# Pólya-type polynomial inequalities in Orlicz spaces and best local approximation

## Abstract

We obtain an extension of Pólya-type inequalities for univariate real polynomials in Orlicz spaces. We also give an application to a best local approximation problem.

MSC 2010: 41A10; 41A17.

## 1 Introduction

Let X be a bounded open subset of . Consider the measure space (X, $ℬ$, μ), where μ is the Lebesgue measure, and denote $ℳ=ℳ\left(X\right)$ the system of all equivalence classes of Lebesgue measurable real valued functions on X. Let Φ be the set of convex functions ϕ : ++, with ϕ(x) > 0 for x > 0, and ϕ(0) = 0.

Given ϕ Φ, we define

${L}^{\varphi }={L}^{\varphi }\left(X\right):=\left\{f\in ℳ:\underset{X}{\int }\varphi \left(\alpha \left|f\left(x\right)\right|\right)dx<\infty ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{some}\phantom{\rule{2.77695pt}{0ex}}\alpha >0\right\}.$

The space Lϕis called the Orlicz space determined by ϕ. This space is endowed with the Luxemburg norm,

${∥f∥}_{\varphi ,X}=\text{inf}\left\{\lambda >0:\underset{X}{\int }\varphi \left(\frac{\left|f\left(x\right)\right|}{\lambda }\right)\frac{dx}{\mu \left(X\right)}\le 1\right\}.$

The space Lϕwith this norm is a Banach space (see [1]). If $E\in ℬ$ and μ(E) > 0, then · ϕ,Eis a seminorm on Lϕ(X). In the particular case, ϕ (t) = tp, we will use the notation · p,Einstead of · ϕ,E.

Let ${\Pi }^{N}\subset ℳ$, N , be the class of all algebraic polynomials of degree at most N, with real coefficients.

Given $E\in ℬ$, we recall that a polynomial g E ΠNis a best approximation of f Lϕ(X) from ΠNrespect to · ϕ,E, if

${∥f-{g}_{E}∥}_{\varphi ,E}=\text{inf}\left\{{∥f-P∥}_{\varphi ,E}:\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}P\in {\prod }^{N}\right\}.$

Let x k , 1 ≤ kn, be n points in X. We consider a net of measurable sets $\left\{E\right\}\subset ℬ$ such that $E={\bigcup }_{k=1}^{n}{E}_{k}$, with μ(E k ) > 0 and

$\underset{1\le k\le n}{\text{sup}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\underset{y\in {E}_{k}}{\text{sup}}\left|{x}_{k}-y\right|\to 0,\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\mu \left(E\right)\to 0.$

Given f Lϕ(X) and ΠN, we consider a net of best approximation functions {g E }. If it has a limit in ΠNas μ(E) → 0, this limit is called the best local approximation of f from ΠNon {x1,..., x n }. If the points in our approximation problem have not the same importance the neighborhoods E k can be adjusted to reflect it. In [2], Chui et al. introduced the balanced neighborhood concept and they studied existence and characterization of best local approximation in Lp-spaces for several points with different size neighborhoods. In [3, 4], the last problem was considered for ϕ-approximation and · ϕ -approximation, respectively, in Orlicz spaces. Other results in these spaces about best local approximation with non balanced neighborhoods were considered in [5].

Polynomial inequalities on measurable sets have been studied extensively in the literature (see [68]). In [9], the authors proved the following extension of the Pólya inequality in Lp-spaces, 0 < p ≤ ∞.

Theorem 1.1. Let 0 < p ≤ ∞ and n, N . Let i k , 1 ≤ kn, be n positive integers such that${\sum }_{k=1}^{n}{i}_{k}=N+1$. Let B k , 1 ≤ kn, be disjoint pairwise compact intervals in with 0 < μ(B k ) ≤ 1. Then there exists a constant K depending on p, i k and B k , for 1 ≤ kn, such that

$\left|{c}_{j}\right|\le \frac{K}{\underset{1\le k\le n}{\text{min}}\mu {\left(E\cap {B}_{k}\right)}^{{i}_{k}-1+1/p}}{∥P∥}_{p,E,}\phantom{\rule{1em}{0ex}}0\le j\le N,$

for all$P\left(x\right)={\sum }_{j=0}^{N}{c}_{j}{x}^{j},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}E\subset {\bigcup }_{k=1}^{n}{B}_{k},\phantom{\rule{2.77695pt}{0ex}}\mu \left(E\cap {B}_{k}\right)>0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\le k\le n.$.

They gave an application of this theorem to the existence of the best multipoint local approximation in Lpspaces, with balanced neighborhoods.

In this article, we generalize Theorem 1.1 and the balanced neighborhood concept to Lϕ. As a consequence of this extension we prove the existence of the best local approximation of a function from ΠNon {x1,..., x n }, with balanced neighborhoods, following the pattern used in [9]. Moreover, we prove that the best local approximation polynomial is the Hermite interpolating polynomial.

We say that a function ϕ Φ satisfies the Δ2-condition if there exists a constant k > 0 such that ϕ(2x) ≤ (x), for x ≥ 0, and we say that ϕ satisfies the Δ'-condition if there exists a constant c > 0 such that ϕ(xy) ≤ (x)ϕ (y) for x, y ≥ 0. We point out that the Δ'-condition implies the Δ2-condition. A detailed treatment about these subjects may be found in [1].

