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

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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, , μ), where μ is the Lebesgue measure, and denote = ( X ) 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 ϕ = L ϕ ( X ) : = f : X ϕ α f ( x ) d x < , for some α > 0 .
The space L ϕ is called the Orlicz space determined by ϕ. This space is endowed with the Luxemburg norm,
f ϕ , X = inf λ > 0 : X ϕ f ( x ) λ d x μ ( X ) 1 .

The space L ϕ with this norm is a Banach space (see [1]). If E 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 Π N , N , be the class of all algebraic polynomials of degree at most N, with real coefficients.

Given E , 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 ϕ , E = inf f - P ϕ , E : P N .
Let x k , 1 ≤ kn, be n points in X. We consider a net of measurable sets { E } such that E = k = 1 n E k , with μ(E k ) > 0 and
sup 1 k n sup y E k x k - y 0 , as μ ( E ) 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 [2], 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 [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 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 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
c j K min 1 k n μ ( E B k ) i k - 1 + 1 / p P p , E , 0 j N ,

for all P ( x ) = j = 0 N c j x j , E k = 1 n B k , μ ( E B k ) > 0 , 1 k 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 [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
ϕ - 1 ( x ) ϕ - 1 ( y ) K ϕ - 1 ( x y ) , for all x , y 0 .
(1)

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

2 Preliminary results

Let 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) X E ϕ , E = 1 ϕ - 1 ( 1 ) .

(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 ϕ , G M ϕ - 1 μ ( G ) μ ( D ) f ϕ , D , f L ϕ ( X ) ,
(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
E ϕ X E λ d x μ ( E ) = E d x μ ( E ) = 1 .
Now, the Δ2- condition implies X E ϕ , E = 1 / ϕ - 1 ( 1 ) .
  1. (b)
    If |f| ≤ |g| on E, then
    E ϕ f λ d x μ ( E ) E ϕ g λ d x μ ( E ) , λ > 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
    U ( λ ) : = G ϕ f λ d x μ ( G ) and B ( λ ) : = D ϕ f λ d x μ ( D ) .
     
We consider λ > 0 such that B ( λ ) 1 . By the Δ'-condition we obtain
U λ ϕ - 1 μ ( G ) c μ ( D ) D c μ ( G ) c μ ( D ) ϕ f λ d x μ ( G ) = B ( λ ) 1 .

Then f ϕ , G λ ϕ - 1 μ ( G ) c μ ( D ) | , for all λ > 0 with B ( λ ) 1 . So, the definition of fϕ,Dand (1) imply f ϕ , G M ϕ - 1 μ ( G ) μ ( D ) f ϕ , D with M = K ϕ - 1 ( c - 1 ) .

Lemma 2.2. There exists a constant M > 0 such that
P ( j ) ( a ) M ε j P ϕ , [ a - ε , a + ε ] ,

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 ϕ , J ε M P ϕ , [ a - ε , a + ε ] , where M is independent on P, a, and ϵ. In addition, there exists yϵ Jϵ such that P ( y ε ) ϕ - 1 ( 1 ) P ϕ , J ε . In fact, if ϕ - 1 ( 1 ) P ϕ , J ε < P ( y ) , for all y Jϵ, then Proposition 2.1 (a) and (b) yield P ϕ , J ε > P ϕ , J ε . 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
P ( y i , ε ) M ϕ - 1 ( 1 ) P ϕ , [ a - ε , a + ε ] , 1 i N + 1 .
(3)
If t i , ε : = y i , ε - a ε [ - 1 , 1 ] , we have
P ( y i , ε ) = j = 0 N P ( j ) ( a ) j ! ( y i , ε - a ) j = j = 0 N P ( j ) ( a ) j ! ε j t i , ε j , 1 i N + 1 .
(4)
The matrix of the linear system (4), t i , ε j , 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
P ( j ) ( a ) ε j M P ϕ , [ a , ε , a + ε ] 0 j 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 Π N , there exists an interval F := F(E,P) C such that

a) μ ( F ) μ ( E ) 2 N ,

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 lim a G ( a ) = μ ( C ) . Therefore, there exists a constant a* + such that μ ( D a * ) = μ ( E ) / 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 * = j = 1 k E j .

We denote A ¯ = C \ A , for any set A. Then
μ ( E D ¯ a * ) = μ ( E ) - μ ( E D a * ) μ ( E ) - μ ( D a * ) = μ ( E ) 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 μ ( D a * ) < k / S μ ( E ) μ ( E ) / 2 , which is a contradiction. So, we have proved a) with F := E j .

Using (5), we obtain
μ ( E D ¯ a * ) μ ( E ) 2 = μ ( D a * ) μ ( F ) μ ( F E ¯ ) .
Therefore
F ϕ P λ d x μ ( F ) F E ϕ P λ d x μ ( F ) + ϕ a * λ μ ( E D ¯ a * ) μ ( F ) E D a * ϕ P λ d x μ ( F ) + E D ¯ a * ϕ P λ d x μ ( F ) = E ϕ P λ d x μ ( F ) .
So, (a) implies
A F ( λ ) : = F ϕ P λ d x μ ( F ) S E ϕ P λ d x μ ( E ) = : S A E ( λ ) .

