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Weighted inequalities for generalized polynomials with doubling weights
Journal of Inequalities and Applications volume 2017, Article number: 91 (2017)
Abstract
Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case \(1 \leq p < \infty\), and by Tamás Erdélyi for \(0< p \leq1\). In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for generalized trigonometric polynomials with doubling weights.
1 Introduction
A generalized nonnegative trigonometric polynomial is a function of the type
with \(r_{j}\in\mathbb{R}^{+}\), \(z_{j}\in\mathbb{C}\), and the number
is called the degree of f.
We denote by \(\mathbb{GT}_{n}\) (\(n\in\mathbb{R}^{+}\)) the set of all generalized nonnegative trigonometric polynomials of degree at most n and we denote by \(\mathbb{T}_{n}\) (\(n \in\mathbb{N}\)) the set of all real trigonometric polynomials of degree at most n.
In this paper we work on the real line. If \(x \in\mathbb{R}\), then
therefore, \(f \in\mathbb{GT}_{n}\) can be written as
where \(T_{j}\) is a nonnegative real trigonometric polynomial of degree 1. Many inequalities for generalized nonnegative polynomials are known; see [1].
Note that if \(f \in\mathbb{GT}_{n}\) with each \(r_{j} \geq2\) in its representation (1.1), then f is differentiable for all \(x \in\mathbb{R}\).
In this paper we deal with doubling weights and \(A_{\infty}\) weights. An integrable, 2π-periodic weight function W is called a doubling weight if there is a positive constant L such that
for any interval \(J\subset\mathbb{R}\), where 2J is the interval with length \(2\vert J\vert \) (\(\vert J\vert \) denotes the Lebesgue measure of the set J) and with midpoint at the midpoint of J. The constant L in (1.2) will be called the doubling constant. A periodic weight function W on \(\mathbb{R}\) is an \(A_{\infty}\) weight if for every \(\epsilon>0\), there is a \(\delta>0\) such that
for any interval \(J \subset\mathbb{R}\) and any measurable set \(E\subset J\) with \(\vert E\vert \geq\epsilon \vert J\vert \). Obviously \(A_{\infty}\) weights are doubling weights. Many properties of doubling and \(A_{\infty}\) weights are studied; see [2].
Weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Remez, Schur, Nikolskii inequalities, with doubling and \(A_{\infty}\) weights were proved by G. Mastroianni and V. Totik in [2], where \(L_{p}\) norm is considered for \(1\leq p <\infty\). For \(0< p\leq1\), Tamás Erdélyi [3] proved such inequalities for the trigonometric case. Recently, it has been proved that inequalities of this kind hold also for more general weight functions, namely for the product of a doubling and an exponential weight (see [4]) and for a class of nondoubling weights (see [5]).
In this paper we show that many weighted polynomial inequalities hold for generalized nonnegative trigonometric polynomials as well. We also prove the large sieve for generalized trigonometric polynomials with doubling weights.
The rest of this paper is organized as follows. In Section 2, we prove the basic theorems which will be used in the proof of weighted inequalities for generalized trigonometric polynomials. In Section 3, we prove Bernstein, Marcinkiewicz, and Schur inequalities for generalized trigonometric polynomials with doubling weights and in Section 4 we prove Remez and Nikolskii inequalities for generalized trigonometric polynomials with \(A_{\infty}\) weights.
2 The basic theorems
The following theorem is a basic tool in proving the weighted inequalities for generalized trigonometric polynomials. For ordinary polynomials the theorem is proved by Mastroianni and Totik in [2] for \(1 \leq p< \infty\), and by Tamás Erdélyi in [3] for \(0< p \leq1\). The proof is a modification of their arguments.
Theorem 2.1
Let \(0< p<\infty\). Let W be a doubling weight, and let
Then there is a constant \(C>0\) depending only on p and on the doubling constant L such that for every \(f \in\mathbb{GT}_{n}\) (\(1\leq n \in\mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
The function \(W_{n}\) in (2.1) is continuous and can be approximated by polynomials as follows. If \(1 \leq n \in\mathbb{R} ^{+}\), thenFootnote 1
uniformly in \(x \in\mathbb{R}\), hence, by Theorem 2.2 in [3], for \(0< p<\infty\), for each \(n \in\mathbb{R}^{+}\) (\(n \geq1\)) there is a nonnegative real trigonometric polynomial \(P_{n}\) of degree at most \(( \frac{\log_{2} L}{p} +4) n\) such thatFootnote 2
and
uniformly in \(x \in\mathbb{R}\).