If ϕ satisfies the Δ'-condition, it is easy to see that there exists a constant K > 0 such that

${\varphi }^{-1}\left(x\right){\varphi }^{-1}\left(y\right)\le K{\varphi }^{-1}\left(xy\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}x,y\ge 0.$
(1)

We assume in this article that ϕ Φ and it satisfies the Δ'-condition.

## 2 Preliminary results

Let ${\mathcal{X}}_{A}$ denotes the characteristic function on the measurable set A X.

Proposition 2.1. The family of all seminorms · ϕ.Ewith μ(E) > 0, has the following properties:

(a)${∥{\mathcal{X}}_{E}∥}_{\varphi ,E}=\frac{1}{{\varphi }^{-1}\left(1\right)}$.

(b) if f, g Lϕ(X) satisfy |f| ≤ |g| on E, then fϕ,Egϕ,E. The inequality is strict if |f| < |g| on some subset of E with positive measure.

(c) There exists a constant M > 0 such that

${∥f∥}_{\varphi ,G}\le \frac{M}{{\varphi }^{-1}\left(\frac{\mu \left(G\right)}{\mu \left(D\right)}\right)}{∥f∥}_{\varphi ,D,}\phantom{\rule{1em}{0ex}}f\in {L}^{\varphi }\left(X\right),$
(2)

for all pair of measurable sets G, D, with G D and μ(G) > 0.

Proof (a) For λ := 1/ϕ-1(1) we have

$\underset{E}{\int }\varphi \left(\frac{\left|{\mathcal{X}}_{E}\right|}{\lambda }\right)\frac{dx}{\mu \left(E\right)}=\underset{E}{\int }\frac{dx}{\mu \left(E\right)}=1.$

Now, the Δ2- condition implies ${∥{\mathcal{X}}_{E}∥}_{\varphi ,E}=1/{\varphi }^{-1}\left(1\right)$.

1. (b)

If |f| ≤ |g| on E, then

$\underset{E}{\int }\varphi \left(\frac{\left|f\right|}{\lambda }\right)\frac{dx}{\mu \left(E\right)}\le \underset{E}{\int }\varphi \left(\frac{\left|g\right|}{\lambda }\right)\frac{dx}{\mu \left(E\right)},\phantom{\rule{1em}{0ex}}\lambda >0,$

and so fϕ,Egϕ,E. In addition, if |f| < |g| on some subset of E with positive measure, the above inequality is strict. So, the Δ2-condition implies the assertion.

1. (c)

Given G D, μ(G) > 0, and f L ϕ(X), for each λ > 0, we denote

$\mathfrak{U}\left(\lambda \right):=\underset{G}{\int }\varphi \left(\frac{\left|f\right|}{\lambda }\right)\frac{dx}{\mu \left(G\right)}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathfrak{B}\left(\lambda \right):=\underset{D}{\int }\varphi \left(\frac{\left|f\right|}{\lambda }\right)\frac{dx}{\mu \left(D\right)}.$

We consider λ > 0 such that $\mathfrak{B}\left(\lambda \right)\le 1$. By the Δ'-condition we obtain

$\mathfrak{U}\left(\frac{\lambda }{{\varphi }^{-1}\left(\frac{\mu \left(G\right)}{c\phantom{\rule{2.77695pt}{0ex}}\mu \left(D\right)}\right)}\right)\le \underset{D}{\int }c\frac{\mu \left(G\right)}{c\phantom{\rule{2.77695pt}{0ex}}\mu \left(D\right)}\varphi \left(\frac{\left|f\right|}{\lambda }\right)\frac{dx}{\mu \left(G\right)}=\mathfrak{B}\left(\lambda \right)\le 1.$

Then ${∥f∥}_{\varphi ,G}\le \frac{\lambda }{{\varphi }^{-1}\left(\frac{\mu \left(G\right)}{c\phantom{\rule{2.77695pt}{0ex}}\mu \left(D\right)|}\right)}$, for all λ > 0 with $\mathfrak{B}\left(\lambda \right)\le 1$. So, the definition of fϕ,Dand (1) imply ${∥f∥}_{\varphi ,G}\le \frac{M}{{\varphi }^{-1}\left(\frac{\mu \left(G\right)}{\mu \left(D\right)}\right)}{∥f∥}_{\varphi ,D}$ with $M=\frac{K}{{\varphi }^{-1}\left({c}^{-1}\right)}$.

Lemma 2.2. There exists a constant M > 0 such that

$\left|{P}^{\left(j\right)}\left(a\right)\right|\le \frac{M}{{\epsilon }^{j}}{∥P∥}_{\varphi ,\left[a-\epsilon ,\phantom{\rule{2.77695pt}{0ex}}a+\epsilon \right],}$

for all P ΠN, [a - ϵ, a + ϵ] X, and 0 ≤ jN.

Proof. Given P ΠNand [a - ϵ, a + ϵ] X, we divide that interval in 2(N + 1) close subintervals with the same size. Let Jϵ be one of them. From Proposition 2.1 (c), we get ${∥P∥}_{\varphi ,{J}_{\epsilon }}\le M{∥P∥}_{\varphi ,\left[a-\epsilon ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}a+\epsilon \right]}$, where M is independent on P, a, and ϵ. In addition, there exists yϵ Jϵ such that $\left|P\left({y}_{\epsilon }\right)\right|\le {\varphi }^{-1}\left(1\right){∥P∥}_{\varphi ,{J}_{\epsilon }}$. In fact, if ${\varphi }^{-1}\left(1\right){∥P∥}_{\varphi ,{J}_{\epsilon }}<\left|P\left(y\right)\right|$, for all y Jϵ, then Proposition 2.1 (a) and (b) yield ${∥P∥}_{\varphi ,{J}_{\epsilon }}>{∥P∥}_{\varphi ,{J}_{\epsilon }}$. A contradiction.