Let λ be such that A E ( λ ) = 1 . The convexity of ϕ implies A F ( S λ ) 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 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
c j M min 1 k n μ ( E B k ) i k - 1 ϕ - 1 μ ( E B k ) μ ( E ) P ϕ , E , 0 j N ,
(6)

for all P ( x ) = j = 0 N c j x j , E 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 ( x ) = j = 0 N c j x j Π N , and let E 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 ϕ , F k 2 N P ϕ , E 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
P ( j ) ( a k ) M μ ( F k ) j P ϕ , F k M μ ( F k ) i k - 1 P ϕ , F k M μ ( E B k ) i k - 1 P ϕ , E B k .
(7)
From (7) and (2), there is a constant M such that
P ( j ) ( a k ) M μ ( E B k ) i k - 1 ϕ - 1 μ ( E B k ) μ ( E ) P ϕ , E
for 0 ≤ ji k - 1, 1 ≤ kn. So
P ( j ) ( a k ) M min 1 s n μ ( E B s ) i s - 1 ϕ - 1 μ ( E B s ) μ ( E ) P ϕ , E ,
for 0 ≤ ji k - 1, 1 ≤ kn. From the equivalence of the norms 1 and 2 on Π N ,
P 1 = max 1 k n sup a k B k max 0 j i k - 1 P ( j ) ( a k ) and P 2 = max 0 j N c j ,

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
A k ( α ) : = μ ( E k ) α ϕ - 1 μ ( E ) μ ( E k ) .

We assume the following condition, which allows us that A k ( α ) 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,
either A k ( α ) = O ( A j ( β ) ) or A j ( β ) = o ( A k ( α ) ) , as μ ( E ) 0 .
(8)
Let < i k > be an ordered n-tuple of nonnegative integers. We say that A j ( i j ) is a maximal element of < A k ( i k ) > if A k ( i k ) = O ( A j ( i j ) ) for all 1 ≤ kn. We denote it by
A j ( i j ) = max A k ( i k ) .

Observe that k = 1 n A k ( i k ) = O ( max { A k ( i k ) } ) . .

Definition 4.1. An n-tuple < i k > of nonnegative integers is balanced if
k = 1 n A k ( i k ) = o min 1 k n μ ( E k ) i k - 1 ϕ - 1 μ ( E k ) μ ( E ) .

In this case, we say that 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 > is distinct from < i k > and k = 1 n i k = k = 1 n i k , there exist indices j and s such that i j i j + 1 and i s i s + 1 . From definition of balanced neighborhood, we have
A : = k = 1 n A k ( i k ) = o μ ( E j ) i j - 1 ϕ - 1 μ ( E j ) μ ( E ) .
In addition, by (1) we get μ ( E j ) i j - 1 ϕ - 1 μ ( E j ) μ ( E ) μ ( E j ) i j ϕ - 1 μ ( E j ) μ ( E ) K ϕ - 1 ( 1 ) A j ( i j ) . So, A = o k = 1 n A k ( i k ) . Again, by (1) we get
k = 1 n A k ( i k ) μ ( E s ) i s - 1 ϕ - 1 μ ( E s ) μ ( E ) k = 1 n A k ( i k ) μ ( E s ) i s ϕ - 1 μ ( E s ) μ ( E ) k = 1 n A k ( i k ) K ϕ - 1 ( 1 ) A s ( i s ) .

Then < i k > 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 > be two balanced n-tuples with k = 1 n i k < k = 1 n i k . Let A = A ( < i k > ) : = { j : A j ( i j ) = max { A k ( i k ) } } and B = B(< i k >):= {1,2,...,n}\ A. Then

(a) for j A i j i j + 1 .

(b) for j A i j i j .

Proof. (a) Suppose i j i j for some j A. For any l B, from (8) we get A l ( i l ) = o A j ( i j ) . Assume now i l i l + 1 for some l B. By (1), there exists a constant M > 0 such that
A j ( i j ) μ ( E l ) i l - 1 ϕ - 1 μ ( E l ) μ ( E ) A j ( i j ) μ ( E l ) i l ϕ - 1 μ ( E l ) μ ( E ) A j ( i j ) M A l ( i l ) ,
as μ(E) → 0. Thus < i k > cannot be balanced, a contradiction. Therefore, either B = or i l i l , for all l B. On the other hand, since k = 1 n i k < k = 1 n i k , there is s A such that i s i s + 1 . According to (1) and the definition of A we obtain
A j ( i j ) μ ( E s ) i s - 1 ϕ - 1 μ ( E s ) μ ( E ) A j ( i j ) μ ( E s ) i s ϕ - 1 μ ( E s ) μ ( E ) A j ( i j ) M A s ( i s ) M ,
as μ(E) → 0, for some constant M' > 0. Therefore, < i k > cannot be balanced.
  1. (b)
    Suppose i j < i j for some j B. From (a), (1) and the definition of balanced n-tuple, we obtain for each l A,
    A j ( i j ) μ ( E l ) i l - 1 ϕ - 1 μ ( E l ) μ ( E ) A j ( i j - 1 ) M A l ( i l ) M μ ( E j ) i l - 1 ϕ - 1 μ ( E j ) μ ( E ) A l ( i l ) ,
     