The following lemma plays a crucial role in proving Theorem 2.1.
Lemma 2.2
Let \(0< p<\infty\) and let W be a doubling weight, and
Let
and
Then there is a constant \(C>0\) depending only on p and on the weight W such that for every \(f \in\mathbb{GT}_{n}\) (\(n \in\mathbb{R} ^{+}\)) we have
where \(b:= ( \frac{\log_{2} L}{p} +5) \).
Proof
Applying Theorem 2.2 in [6] to \(f^{p} \in\mathbb{GT}_{np}\) with \(r=p\) and \(\Psi(x)=x\), we have
The polynomial \(P_{n}\) in (2.3) has degree at most \(( \frac{\log_{2} L}{p} +4) n\), hence, \(fP_{n} \in \mathbb{GT}_{bn}\) where \(b= ( \frac{\log_{2} L}{p} +5) \). Applying (2.3) and (2.5) we have
which completes the proof. □
As an application of Theorem 2.1 we have the following weighted analog of a large sieve.
Theorem 2.3
Let \(0< p<\infty\) and let W be a doubling weight. With the same notations as in Lemma 2.2, there is a constant \(B>0\) depending only on p and on the weight W such that for every \(f \in\mathbb{GT}_{n}\) (\(1\leq n \in\mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
where \(b:= ( \frac{\log_{2} L}{p} +5) \).
Proof
Applying Lemma 2.2 and Theorem 2.1, we have
which completes the proof. □
We now prove Theorem 2.1.
Proof of Theorem 2.1
We closely follow the proof of Theorem 2.1 in [3]. Let \(0< p<\infty\). First we show that there is a constant \(C>0\) depending only on p and on the doubling constant L such that for every \(f \in\mathbb{GT}_{n}\) (\(1 \leq n \in \mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
In fact by (2.3), there is a polynomial \(P_{n}\) of degree at most \(N = ( \frac{\log_{2} L}{p} +4) n\) such that
Using
and \((a+b)^{p} \leq2^{p} (a^{p} +b^{p})\) for any \(a, b, p>0\), we have
For the first term in the right hand side of the above inequality, we use Bernstein’s inequality (Theorem 10 and its Remark in [1]) for generalized trigonometric polynomials of degree at most \((n+N)\), and (2.3), then we have
For the second term, we use (2.4), then we have
Since
we have, for \(0< p<\infty\),
Thus the proof of (2.6) is complete.
Note that the case \(1 < p < \infty\) of the theorem follows from the case \(0< p\leq1\). In fact, if \(1 < p < \infty\) then we may apply the theorem for the case \(0< p\leq1\) with f and p replaced by \(f^{p}\) and 1, respectively. Since
uniformly in \(x \in\mathbb{R}\), the case \(1 < p < \infty\) of the theorem follows.
So from now on we assume that \(0< p\leq1\). Now let K be a large positive even integer which will be chosen later, and set \(n^{*} = [n]\) and
Let \(\alpha_{i} \in J_{i}\) be a point such that \(f(\alpha_{i}) = \max_{x \in J_{i}} f(x)\) and let \(\beta_{i} \in J_{i}\) be a point such that \(W_{n}(\beta_{i})=\max_{x \in J_{i}} W_{n}(x)\). Let
where the summation is taken for \(i=0, 1, \ldots, Kn^{*} -1\), unless stated otherwise. Now let \(\xi_{i} \in J_{i}\) be arbitrary. Using \(a^{p}-b^{p} \leq(a-b)^{p}\) for \(a \geq b \geq0\), \(0< p\leq1\), we have
with some \(\tau_{i} \in J_{i}\). Since, uniformly for \(x, y \in J_{i}\),
we can continue this:
Now we write
and then applying Lemma 2.2, we have
and
hence
where we assume that \(K \geq ( \frac{\log_{2}L}{p}+6) \) so that \(bn+1 \leq Kn\) (b is defined in Lemma 2.2). Thus, by using the above inequality and (2.6), we can continue the inequality (2.8) thus:
Since
we have
from which it follows that
or, equivalently,
provided
Using
uniformly whenever \(\eta_{i} \in J_{i}\), we have, for any \(\xi_{i}\), \(\eta_{i} \in J_{i}\),
In particular, this is true for the points \(\gamma_{i} \in J_{i}\) and \(\delta_{i} \in J_{i}\) where \(f(\gamma_{i})=\min_{x \in J_{i}}f(x)\) and \(W_{n}(\delta_{i})=\min_{x \in J_{i}}W_{n}(x)\); hence, we have, for any \(x_{i} , y_{i} \in J_{i} \),
If we also note that \(y_{i} \in J_{i}\) implies
it follows that
whenever \(x_{i},y_{i} \in J_{i}\). Letting \(x_{i}=y_{i}=2i\pi/(Kn ^{*}) +x\) and integrating the above inequality with respect to \(x \in[0, (2\pi)/(Kn^{*})]\), we obtain
Since \(f(\alpha_{i})=\max_{x \in J_{i}}f(x)\) and \(f(\gamma_{i})= \min_{x \in J_{i}}f(x)\), we obtain
which proves the theorem. □
3 Results on weighted inequalities for generalized trigonometric polynomials with doubling weights
In this section we apply the basic theorem to prove the weighted inequalities for generalized trigonometric polynomials with doubling weights.
