb{N} \) are true, then
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
and (9) with \(r\in \mathbb{N} \) are true, then
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
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:
when \(r=1\) or
when \(r\geq2\), and
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
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
Remark 1
The Hölder inequality gives
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
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:
for \(\gamma\in ( \frac{1}{p},\frac{1}{p}+\beta ) \) and \(r\in \mathbb{N} \) (instead of (13)), and
when \(r=1\) or
when \(r\geq2\) (instead of (11) and (12), respectively). If a matrix A is such that (9) and (8) are true, then
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
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
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
or
3 Auxiliary results
We begin this section by some notations from [5] and [4, Sect. 5 of Chapter II]. Let for \(r=1,2,\ldots \)
and
It is clear by [4] that
and
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:
Next, we present the following well-known estimates.
Lemma 2
([4])
If \(0< \vert t \vert \leq\pi\) then
and, for any real t, we have
Lemma 3
Let \(r\in N\), \(l\in Z\) and \((a_{n})\subset \mathbb{C} \). If \(t \neq \frac{2l\pi}{r}\), then for every \(m\geq n\)
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
Proof
By Lemma 3,
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
where \(m\in \{ 0,\ldots [ \frac{r}{2} ] -1 \} \).
Proof
By the substitution \(t=\frac{2 ( m+1 ) \pi}{r}-u\), we obtain
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
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
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
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
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
and for even r
whence
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
for \(0\leq\beta<1-\frac{1}{p}\). We note that applying the condition (9) we have
whence
By Lemma 2
and using the Hölder inequality with \(p>1\) and \(q=\frac{p}{p-1}\)
Hence, by Lemmas 5 and 6 with (12) and (9),
for \(0\leq\beta<1-\frac{1}{p}\).
In the case of the last integrals, applying Lemma 4 we obtain
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
By the Hölder inequality with \(p>1\) and \(q=\frac{p}{p-1}\) we have
Further, using Lemmas 7 and 8 with (13) and Lemma 1 we get
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.
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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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DOI: https://doi.org/10.1186/s13660-018-1684-0