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Weighted inequalities for generalized polynomials with doubling weights
- Haewon Joung^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-017-1369-0
© The Author(s) 2017
- Received: 1 November 2016
- Accepted: 19 April 2017
- Published: 27 April 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.
Keywords
- generalized polynomials
- Bernstein inequality
- Marcinkiewicz inequality
- Schur inequality
- Remez inequality
- Nikolskii inequality
- Doubling weights
- large sieve
MSC
- 26D05
- 42A05
1 Introduction
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.
Note that if \(f \in\mathbb{GT}_{n}\) with each \(r_{j} \geq2\) in its representation (1.1), then f is differentiable for all \(x \in\mathbb{R}\).
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
The following lemma plays a crucial role in proving Theorem 2.1.
Lemma 2.2
Proof
As an application of Theorem 2.1 we have the following weighted analog of a large sieve.
Theorem 2.3
Proof
We now prove Theorem 2.1.
Proof of Theorem 2.1
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
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
Proof
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
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
Proof
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
Proof
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.
In what follows \(A \sim B\) means that there are two positive constants \(C_{1}\) and \(C_{2}\) such that \(C_{1} \leq B/A \leq C_{2}\).
Declarations
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Authors’ Affiliations
References
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