as μ(E) → 0. Therefore < i k > 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
min 1 k 2 μ ( E k ) i k - 1 μ ( E k ) μ ( E ) = min ϕ - 1 ( δ ) δ 4 / 3 , ϕ - 1 ( 1 ) + o ( 1 ) ϕ - 1
as δ → 0. Since A 2 ( i 2 ) = o ( A 1 ( i 1 ) ) and A 1 ( i 1 ) = o ( 1 ) , as δ → 0, we have
k = 1 2 A k ( i k ) min 1 k 2 μ ( E k ) i k - 1 ϕ - 1 μ ( E k ) μ ( E ) = o ( 1 ) , as δ - 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
A 2 ( i 2 ) min 1 k 2 μ ( E k ) i k - 1 ϕ - 1 μ ( E k ) μ ( E ) A 2 ( i 2 ) ϕ - 1 μ ( E 1 ) μ ( E ) , as δ 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
μ ( E j ) ϕ - 1 μ ( E ) μ ( E k ) ϕ - 1 μ ( E j ) μ ( E ) μ ( E ) ( ϕ - 1 ( 1 ) ) 2 , 1 j , k n .

Algorithm. Let v q be a balanced integer and let < i k ( v q ) > be the corresponding balanced n-tuple. To build the next n-tuple, < i k ( v q + 1 ) > , put i k ( v q + 1 ) = i k ( v q ) + 1 for k A < i k ( v q ) > and i k ( v q + 1 ) = i k ( v q ) for k B < i k ( v q ) > .

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 k = 1 n i k = q , then the algorithm generates all the balanced n-tuple < i k * > with k = 1 n i k * > q .

Proof. Suppose < i k * > is a balanced n-tuple with k = 1 n i k * = m > q , and the n-tuple < i k ( m ) > is not balanced. Since k = 1 n i k * = k = 1 n i k ( m ) , there exist r and s such that i r ( m ) > i r * and i s * > i s ( m ) . By definition of balanced integer we have
A r i r ( m ) - 1 = O A r i r * = o μ ( E s ) i s * - 1 ϕ - 1 μ ( E s ) μ ( E ) ,
(9)

and (1) implies μ ( E s ) i s * - 1 ϕ - 1 μ ( E s ) μ ( E ) K ϕ - 1 ( 1 ) A s i s * - 1 . So, A r i r ( m ) - 1 = o A s i s ( m ) .

On the other hand, since m > q, Lemma 4.3 implies i r * i r , so i r ( m ) > i r . Therefore A r i r ( m ) - 1 is maximal in a previous step of the algorithm, i.e., there exists m', qm' < m, such that A r i r ( m ) - 1 is maximal of < A k i k ( m ) > . Since the exponents i k ( m ) are nondecreasing,
A s i s ( m ) = O A s i s ( m ) = O A r i r ( m ) - 1 ,

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 > defined by i k = i k + 1 for k A < i k ( v q ) > , and i k = i k for k B < i k ( v q ) > , 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 ϕ , E = O max { A k ( i k ) } .
Proof. Expanding h by the Taylor polynomial at x k up to the order n, we obtain
h ( x ) = k = 1 n h ( i k ) ( ξ k ) ( x - x k ) i k i k ! χ E k ( x ) , x E ,
where ξ k is between x and x k . The change of variable x - x k = ϵy, y A k , yields
h ϕ , E = inf λ > 0 : k = 1 n A k μ ( E k ) ϕ h ( i k ) ( ξ k ) μ ( E k ) i k y i k i k ! λ d y μ ( E ) 1 .
For
λ : = M j = 1 n μ ( E j ) i j ϕ - 1 μ ( E ) n μ ( E j ) ,
where M = max 1 k n 1 i k ! max x X h ( i k ) ( x ) max y A k y i k , we obtain
k = 1 n A k μ ( E k ) ϕ h ( i k ) ( ξ k ) μ ( E k ) i k y i k i k ! λ d y μ ( E ) 1 .

Therefore h ϕ , E = O k = 1 n μ ( E k ) i k ϕ - 1 μ ( E ) n μ ( E k ) . Using the convexity of ϕ, we have ϕ - 1 ( x ) n ϕ - 1 x n , x ≥ 0. So, h ϕ , E = O max { A k ( i k ) } .

If a polynomial P Π N , N + 1 = 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 = 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 ϕ , E = O max { A k ( i k ) } .
Using Theorem 3.1 and the equivalence of the norms in Π N , we get
g E - H K min 1 k n μ ( E k ) i k - 1 ϕ - 1 μ ( E k ) μ ( E ) g E - H ϕ , 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

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Copyright

© Cuenya et al.; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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