Weighted approximation of continuous positive functions

Weighted approximation of continuous positive functions

M.S. Kashimoto

Departamento de Matemática e Computação, IMC, UNIFEI, Universidade Federal de Itajubá, 37500-903 Itajubá, MG, Brasil. E-mails: [e-mail]; [e-mail]

ABSTRACT

We investigate the density of convex cones of continuous positive functions in weighted spaces and present some applications.

Keywords: convex cone, weighted space, Bernstein's Theorem.

RESUMO

Investigamos a densidade de cones convexos de funções contínuas positivas em espaços ponderados e apresentamos algumas aplicações.

Palavras-chave: cone convexo, espaço ponderado, Teorema de Bernstein.

INTRODUCTION AND PRELIMINARIES

Throughout this paper we shall assume, unless stated otherwise, that X is a locally compact Hausdorff space. We shall denote by C (X; ) the space of all continuous real-valued functions on X and by C_b(X; ) the space of continuous and bounded real-valued functions on X.The vector subspace of all functions in C (X; ) with compact support is denoted by C_c (X; ).

An upper semicontinuous real-valued function f on X is said to vanish at infinity if, for every ε > 0, the closed subset {x ∈X :| f (x)| > ε} is compact.

In what follows, we shall present the concept of weighted spaces as developed by Nachbin in [4]. We introduce a set V of non-negative upper semicontinuous functions on X, whose elements are called weights. We assume that V is directed, in the sense that, given v₁,v₂∈ V , there exist λ > 0 and v ∈V such that v₁< λv and v₂< λv.

Let V be a directed set of weights. The vector subspace of C (X; ) of all functions f such that vf vanishes at infinity for each v ∈V will be denoted by CV _∞ (X; ).

When CV_∞ (X; ) is equipped with the locally convex topology ω_Vgenerated by the seminorms

for each v ∈V , we call CV _∞ (X; ) a weighted space.

We assume that for each x ∈X, there is v ∈V such that v(x) > 0.

In the following we present some examples of weighted spaces.

(a) If V consists of the constant function 1, defined by 1(x) = 1 for all x ∈ X, then CV_∞ (X; ) is C₀ (X; ), the vector subspace of all functions in C (X; ) that vanish at infinity. In particular, if X is compact then CV_∞ (X; ) = C (X; ). The corresponding weighted topology is the topology of uniform convergence on X.

(b) Let V be the set of characteristic functions of all compact subsets of X. Then the weighted space CV_∞ (X; ) is C (X; ) endowed with the compact-open topology.

(c) If V consists of characteristic functions of all finite subsets of X,then CV_∞ (X; ) is C (X; ) endowed with the topology of pointwise convergence.

(d) If V ={v ∈C₀ (X; ) : v > 0}, then CV_∞ (X; ) is the vector space C_b(X; ). The corresponding weighted topology is the strict topology β (see Buck [1]).

For more information on weighted spaces we refer the reader to [4, 5].

We set (X; ) = {f ∈CV_∞ (X; ) : f > 0}

A subset W ⊂ (X; ) is a convex cone if λW ⊂ W, for each λ > 0 and W + W ⊂ W.

We denote by the subset of (X × Y; ) consisting of all functions of the form

where g_i∈ (X; ), h_i ∈ (Y; ), i = 1, ..., n, n ∈.

Let W ⊂ (X; ) be a nonempty subset. A function ϕ ∈ C (X; ), 0 < ϕ <1, is called a multiplier of W if ϕ f + (1 − ϕ)g ∈ W for every pair f and g of elements of W. The set of all multipliers of W is denoted by M(W). The notion of a multiplier of W is due to Feyel and De La Pradelle [3] and Chao-Lin [2].

For any x ∈ X, [x]_M(W) denotes the equivalence class of x, when one defines the following equivalence relation on X : x ≡ t (modM(W)) if, and only if, ϕ(x) = ϕ(t) for all ϕ ∈M (W).

A subset A ⊂ C (X; ) separates the points of X if, given any two distinct points s and t of X, there is a function ϕ ∈ A such that ϕ(s) ≠ ϕ(t).

Weierstrass' first theorem states that any real-valued continuous function f defined on the closed interval [0,1] is the limite of a uniformly convergent sequence of algebraic polynomials. One of the most elementary proofs of this classic result is that which uses the Bernstein polynomials of f

for each natural number n. Bernstein's theorem states that B_n( f ) → f uniformly on [0,1] and, since each B_n( f ) is a polynomial, we have as a consequence the Weierstrass approximation theorem. The operator B_ndefined on the space C ([0, 1]) with values in the vector subspace of all polynomials of degree at most n has the property that B_n( f ) > 0 whenever f > 0. Thus Bernstein's theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials.

