# Weighted inequalities for generalized polynomials with doubling weights

## 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

*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*.

Note that if \(f \in\mathbb{GT}_{n}\) with each \(r_{j} \geq2\) in its representation (1.1), then *f* is differentiable for all \(x \in\mathbb{R}\).

*π*-periodic weight function

*W*is called a doubling weight if there is a positive constant

*L*such that

*J*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

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*

^{1}

^{2}

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

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

We now prove Theorem 2.1.

### Proof of Theorem 2.1

*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

*f*and

*p*replaced by \(f^{p}\) and 1, respectively. Since

*K*be a large positive even integer which will be chosen later, and set \(n^{*} = [n]\) and

*b*is defined in Lemma 2.2). Thus, by using the above inequality and (2.6), we can continue the inequality (2.8) thus:

## 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*

### 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

*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

### 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

*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

## 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

*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

*K*be a large positive even integer which will be chosen later, and set \(n^{*} = [n]\) and

*J*by

*s*and

*D*such that, for \(y_{i} \in J_{i}\), \(i \notin J\),

*K*in (4.3), we have

### 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

*E*by

*p*th 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.

## Footnotes

## Notes

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