From the family of intervals Jϵ, we choose pairwise disjoint (N + 1) intervals, and we denote them with Ji, 1 ≤ iN + 1. Let yi Jibe such that

$\left|P\left({y}_{i,\epsilon }\right)\right|\le M{\varphi }^{-1}\left(1\right){∥P∥}_{\varphi ,\left[a-\epsilon ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}a+\epsilon \right],}\phantom{\rule{1em}{0ex}}1\le i\le N+1.$
(3)

If ${t}_{i,\epsilon }:=\frac{{y}_{i,\epsilon }-a}{\epsilon }\in \left[-1,1\right]$, we have

$P\left({y}_{i,\epsilon }\right)=\sum _{j=0}^{N}\frac{{P}^{\left(j\right)}\left(a\right)}{j!}{\left({y}_{i,\epsilon }-a\right)}^{j}=\sum _{j=0}^{N}\frac{{P}^{\left(j\right)}\left(a\right)}{j!}{\epsilon }^{j}{t}_{i,\epsilon }^{j},\phantom{\rule{1em}{0ex}}1\le i\le N+1.$
(4)

The matrix of the linear system (4), $\left({t}_{i,\epsilon }^{j}\right)$, is a Vandermonde matrix whose determinant has a positive lower bound, because ti- ti'≥ 1/N + 1 for i > i'. Using Cramer's rule and (3), there is a constant which we again denote by M such that

$\left|{P}^{\left(j\right)}\left(a\right){\epsilon }^{j}\right|\le M{∥P∥}_{\varphi ,\left[a,\epsilon ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}a+\epsilon \right]}\phantom{\rule{1em}{0ex}}0\le j\le N.$

The proof of the following lemma is analogous to the one of Lemma 2.3 in [9], however we give it for sake of completeness.

Lemma 2.3. Let C X be an interval, E C, μ(E) > 0. For all P ΠN, there exists an interval F := F(E,P) C such that

a)$\mu \left(F\right)\ge \frac{\mu \left(E\right)}{2N}$,

b) Pϕ,F≤ 2NPϕ,E.

Proof. Let P ΠN, S = 2N, and let D a := {x C : |P(x)| < a}. It easy to see that the function G(a): = μ(D a ) is continuous, G(0) = 0 and $\underset{a\to \infty }{\text{lim}}G\left(a\right)=\mu \left(C\right)$. Therefore, there exists a constant a* + such that $\mu \left({D}_{{a}_{*}}\right)=\mu \left(E\right)/2$. Since {x C : |P(x)| = a*} has at most 2N elements, there exists k, 1 ≤ kN, and pairwise disjoint intervals E j , 1 ≤ jk, such that ${D}_{{a}_{*}}={\bigcup }_{j=1}^{k}{E}_{j}$.

We denote $\overline{A}=C\A$, for any set A. Then

$\mu \left(E\cap {\overline{D}}_{{a}_{*}}\right)=\mu \left(E\right)-\mu \left(E\cap {D}_{{a}_{*}}\right)\ge \mu \left(E\right)-\mu \left({D}_{{a}_{*}}\right)=\frac{\mu \left(E\right)}{2}.$
(5)

There exists j, 1 ≤ jk, such that μ(E j ) ≥ μ(E)/S. In fact, if μ(E j ) < μ(E)/S for all j , 1 ≤ jk, we obtain $\mu \left({D}_{{a}_{*}}\right), which is a contradiction. So, we have proved a) with F := E j .

Using (5), we obtain

$\mu \left(E\cap {\overline{D}}_{{a}_{*}}\right)\ge \frac{\mu \left(E\right)}{2}=\mu \left({D}_{{a}_{*}}\right)\ge \mu \left(F\right)\mu \left(F\cap \overline{E}\right).$

Therefore

$\begin{array}{ll}\hfill \underset{F}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(F\right)}& \le \underset{F\cap E}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(F\right)}+\varphi \left(\frac{{a}_{*}}{\lambda }\right)\frac{\mu \left(E\cap {\overline{D}}_{{a}_{*}}\right)}{\mu \left(F\right)}\phantom{\rule{2em}{0ex}}\\ \le \underset{E\cap {D}_{{a}_{*}}}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(F\right)}+\underset{E\cap {\overline{D}}_{{a}_{*}}}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(F\right)}\phantom{\rule{2em}{0ex}}\\ =\underset{E}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(F\right)}.\phantom{\rule{2em}{0ex}}\end{array}$

So, (a) implies

${\mathcal{A}}_{F}\left(\lambda \right):=\underset{F}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(F\right)}\le S\underset{E}{\int }\varphi \left(\frac{\left|P\right|}{\lambda }\right)\frac{dx}{\mu \left(E\right)}=:S{\mathcal{A}}_{E}\left(\lambda \right).$

Let λ be such that ${\mathcal{A}}_{E}\left(\lambda \right)=1$. The convexity of ϕ implies ${\mathcal{A}}_{F}\left(S\lambda \right)\le 1$. So, Pϕ,FSPϕ,E.