3.1 Bernstein inequality
Bernstein type inequalities have numerous applications in approximation theory. The following is a Bernstein type inequality for generalized trigonometric polynomials with doubling weights.
Theorem 3.1
Let W be a doubling weight and let \(0< p<\infty\). Then there is a constant \(C>0\) depending only on p and on the weight W such that for every \(f \in\mathbb{GT}_{n}\) (\(1 \leq n \in\mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
Proof
By Theorem 2.1 we can replace \(W_{n}\) by W in (2.6). □
3.2 Marcinkiewicz inequality
A Marcinkiewicz type inequality is useful when we need to estimate \(L_{p}\) norms of a trigonometric polynomials by a finite sum. The following theorem describes such inequalities for generalized trigonometric polynomials with doubling weights.
Theorem 3.2
Let W be a doubling weight and let \(0< p<\infty\). Then there are two constants \(K>0\) and \(C>0\) depending only on p and on the weight W such that for every \(f \in\mathbb{GT}_{n}\) (\(1 \leq n \in \mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
provided the points \(\tau_{0}<\tau_{1}<\cdots<\tau_{m}\) satisfy \(\tau_{j+1}-\tau_{j} \leq2\pi/(Kn)\) and \(\tau_{m}\geq\tau_{0} + 2 \pi\).
Proof
Let \(n^{*}=[n]\). In the proof of Theorem 2.1 we have proved in (2.10) that there exists a positive integer K such that if \(J_{i}= [ \frac{2i\pi}{Kn^{*}}, \frac{2(i+1)\pi}{ Kn^{*}} ] \), \(i=0, 1, \ldots, Kn^{*}-1\), and \(x_{i} \in J_{i}\) arbitrary, then
Since \(f(\alpha_{i})=\max_{x \in J_{i}}f(x)\), we have
Thus, the theorem is true if there is at least one point \(\tau_{j}\) (\(\operatorname{mod}\ 2\pi\)) in every \(J_{i}\), \(i=0, 1, \ldots, K n^{*}-1\), or if the points \(\tau_{0}<\tau_{1}<\cdots<\tau_{m}\) satisfy \(\tau _{j+1}-\tau _{j} \leq2\pi/(Kn)\) and \(\tau_{m}\geq\tau_{0} + 2\pi\). □
3.3 Schur inequality
The following is a Schur type inequality for generalized trigonometric polynomials with doubling weights involving generalized Jacobi weights.
Theorem 3.3
Let W be a doubling weight and let \(0< p<\infty\). Let V be a generalized Jacobi weight of the form
where v is a positive measurable function bounded away from 0 and ∞. Then there is a constant \(C>0\) independent of n such that for every \(f \in\mathbb{GT}_{n}\) (\(1\leq n \in\mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
where \(\Gamma=\max_{1\le i \le m}\{\gamma_{i}\} \).
Proof
By the Lemma 4.5 in [2], WV is also a doubling weight and it is easy to see that \((WV)_{n}(x) \sim W_{n}(x)V_{n}(x)\) and \(V_{n}(x) \geq cn^{-\Gamma}\). Thus, by Theorem 2.1, we have
which completes the proof. □
4 Results on weighted inequalities for generalized trigonometric polynomials with \(A_{\infty}\) weights
In this section we prove the weighted inequalities for generalized trigonometric polynomials with \(A_{\infty}\) weights.
4.1 Remez inequality
The Remez inequality is useful because we can exclude exceptional sets of measure at most 1. The following describes such inequalities for generalized trigonometric polynomials with \(A_{\infty}\) weights.
