Weighted inequalities for generalized polynomials with doubling weights

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≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1 \leq p < \infty$\end{document}, and by Tamás Erdélyi for 0<p≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< p \leq1$\end{document}. 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.


Introduction
A generalized nonnegative trigonometric polynomial is a function of the type with r j ∈ R + , z j ∈ C, and the number n def =   m j= r j is called the degree of f . We denote by GT n (n ∈ R + ) the set of all generalized nonnegative trigonometric polynomials of degree at most n and we denote by T n (n ∈ N) the set of all real trigonometric polynomials of degree at most n.
In this paper we work on the real line. If x ∈ R, then cosh(Im z j ) -cos(x -Re z j ) / , therefore, f ∈ GT n can be written as where T j is a nonnegative real trigonometric polynomial of degree . Many inequalities for generalized nonnegative polynomials are known; see [].
Note that if f ∈ GT n with each r j ≥  in its representation (.), then f is differentiable for all x ∈ R.
In this paper we deal with doubling weights and A ∞ weights. An integrable, π -periodic weight function W is called a doubling weight if there is a positive constant L such that 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 []) and for a class of nondoubling weights (see []).
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 , we prove the basic theorems which will be used in the proof of weighted inequalities for generalized trigonometric polynomials. In Section , we prove Bernstein, Marcinkiewicz, and Schur inequalities for generalized trigonometric polynomials with doubling weights and in Section  we prove Remez and Nikolskii inequalities for generalized trigonometric polynomials with A ∞ weights.

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 [] for  ≤ p < ∞, and by Tamás Erdélyi in [] for  < p ≤ . The proof is a modification of their arguments.
Then there is a constant C >  depending only on p and on the doubling constant L such that for every f ∈ GT n ( ≤ n ∈ R + ) with each r j ≥  in its representation (.) we have The function W n in (.) is continuous and can be approximated by polynomials as fol- there is a nonnegative real trigonometric polynomial P n of degree at most ( and The following lemma plays a crucial role in proving Theorem ..
Lemma . Let  < p < ∞ and let W be a doubling weight, and Then there is a constant C >  depending only on p and on the weight W such that for every f ∈ GT n (n ∈ R + ) we have The polynomial P n in (.) has degree at most ( which completes the proof.
As an application of Theorem . we have the following weighted analog of a large sieve.
Proof Applying Lemma . and Theorem ., we have which completes the proof.
We now prove Theorem ..
Proof of Theorem . We closely follow the proof of Theorem . in []. Let  < p < ∞.
First we show that there is a constant C >  depending only on p and on the doubling constant L such that for every f ∈ GT n ( ≤ n ∈ R + ) with each r j ≥  in its representation In fact by (.), there is a polynomial P n of degree at most N = ( Using f P n = (fP n ) -fP n For the first term in the right hand side of the above inequality, we use Bernstein's inequality (Theorem  and its Remark in []) for generalized trigonometric polynomials of degree at most (n + N), and (.), then we have For the second term, we use (.), then we have π -π fP n p ≤ C  n p π -π |f | p W n . Since Thus the proof of (.) is complete. Note that the case  < p < ∞ of the theorem follows from the case  < p ≤ . In fact, if  < p < ∞ then we may apply the theorem for the case  < p ≤  with f and p replaced by f p and , respectively. Since uniformly in x ∈ R, the case  < p < ∞ of the theorem follows.
So from now on we assume that  < p ≤ . Now let K be a large positive even integer which will be chosen later, and set n * = [n] and Let α i ∈ J i be a point such that f (α i ) = max x∈J i f (x) and let β i ∈ J i be a point such that W n (β i ) = max x∈J i W n (x). Let where the summation is taken for i = , , . . . , Kn * -, unless stated otherwise. Now let we can continue this: Now we write and then applying Lemma ., we have where we assume that K ≥ ( log  L p + ) so that bn +  ≤ Kn (b is defined in Lemma .). Thus, by using the above inequality and (.), we can continue the inequality (.) thus: from which it follows that or, equivalently, Using In particular, this is true for the points γ i ∈ J i and δ i ∈ J i where f (γ i ) = min x∈J i f (x) and W n (δ i ) = min x∈J i W n (x); hence, we have, for any x i , y i ∈ J i , If we also note that y i ∈ J i implies which proves the theorem.

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.

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 . Let W be a doubling weight and let  < p < ∞. Then there is a constant C >  depending only on p and on the weight W such that for every f ∈ GT n ( ≤ n ∈ R + ) with each r j ≥  in its representation (.) we have Proof By Theorem . we can replace W n by W in (.).

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 . Let W be a doubling weight and let  < p < ∞. Then there are two constants K >  and C >  depending only on p and on the weight W such that for every f ∈ GT n ( ≤ n ∈ R + ) with each r j ≥  in its representation (.) we have provided the points τ  < τ  < · · · < τ m satisfy τ j+τ j ≤ π/(Kn) and τ m ≥ τ  + π .

Schur inequality
The following is a Schur type inequality for generalized trigonometric polynomials with doubling weights involving generalized Jacobi weights.
Theorem . Let W be a doubling weight and let  < p < ∞. Let V be a generalized Jacobi weight of the form where v is a positive measurable function bounded away from  and ∞. Then there is a constant C >  independent of n such that for every f ∈ GT n ( ≤ n ∈ R + ) with each r j ≥  in its representation (.) we have Proof By the Lemma . in [], WV is also a doubling weight and it is easy to see that (WV ) n (x) ∼ W n (x)V n (x) and V n (x) ≥ cn -. Thus, by Theorem ., we have which completes the proof.

Results on weighted inequalities for generalized trigonometric polynomials with A ∞ weights
In this section we prove the weighted inequalities for generalized trigonometric polynomials with A ∞ weights.

Remez inequality
The Remez inequality is useful because we can exclude exceptional sets of measure at most . The following describes such inequalities for generalized trigonometric polynomials with A ∞ weights.