## 3 Pólya inequality

Now, we present the main result concerning to Pólya inequality in Lϕ.

Theorem 3.1. Let ϕ Φ, and n, N . Let i k , 1 ≤ kn, be n positive integers such that${\sum }_{k=1}^{n}{i}_{k}=N+1$. Let B k , 1 ≤ kn, be disjoint pairwise compact intervals in , with 0 < μ(B k ) ≤ 1. Then there exists a positive constant M depending on ϕ, i k , and B k , 1 ≤ kn, such that

$\left|{c}_{j}\right|\le \frac{M}{{\text{min}}_{1\le k\le n}\left\{\mu {\left(E\cap {B}_{k}\right)}^{{i}_{k}-1}{\varphi }^{-1}\left(\frac{\mu \left(E\cap {B}_{k}\right)}{\mu \left(E\right)}\right)\right\}}{∥P∥}_{\varphi ,E,}\phantom{\rule{1em}{0ex}}0\le j\le N,$
(6)

for all$P\left(x\right)={\sum }_{j=0}^{N}{c}_{j}{x}^{j},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}E\subset {\bigcup }_{k=1}^{n}{B}_{k}$with μ(EB k ) > 0, 1 ≤ kn.

Proof. In the following proof, the constant M can be different in each occurrence. Let $P\left(x\right)={\sum }_{j=0}^{N}{c}_{j}{x}^{j}\in {\Pi }^{N}$, and let $E\subset {\bigcup }_{k=1}^{n}{B}_{k}$ be a measurable set with μ(EB k ) > 0, 1 ≤ kn. By Lemma 2.3 for C = B k , there exist n intervals F k = [a k - r k , a k + r k ] B k , 1 ≤ kn, such that μ(F k ) ≥ μ(EB k )/2N and ${∥P∥}_{\varphi ,{F}_{k}}\le 2N{∥P∥}_{\varphi ,E\cap {B}_{k}}$. From Lemma 2.2, there exists a positive constant M depending on p, i k , and B k , 1 ≤ kn, such that for all j, 0 ≤ ji k - 1, 1 ≤ kn, it verifies

$\left|{P}^{\left(j\right)}\left({a}_{k}\right)\right|\le \frac{M}{\mu {\left({F}_{k}\right)}^{j}}{∥P∥}_{\varphi ,{F}_{k}}\le \frac{M}{\mu {\left({F}_{k}\right)}^{{i}_{k}-1}}{∥P∥}_{\varphi ,{F}_{k}}\le \frac{M}{\mu {\left(E\cap {B}_{k}\right)}^{{i}_{k}-1}}{∥P∥}_{\varphi ,E\cap {B}_{k}}.$
(7)

From (7) and (2), there is a constant M such that

$\left|{P}^{\left(j\right)}\left({a}_{k}\right)\right|\le \frac{M}{\mu {\left(E\cap {B}_{k}\right)}^{{i}_{k}-1}{\varphi }^{-1}\left(\frac{\mu \left(E\cap {B}_{k}\right)}{\mu \left(E\right)}\right)}{∥P∥}_{\varphi ,E}$

for 0 ≤ ji k - 1, 1 ≤ kn. So

$\left|{P}^{\left(j\right)}\left({a}_{k}\right)\right|\le \frac{M}{{\text{min}}_{1\le s\le n}\left\{\mu {\left(E\cap {B}_{s}\right)}^{{i}_{s}-1}{\varphi }^{-1}\left(\frac{\mu \left(E\cap {B}_{s}\right)}{\mu \left(E\right)}\right)\right\}}{∥P∥}_{\varphi ,E},$

for 0 ≤ ji k - 1, 1 ≤ kn. From the equivalence of the norms 1 and 2 on ΠN,

${∥P∥}_{1}=\underset{1\le k\le n}{\text{max}}\underset{{a}_{k}\in {B}_{k}}{\text{sup}}\underset{0\le j\le {i}_{k}-1}{\text{max}}\left|{P}^{\left(j\right)}\left({a}_{k}\right)\right|\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{∥P∥}_{2}=\underset{0\le j\le N}{\text{max}}\left|{c}_{j}\right|,$

we obtain (6).

## 4 Best local approximation

In this section, we introduce a concept of balanced neighborhood in Lϕand we prove the existence of the best local approximation using the neighborhoods E k , 1 ≤ kn, mentioned in the Section 1.

It is easy to see that E k = x k + μ(E k )A k , where A k is a measurable set with measure 1. Henceforward, we assume the sets A k are uniformly bounded.

For each α and k, 1 ≤ kn, we denote

${\mathcal{A}}_{k}\left(\alpha \right):=\frac{\mu {\left({E}_{k}\right)}^{\alpha }}{{\varphi }^{-1}\left(\frac{\mu \left(E\right)}{\mu \left({E}_{k}\right)}\right)}.$

We assume the following condition, which allows us that ${\mathcal{A}}_{k}\left(\alpha \right)$ can be compared with each other as functions of α when μ(E) → 0.