Theorem 4.1
Let \(0< p<\infty\) and let W be an \(A_{\infty}\) weight. Then there is a constant \(C>0\) depending only on p and on the weight W such that if \(f \in\mathbb{GT}_{n}\) (\(1 \leq n \in\mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) and E is a measurable subset of \([0, 2\pi]\) of measure at most \(\lambda\in(0, 1]\), then
Proof
First we show that if we replace W by \(W_{n}\) in (4.1), then inequality holds. By (2.3), we have a trigonometric polynomial \(P_{n}\) of degree at most \(( \frac{\log_{2} L}{p} +4) n\) such that
Then we apply the Remez inequality for generalized trigonometric polynomials (see Theorem 8 in [1]) to \(fP_{n} \in\mathbb{GT} _{bn}\) where \(b= ( \frac{\log_{2} L}{p} +5) \) as follows:
Note that the case \(1 < p < \infty\) of the theorem follows from the case \(0< p\leq1\). So from now on we assume that \(0< p\leq1\). Next we follow the proof of Theorem 2.1. Let K be a large positive even integer which will be chosen later, and set \(n^{*} = [n]\) and
Define the set J by
and let
Then
Let \(\alpha_{i} \in J_{i}\) be a point such that \(f(\alpha_{i}) = \max_{x \in J_{i}} f(x)\) and let \(\beta_{i} \in J_{i}\) be a point such that \(W_{n}(\beta_{i})=\max_{x \in J_{i}} W_{n}(x)\). Let
Now let \(\xi_{i} \in J_{i}\) be arbitrary. Using exactly the same method as in the proof of Theorem 2.1 (from (2.7) to (2.9)), we have
By (4.2) we have
hence,
from which it follows that
provided
Using
uniformly whenever \(\eta_{i} \in J_{i}\), we have, for any \(\xi_{i}\), \(\eta_{i}\in J_{i}\),
In particular, this is true for the points \(\gamma_{i} \in J_{i}\) and \(\delta_{i} \in J_{i}\) where \(f(\gamma_{i})=\min_{x \in J_{i}}f(x)\) and \(W_{n}(\delta_{i})=\min_{x \in J_{i}}W_{n}(x)\), hence, we have, for any \(x_{i} , y_{i} \in J_{i} \),
Now we use the property of the \(A_{\infty}\) weight. If \(i \notin J\), then \(\vert J_{i} \setminus E\vert \geq \vert J_{i}\vert /2= \pi/(Kn^{*})\), hence, by Lemma 5.1(vi)′ in [2], there are constants s and D such that, for \(y_{i} \in J_{i}\), \(i \notin J\),
Similarly to (2.10), we have
whenever \(x_{i}\), \(y_{i} \in J_{i}\). Letting \(x_{i}=y_{i}=2i\pi/(Kn ^{*}) +x\) and integrating the above inequality with respect to \(x \in[0, (2\pi)/(Kn^{*})]\), we obtain
Applying Theorem 2.1, (4.2), and the definition of K in (4.3), we have
which proves the theorem. □
4.2 Nikolskii inequality
Nikolskii inequality is used to compare the \(L_{p}\) and \(L_{q}\) norms of polynomials. The following theorem describes such inequalities for generalized trigonometric polynomials with \(A_{\infty}\) weights.
Theorem 4.2
Let W be an \(A_{\infty}\) weight and let \(0< p< q<\infty\). Then there is a constant \(C>0\) depending only on p and q and on the weight W such that for every \(f \in\mathbb{GT}_{n}\) (\(1 \leq n \in \mathbb{R}^{+}\)) with each \(r_{j} \geq2\) in its representation (1.1) we have
Proof
Define the set E by
Then \(\int_{E} f^{q} W \geq n \vert E\vert \int_{-\pi}^{\pi}f^{q} W\), hence, \(\vert E\vert \leq1/n\). Now applying the Theorem 4.1, we have
Taking pth root yields the theorem. □
5 Conclusions
In this paper, we have established weighted inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, for generalized trigonometric polynomials with doubling weights. We also have established the large sieve for generalized trigonometric polynomials with doubling weights.
Notes
Here, and in what follows, \([x]\) denotes the integer part of x.
In what follows \(A \sim B\) means that there are two positive constants \(C_{1}\) and \(C_{2}\) such that \(C_{1} \leq B/A \leq C_{2}\).
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Joung, H. Weighted inequalities for generalized polynomials with doubling weights. J Inequal Appl 2017, 91 (2017). https://doi.org/10.1186/s13660-017-1369-0
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DOI: https://doi.org/10.1186/s13660-017-1369-0