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Local saturation of a positive linear convolution operator
Journal of Inequalities and Applications volume 2014, Article number: 329 (2014)
Let be a sequence of non-negative, even, and continuous functions on ℝ. In this paper, we consider a convolution operator , , and then investigate the local saturation of .
1 Introduction and theorems
Through this paper we let , , be a sequence of non-negative, even, and continuous functions on , and there exist , and such that satisfied uniformly
and there exist two positive constants α, β with such that uniformly for n,
As an example for we give the following:
Then it satisfies (1.2),
Let us denote . In what follows we assume that satisfies the conditions (1.1)-(1.4). Using , we define the convolution operators for ,
Swetits and Wood  studied the operators on a finite interval ;
where is defined on as , and then they gave a local saturation theorem. Furthermore, there is a rich bibliography concerning the convergence of positive linear operators on (e.g. see [2–4] and the references cited therein).
In this paper we extend  to the infinite interval . Then we use a similar methods as . For and , we define as the space of those functions f such that and is a locally absolutely continuous function on , with and . Let be the space of continuous, compactly supported and continuously second differentiable functions on . Furthermore, the total variation of a real-valued function f defined on an interval is the quantity
Here the supremum is taken over the set of all partitions of the interval considered. If and , we define
If is finite, we say that F is of bounded variation on , and is called the total variation of F on . We define to be the set of all functions on whose total variation on is finite.
Then we first give the pointwise convergence theorem.
Theorem 1.1 (cf. )
Let , and let , . Then we have a pointwise convergence as follows:
Equation (1.6) holds uniformly on .
Then the following is a direct convergence theorem.
Theorem 1.2 Let .
If , then we have for(1.7)
If , then we have for with(1.8)
Let and f be linear on . Then we have(1.9)
Finally, we give an inverse theorem as follows.
Theorem 1.3 Let . Let and .
For , the condition (1.7) implies .
For , the condition (1.8) implies .
For , the condition (1.9) implies that f is linear on .
This paper is organized as follows. In Section 2, we will give some fundamental lemmas in order to prove the main results and we will prove the results in Section 3.
2 Fundamental lemmas
Throughout this paper denote positive constants independent of n, x, t or function . The same symbol does not necessarily denote the same constant in different occurrences.
To prove the theorems we need some lemmas.
Lemma 2.1 Let and . Then for defined in (1.4)
Proof Let . Then we have from (1.4)
Lemma 2.2 Let , and let δ be a positive constant. Then for , we have
Moreover, we have for
Here, α and β are defined in (1.4).
Proof For , we have by (2.1)
For (2.3), we have from (1.3)
Since we know for
and from (2.1)
we have for ,
Next, we give an estimate for . Since
and by (2.1), for ,
we have for ,
Let and then we define
Lemma 2.3 Let , and let . If , then
Proof Let . Then we have
For and , we see . From (1.4), we have the following:
Hence we have
Let , and . By Hölder’s inequality,
Here, from (2.1) we have
Also, we see by (2.6)
Thus, by (2.8) we conclude. □
3 Proof of theorems
Proof of Theorem 1.1 For and , we set
Then we see
For , we have from (2.4)
and for , we have from (2.2)
Now, we will estimate . We have
For the second term, we have by (2.1)
For a given , there exists a positive constant (depending only on ε) such that for
For the first term, we have using (1.3)
and by (2.1)
Then we have for some positive constants , , and
Therefore, we have for an arbitrary
Thus, (1.6) is proved. Moreover, noting (3.1), we see that (1.6) holds uniformly on . □
Proof of Theorem 1.2 Let . (i) We start under the condition for the convenience of considering (ii). On the way of the proof we switch over to the assumption with . Let and let and let . Since
By Lemma 2.3,
We estimate . Let . Then
Here, for the first term, using
which is shown in (2.7), we have
From this we suppose . By
Here we note that is an odd function. Then we have by (2.1)
and the first term is estimated as
Now we set
and denote the Hardy-Littlewood majorant of at x. Since and , we have
where depend only on p [, Theorem 1, p.201]. Since
where , we have by (3.7) and (1.3)
Hence, from (3.6) and (3.8), we have for some positive constant
So, from (3.5) and (3.9),
Now, we see by (3.10) and (2.4)
Consequently, with (3.2) and (3.3) we conclude (i).
(ii) Let and . Then we have for ,
On the proof of (i) we recall the part which we assumed as . From (3.3) and (3.5) we may only estimate
(see (3.4)). By (3.11) we see
Now, we see that
by (1.3). We estimate . We fix an arbitrary . Then we have by means of the substitution with a new variable y
Since we can see
Now, we estimate for a non-negative integer j. Let . Then from (1.2) and (1.3), we have
Let . Then by (2.1) we have
Therefore, we have
If we let , then we have , because . Consequently, (ii) is proved.
(iii) It follows from (2.4). Consequently, for a linear function f on
Proof of Theorem 1.3 (i), (ii) hold as follows: Let . First, we choose with (for any ) such that
We use the bilinear functional;
We will show that for fixed ψ, is uniformly bounded on . We see
For we can write
Since for (so, we may take ), by (2.1) we have
where . Thus, we have
We estimate . We may , because for . Noting (1.4) and ,
Finally, we estimate . Let and . As the estimation for , we have by (1.3)
We estimate for . There exists η between x and t such that
Here we used (1.3). Consequently, it follows that is uniformly bounded on . Next, from Theorem 1.1 we see that for ,
Since is uniformly bounded on , and is dense in , (3.14) yields
for any . Now, for any fixed , we consider the sequence of linear functional . Since , , there exist () and () and a subsequence such that
From (3.15) and (3.16) we obtain
A particular solution to (3.17) is
The homogeneous problem
has the general solution for , since we can take arbitrarily as , , , . Hence, if , , and if then . Hence, (i) and (ii) hold. We will show (iii). Now, if
where is independent of n. Hence
Considering (3.15), (3.16), (3.17), and (3.18), we obtain
consequently, we have , that is, f is linear on . □
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The authors thank the referees for many valuable comments and corrections.
The authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript and participated in the sequence alignment. All authors read and approved the final manuscript.
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Jung, H.S., Sakai, R. Local saturation of a positive linear convolution operator. J Inequal Appl 2014, 329 (2014). https://doi.org/10.1186/1029-242X-2014-329
- convolution operator
- local saturation