For any nonnegative integers α and β, and any pair j, k, 1 ≤ j, kn,

$\text{either}\phantom{\rule{2.77695pt}{0ex}}{\mathcal{A}}_{k}\left(\alpha \right)=O\left({\mathcal{A}}_{j}\left(\beta \right)\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{or}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\mathcal{A}}_{j}\left(\beta \right)=o\left({\mathcal{A}}_{k}\left(\alpha \right)\right),\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\mu \left(E\right)\to 0.$
(8)

Let < i k > be an ordered n-tuple of nonnegative integers. We say that ${\mathcal{A}}_{j}\left({i}_{j}\right)$ is a maximal element of $<{\mathcal{A}}_{k}\left({i}_{k}\right)>$ if ${\mathcal{A}}_{k}\left({i}_{k}\right)=O\left({\mathcal{A}}_{j}\left({i}_{j}\right)\right)$ for all 1 ≤ kn. We denote it by

${\mathcal{A}}_{j}\left({i}_{j}\right)=\text{max}\left\{{\mathcal{A}}_{k}\left({i}_{k}\right)\right\}.$

Observe that ${\sum }_{k=1}^{n}{\mathcal{A}}_{k}\left({i}_{k}\right)=O\left(\text{max}\left\{{\mathcal{A}}_{k}\left({i}_{k}\right)\right\}\right).$.

Definition 4.1. An n-tuple < i k > of nonnegative integers is balanced if

$\sum _{k=1}^{n}{\mathcal{A}}_{k}\left({i}_{k}\right)=o\left(\underset{1\le k\le n}{\text{min}}\left\{\mu {\left({E}_{k}\right)}^{{i}_{k}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{k}\right)}{\mu \left(E\right)}\right)\right\}\right).$

In this case, we say that${\sum }_{k=1}^{n}{i}_{k}$is a balanced integer, and < E k > are balanced neighborhoods.

Lemma 4.2. To each balanced integer there corresponds exactly one balanced n-tuple.

Proof. Let < i k > be a balanced n-tuple. If $<{i}_{k}^{\prime }>$ is distinct from < i k > and ${\sum }_{k=1}^{n}{i}_{k}={\sum }_{k=1}^{n}{{i}^{\prime }}_{k}$, there exist indices j and s such that ${i}_{j}\ge {i}_{j}^{\prime }+1$ and ${i}_{s}^{\prime }\ge {i}_{s}+1$. From definition of balanced neighborhood, we have

$\mathcal{A}:=\sum _{k=1}^{n}{\mathcal{A}}_{k}\left({i}_{k}\right)=o\left(\mu {\left({E}_{j}\right)}^{{i}_{j}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{j}\right)}{\mu \left(E\right)}\right)\right).$

In addition, by (1) we get $\mu {\left({E}_{j}\right)}^{{i}_{j}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{j}\right)}{\mu \left(E\right)}\right)\le \mu {\left({E}_{j}\right)}^{{{i}^{\prime }}_{j}}{\varphi }^{-1}\left(\frac{\mu \left({E}_{j}\right)}{\mu \left(E\right)}\right)\le K{\varphi }^{-1}\left(1\right){\mathcal{A}}_{j}\left({i}_{j}^{\prime }\right)$. So, $\mathcal{A}=o\left(\sum _{k=1}^{n}{\mathcal{A}}_{k}\left({{i}^{\prime }}_{k}\right)\right)$. Again, by (1) we get

$\frac{\sum _{k=1}^{n}{\mathcal{A}}_{k}\left({{i}^{\prime }}_{k}\right)}{\mu {\left({E}_{s}\right)}^{{{i}^{\prime }}_{s}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{s}\right)}{\mu \left(E\right)}\right)}\ge \frac{\sum _{k=1}^{n}{\mathcal{A}}_{k}\left({{i}^{\prime }}_{k}\right)}{\mu {\left({E}_{s}\right)}^{{i}_{s}}{\varphi }^{-1}\left(\frac{\mu \left({E}_{s}\right)}{\mu \left(E\right)}\right)}\ge \frac{\sum _{k=1}^{n}{\mathcal{A}}_{k}\left({{i}^{\prime }}_{k}\right)}{K{\varphi }^{-1}\left(1\right){\mathcal{A}}_{s}\left({i}_{s}\right)}\to \infty .$

Then $<{i}_{k}^{\prime }>$ cannot be balanced.

The following lemma allows us to state an algorithm to compute all the balanced integers greater than a given balanced integer.

Lemma 4.3. Let < i k > and$<{i}_{k}^{\prime }>$be two balanced n-tuples with${\sum }_{k=1}^{n}{i}_{k}<{\sum }_{k=1}^{n}{{i}^{\prime }}_{k}$. Let$A=A\left(<{i}_{k}>\right):=\left\{j:{\mathcal{A}}_{j}\left({i}_{j}\right)=\text{max}\left\{{\mathcal{A}}_{k}\left({i}_{k}\right)\right\}\right\}$and B = B(< i k >):= {1,2,...,n}\ A. Then

(a) for j A${i}_{j}^{\prime }\ge {i}_{j}+1$.

(b) for j A${i}_{j}^{\prime }\ge {i}_{j}$.

