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

Journal of Inequalities and Applications20122012:26

https://doi.org/10.1186/1029-242X-2012-26

• Received: 14 July 2011
• Accepted: 10 February 2012
• Published:

## 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.

## Keywords

• algebraic polynomials
• pólya-type inequalities
• best local approximation
• balanced integers

## 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 L ϕ is 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 L ϕ with this norm is a Banach space (see ). If $E\in ℬ$ and μ(E) > 0, then · ϕ,Eis a seminorm on L ϕ (X). In the particular case, ϕ (t) = t p , 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 Π N is a best approximation of f L ϕ (X) from Π N respect 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 L ϕ (X) and Π N , we consider a net of best approximation functions {g E }. If it has a limit in Π N as μ(E) → 0, this limit is called the best local approximation of f from Π N on {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 , Chui et al. introduced the balanced neighborhood concept and they studied existence and characterization of best local approximation in L p -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 .

Polynomial inequalities on measurable sets have been studied extensively in the literature (see ). In , the authors proved the following extension of the Pólya inequality in L p -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 L p spaces, with balanced neighborhoods.

In this article, we generalize Theorem 1.1 and the balanced neighborhood concept to L ϕ . As a consequence of this extension we prove the existence of the best local approximation of a function from Π N on {x1,..., x n }, with balanced neighborhoods, following the pattern used in . 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 .

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 L ϕ (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 Π N , [a - ϵ, a + ϵ] X, and 0 ≤ jN.

Proof. Given P Π N and [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 , 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 Π N , 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 Π N , 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 L ϕ .

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

we obtain (6).

## 4 Best local approximation

In this section, we introduce a concept of balanced neighborhood in L ϕ and 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,
${\mathcal{A}}_{s}\left({i}_{s}^{\left(m\right)}\right)=O\left({\mathcal{A}}_{s}\left({i}_{s}^{\left({m}^{\prime }\right)}\right)\right)=O\left({\mathcal{A}}_{r}\left({i}_{r}^{\left(m\right)}-1\right)\right),$

which contradicts (9).

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 PC m (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 PC m (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 Π N , $N+1={\sum }_{k=1}^{n}{i}_{k}$, 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 PC m (X), then the best local approximation of f from Π N on {x1,...,x k } is the Hermite interpolating polynomial of f on {x1,...,x n }.

Proof Let H Π N be the Hermite interpolating polynomial and let {g E } be a net of best approximations of f from Π N respect 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 Π N , 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.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
Department of Mathematics, UNRC, 5800 Río Cuarto, Argentina
(2)
Department of Mathematics, UNSL, 5700 San Luis, Argentina

## References

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