Proof. (a) Suppose ${i}_{j}^{\prime }\le {i}_{j}$ for some j A. For any l B, from (8) we get ${\mathcal{A}}_{l}\left({i}_{l}\right)=o\left({\mathcal{A}}_{j}\left({i}_{j}\right)\right)$. Assume now ${i}_{l}^{\prime }\ge {i}_{l}+1$ for some l B. By (1), there exists a constant M > 0 such that

$\frac{{\mathcal{A}}_{j}\left({{i}^{\prime }}_{j}\right)}{\mu {\left({E}_{l}\right)}^{{{i}^{\prime }}_{l}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{l}\right)}{\mu \left(E\right)}\right)}\ge \frac{{\mathcal{A}}_{j}\left({i}_{j}\right)}{\mu {\left({E}_{l}\right)}^{{i}_{l}}{\varphi }^{-1}\left(\frac{\mu \left({E}_{l}\right)}{\mu \left(E\right)}\right)}\ge \frac{{\mathcal{A}}_{j}\left({i}_{j}\right)}{M{\mathcal{A}}_{l}\left({i}_{l}\right)}\to \infty ,$

as μ(E) → 0. Thus $<{i}_{k}^{\prime }>$ cannot be balanced, a contradiction. Therefore, either $B=\varnothing$ or ${i}_{l}^{\prime }\le {i}_{l}$, for all l B. On the other hand, since ${\sum }_{k=1}^{n}{i}_{k}<{\sum }_{k=1}^{n}{{i}^{\prime }}_{k}$, there is s A such that ${i}_{s}^{\prime }\ge {i}_{s}+1$. According to (1) and the definition of A we obtain

$\frac{{\mathcal{A}}_{j}\left({{i}^{\prime }}_{j}\right)}{\mu {\left({E}_{s}\right)}^{{{i}^{\prime }}_{s}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{s}\right)}{\mu \left(E\right)}\right)}\ge \frac{{\mathcal{A}}_{j}\left({i}_{j}\right)}{\mu {\left({E}_{s}\right)}^{{i}_{s}}{\varphi }^{-1}\left(\frac{\mu \left({E}_{s}\right)}{\mu \left(E\right)}\right)}\ge \frac{{\mathcal{A}}_{j}\left({i}_{j}\right)}{M{\mathcal{A}}_{s}\left({i}_{s}\right)}\ge {M}^{\prime },$

as μ(E) → 0, for some constant M' > 0. Therefore, $<{i}_{k}^{\prime }>$ cannot be balanced.

1. (b)

Suppose ${i}_{j}^{\prime }<{i}_{j}$ for some j B. From (a), (1) and the definition of balanced n-tuple, we obtain for each l A,

$\frac{{\mathcal{A}}_{j}\left({{i}^{\prime }}_{j}\right)}{\mu {\left({E}_{l}\right)}^{{{i}^{\prime }}_{l}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{l}\right)}{\mu \left(E\right)}\right)}\ge \frac{{\mathcal{A}}_{j}\left({i}_{j}-1\right)}{M{\mathcal{A}}_{l}\left({i}_{l}\right)}\ge {M}^{\prime }\frac{\mu {\left({E}_{j}\right)}^{{i}_{l}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{j}\right)}{\mu \left(E\right)}\right)}{{\mathcal{A}}_{l}\left({i}_{l}\right)}\to \infty ,$

as μ(E) → 0. Therefore $<{i}_{k}^{\prime }>$ cannot be balanced.

Given a balanced integer, the above lemma gives us a necessary condition which must satisfy the next balanced integer. The following example shows that the conditions of Lemma 4.3 are not sufficient to get a balanced n-tuple.

Example 4.4. Define ϕ (x) = x3(1 + |ln x|), x > 0, and ϕ(0) = 0. Consider two points x1, x2 with μ(E1) = δ4/3, μ (E2) = δ1/3, and A1 = A2 = [0,1]. The 2-tuple < 0,1 > is balanced. Here, the set A(< 0,1 >) = {0}, however < 1,1 > is not a balanced 2-tuple. In fact, if < i k >=< 0,1 > we obtain

$\underset{1\le k\le 2}{\text{min}}\left\{\mu {\left({E}_{k}\right)}^{{i}_{k}-1}\left(\frac{\mu \left({E}_{k}\right)}{\mu \left(E\right)}\right)\right\}=\text{min}\left\{\frac{{\varphi }^{-1}\left(\delta \right)}{{\delta }^{4/3}},{\varphi }^{-1}\left(1\right)\right\}+o\left(1\right)\to {\varphi }^{-1}$

as δ → 0. Since ${\mathcal{A}}_{2}\left({i}_{2}\right)=o\left({\mathcal{A}}_{1}\left({i}_{1}\right)\right)$ and ${\mathcal{A}}_{1}\left({i}_{1}\right)=o\left(1\right)$, as δ → 0, we have

$\frac{{\sum }_{k=1}^{2}{\mathcal{A}}_{k}\left({i}_{k}\right)}{\underset{1\le k\le 2}{\text{min}}\left\{\mu {\left({E}_{k}\right)}^{{i}_{k}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{k}\right)}{\mu \left(E\right)}\right)\right\}}=o\left(1\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\delta -0.$

So < 0,1 > is a balanced 2-tuple, A(< 0,1 >) = {0}, and < 1,1 > is the next 2-tuple generated by the algorithm. For < i k > = < 1,1 > we have

$\frac{{\mathcal{A}}_{2}\left({i}_{2}\right)}{\underset{1\le k\le 2}{\text{min}}\left\{\mu {\left({E}_{k}\right)}^{{i}_{k}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{k}\right)}{\mu \left(E\right)}\right)\right\}}\ge \frac{{\mathcal{A}}_{2}\left({i}_{2}\right)}{{\varphi }^{-1}\left(\frac{\mu \left({E}_{1}\right)}{\mu \left(E\right)}\right)}\to \infty ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\delta \to 0.$

Thus < 1,1 > is not a balanced 2-tuple.

Next, we establish an algorithm which gives all balanced n-tuples. First, we observe that < 0 > is a balanced n-tuple. In fact, since ϕ-1 is a concave positive function on + with ϕ-1(0) = 0, we have ϕ-1(x) ≥ ϕ-1(1)x, for x ≤ 1. This yields

$\frac{\mu \left({E}_{j}\right)}{{\varphi }^{-1}\left(\frac{\mu \left(E\right)}{\mu \left({E}_{k}\right)}\right){\varphi }^{-1}\left(\frac{\mu \left({E}_{j}\right)}{\mu \left(E\right)}\right)}\le \frac{\mu \left(E\right)}{{\left({\varphi }^{-1}\left(1\right)\right)}^{2}},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\le j,k\le n.$

Algorithm. Let v q be a balanced integer and let $<{i}_{k}^{\left(vq\right)}>$ be the corresponding balanced n-tuple. To build the next n-tuple, $<{i}_{k}^{\left(vq+1\right)}>$, put ${i}_{k}^{\left(vq+1\right)}={i}_{k}^{\left(vq\right)}+1$ for $k\in A\left(<{i}_{k}^{\left(vq\right)}>\right)$ and ${i}_{k}^{\left({v}_{q}+1\right)}={i}_{k}^{\left({v}_{q}\right)}$ for $k\in B\left(<{i}_{k}^{\left(vq\right)}>\right)$.

The following lemma shows that all balanced n-tuples are contained in the set of n-tuples generated by the algorithm.

Lemma 4.5. if < i k > is a balanced n-tuple with${\sum }_{k=1}^{n}{i}_{k}=q$, then the algorithm generates all the balanced n-tuple$<{i}_{k}^{*}>$with${\sum }_{k=1}^{n}{i}_{k}^{*}>q$.

Proof. Suppose $<{i}_{k}^{*}>$ is a balanced n-tuple with ${\sum }_{k=1}^{n}{i}_{k}^{*}=m>q$, and the n-tuple $<{i}_{k}^{\left(m\right)}>$ is not balanced. Since $\sum _{k=1}^{n}{i}_{k}^{*}=\sum _{k=1}^{n}{i}_{k}^{\left(m\right)}$, there exist r and s such that ${i}_{r}^{\left(m\right)}>{i}_{r}^{*}$ and ${i}_{s}^{*}>{i}_{s}^{\left(m\right)}$. By definition of balanced integer we have

${\mathcal{A}}_{r}\left({i}_{r}^{\left(m\right)}-1\right)=O\left({\mathcal{A}}_{r}\left({i}_{r}^{*}\right)\right)=o\left(\mu {\left({E}_{s}\right)}^{{i}_{s}^{*}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{s}\right)}{\mu \left(E\right)}\right)\right),$
(9)

and (1) implies $\mu {\left({E}_{s}\right)}^{{i}_{s}^{*}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{s}\right)}{\mu \left(E\right)}\right)\le K{\varphi }^{-1}\left(1\right){\mathcal{A}}_{s}\left({i}_{s}^{*}-1\right)$. So, ${\mathcal{A}}_{r}\left({i}_{r}^{\left(m\right)}-1\right)=o\left({\mathcal{A}}_{s}\left({i}_{s}^{\left(m\right)}\right)\right)$.

On the other hand, since m > q, Lemma 4.3 implies ${i}_{r}^{*}\ge {i}_{r}$, so ${i}_{r}^{\left(m\right)}>{i}_{r}$. Therefore ${\mathcal{A}}_{r}\left({i}_{r}^{\left(m\right)}-1\right)$ is maximal in a previous step of the algorithm, i.e., there exists m', qm' < m, such that ${\mathcal{A}}_{r}\left({i}_{r}^{\left(m\right)}-1\right)$ is maximal of $<{\mathcal{A}}_{k}\left({i}_{k}^{\left({m}^{\prime }\right)}\right)>$. Since the exponents ${i}_{k}^{\left(m\right)}$ are nondecreasing,

${\mathcal{A}}_{s}\left({i}_{s}^{\left(m\right)}\right)=O\left({\mathcal{A}}_{s}\left({i}_{s}^{\left({m}^{\prime }\right)}\right)\right)=O\left({\mathcal{A}}_{r}\left({i}_{r}^{\left(m\right)}-1\right)\right),$

Remark 4.6. If we assume the additional condition ϕ-1(x)ϕ-1 (1/x) ≥ c > 0 for x > 0, given a balanced n-tuple < i k >, it is easy to see that the n-tuple $<{i}_{k}^{\prime }>$ defined by ${i}_{k}^{\prime }={i}_{k}+1$ for $k\in A\left(<{i}_{k}^{\left(vq\right)}>\right)$, and ${i}_{k}^{\prime }={i}_{k}$ for $k\in B\left(<{i}_{k}^{\left(vq\right)}>\right)$, is balanced. It give us an algorithm that generates the infinite sequences of all balanced n-tuples.

Let PCm(X) be the class of functions with derivatives up to order m - 1 and with bounded piecewise continuous mth derivative on X.

Next, we prove the following auxiliary lemma.

Lemma 4.7. Let < i k > be an ordered n-tuple of nonnegative integers. Suppose h PCm(X), where m = max{i k } and h(j)(x k ) = 0, 0 ≤ ji k - 1, 1 ≤ kn. Then

${∥h∥}_{\varphi ,E}=O\left(\text{max}\left\{{\mathcal{A}}_{k}\left({i}_{k}\right)\right\}\right).$

Proof. Expanding h by the Taylor polynomial at x k up to the order n, we obtain

$h\left(x\right)=\sum _{k=1}^{n}{h}^{\left({i}_{k}\right)}\left({\xi }_{k}\right)\frac{{\left(x-{x}_{k}\right)}^{{i}_{k}}}{{i}_{k}!}{\chi }_{{E}_{k}}\left(x\right),\phantom{\rule{1em}{0ex}}x\in E,$

where ξ k is between x and x k . The change of variable x - x k = ϵy, y A k , yields

${∥h∥}_{\varphi ,E}=\text{inf}\left\{\lambda >0:\sum _{k=1}^{n}\underset{{A}_{k}}{\int }\mu \left({E}_{k}\right)\varphi \left(\frac{\left|{h}^{\left({i}_{k}\right)}\left({\xi }_{k}\right)\right|\frac{\mu {\left({E}_{k}\right)}^{{i}_{k}}\left|{y}^{{i}_{k}}\right|}{{i}_{k}!}}{\lambda }\right)\frac{dy}{\mu \left(E\right)}\le 1\right\}.$

For

$\lambda :=M\sum _{j=1}^{n}\frac{\mu {\left({E}_{j}\right)}^{{i}_{j}}}{{\varphi }^{-1}\left(\frac{\mu \left(E\right)}{n\phantom{\rule{2.77695pt}{0ex}}\mu \left({E}_{j}\right)}\right)},$

where $M=\underset{1\le k\le n}{\text{max}}\left\{\frac{1}{{i}_{k}!}\underset{x\in X}{\text{max}}\left\{\left|{h}^{\left({i}_{k}\right)}\left(x\right)\right|\right\}\underset{y\in {A}_{k}}{\text{max}}\left\{{\left|y\right|}^{{i}_{k}}\right\}\right\}$, we obtain

$\sum _{k=1}^{n}\underset{{A}_{k}}{\int }\mu \left({E}_{k}\right)\varphi \left(\frac{\left|{h}^{\left({i}_{k}\right)}\left({\xi }_{k}\right)\right|\frac{\mu {\left({E}_{k}\right)}^{{i}_{k}}\left|{y}^{{i}_{k}}\right|}{{i}_{k}!}}{\lambda }\right)\frac{dy}{\mu \left(E\right)}\le 1.$

Therefore ${∥h∥}_{\varphi ,E}=O\left({\sum }_{k=1}^{n}\frac{\mu {\left({E}_{k}\right)}^{{i}_{k}}}{{\varphi }^{-1}\left(\frac{\mu \left(E\right)}{n\phantom{\rule{2.77695pt}{0ex}}\mu \left({E}_{k}\right)}\right)}\right)$. Using the convexity of ϕ, we have $\frac{{\varphi }^{-1}\left(x\right)}{n}\le {\varphi }^{-1}\left(\frac{x}{n}\right)$, x ≥ 0. So, ${∥h∥}_{\varphi ,E}=O\left(\text{max}\left\{{\mathcal{A}}_{k}\left({i}_{k}\right)\right\}\right)$.

If a polynomial P ΠN, $N+1={\sum }_{k=1}^{n}{i}_{k}$, satisfies P(j)(x k ) = f(j)(x k ), 1 ≤ ji k - 1, 1 ≤ kn, we call it the Hermite interpolating polynomial of the function f on {x1,...,x n }.

Now, we are in condition to prove the main result in this Section.

Theorem 4.8. Let < i k > be a balanced n-tuple and$N+1={\sum }_{k=1}^{n}{i}_{k}$. If m = max{i k } and f PCm(X), then the best local approximation of f from ΠNon {x1,...,x k } is the Hermite interpolating polynomial of f on {x1,...,x n }.

Proof Let H ΠNbe the Hermite interpolating polynomial and let {g E } be a net of best approximations of f from ΠNrespect to · ϕ,E. From Lemma 4.7,

${∥{g}_{E}-H∥}_{\varphi ,E}=O\left(\text{max}\left\{{\mathcal{A}}_{k}\left({i}_{k}\right)\right\}\right).$

Using Theorem 3.1 and the equivalence of the norms in ΠN, we get

${∥{g}_{E}-H∥}_{\infty }\le \frac{K}{\underset{1\le k\le n}{\text{min}}\left\{\mu {\left({E}_{k}\right)}^{{i}_{k}-1}{\varphi }^{-1}\left(\frac{\mu \left({E}_{k}\right)}{\mu \left(E\right)}\right)\right\}}{∥{g}_{E}-H∥}_{\varphi ,E.}$

So, the definition of balanced n-tuple implies g E H, as μ(E) → 0.

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## Acknowledgements

This work was supported by Universidad Nacional de Rio Cuarto, Universi-dad Nacional de San Luis and CONICET.

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Correspondence to Claudia V Ridolfi.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

The three authors participated in the preparation of all work. All authors read and approved the final manuscript.

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Cuenya, H.H., Levis, F.E. & Ridolfi, C.V. Pólya-type polynomial inequalities in Orlicz spaces and best local approximation. J Inequal Appl 2012, 26 (2012). https://doi.org/10.1186/1029-242X-2012-26