Consider a compact Hausdorff space X and the convex cone

C⁺ (X; ) = {f ∈C (X; ) : f > 0}.

A generalized Bernstein's theorem would be a theorem stating when a convex cone contained in C⁺ (X; ) is dense in it.

Prolla [6] proved the following result of uniform density of convex cones in C⁺ (X; ).

Theorem 1.1. Let X be a compact Hausdorff space. Let W ⊂ C⁺ (X; ) be a convex cone satisfying the following conditions:

(a) given any two distinct points x and y in X, there is a multiplier ϕ of W such that ϕ(x) ≠ ϕ(y);

(b) given any x ∈ X, there is g ∈W such that g (x) > 0.

Then W is uniformly dense in C⁺ (X; ).

The purpose of this note is to present an extension of this result to weighted spaces and give some applications. The main tool is a Stone-Weierstrass-type theorem for subsets of weighted spaces.

2 THE RESULTS

We need the following lemma, whose proof can be found in [7].

Lemma 2.1. Le t W be a nonempty subset of CV_∞ (X; ). Given any f ∈ CV_∞ (X; ), v ∈V and ε > 0, the following statements are equivalent:

1. there exists h ∈W such that v(x)║f (x) − h(x)║< ε for all x ∈ X ;

2. for each x ∈X, there exists g_x∈W such that v(t) ║f (t) − g_x(t)║< ε for all t ∈ [x]_M(W).

Now we state the main result.

Theorem 2.1. Let W ⊂ (X ; ) be a convex cone satisfying the following conditions:

(a) given any two distinct points x and y in X, there exists a multiplier ϕ of W such that ϕ(x) ≠ ϕ(y);

(b) given any x ∈X, there exists g ∈W such that g(x) > 0.

Then W is ω_V -dense in (X; ).

Proof. Let x be an arbitrary element of X . Condition (a) implies that [x]_{M(W )} = {x}. By condition (b), there exists g ∈W such that g(x) > 0. Then, for any f ∈ (X; ), v ∈V and ε > 0, we have

Since W is a convex cone, ∈ W. Then, it follows from Lemma 2.1 that there exists h ∈ W such that v(t)║f (t) − h(t)║< ε for all t ∈ X.

Corollary 2.1. Let X and Y be locally compact Hausdorff spaces. Then

is dense in (X × Y; ).

Proof. It follows from Urysohn's Lemma [8] that for any two distinct elements (s, t) and (u, v) of X × Y, there exist functions h₁∈ C_c(X; ) and h₂∈ C_c(Y; ), 0 < h₁, h₂< 1, such that φ(x, y) := h₁(x)h₂(y) is a multiplier of (X; ) (Y; ) and φ(s, t) =1 and φ(u, v) = 0. Hence, condition (a) of Theorem 2.1 is satisfied.

By using Urysohn's Lemma again, given (x, y) ∈ X × Y, there exist ϕ ∈ C_c(X; ) and ψ ∈ C_c(Y; ) such that ϕ(x) = 1 and ψ(y) = 1 so that ϕ(x)ψ(y) > 0,

Then, condition (b) of Theorem 2.1 is satisfied. Hence, the assertion follows by Theorem 2.1.

Example 2.1. Consider (; ),where V is the set of characteristic functions of all compact subsets of . Let ψ ∈ C (; ),0 < ψ < 1, be a one-to-one function. Let W be the set of all functions g of the form

where each b_ijis a non-negative real number and i, j, n are non-negative integers numbers. Note that W ⊂ (; ) is a convex cone.

Since ψ ∈ M (W) and W contains positive constant functions, it follows from Theorem 2.1 that W is dense in (; ).

Example 2.2. Let a be a fixed positive real number. Let W be the set of all functions of the form

Clearly, W is a convex cone contained in ([0, ∞); ). The function e^–ax, x ∈ [0, ∞), belongs to W and is a multiplier of W that separates the points of X. Hence, by Theorem 2.1 W is dense in ([0,∞); ).

Received on August 24, 2012 / Accepted on May 17, 2013

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• [4] L. Nachbin. "Elements of Approximation Theory". Van Nostrand, Princeton, NJ, 1967, reprinted by Krieger, Huntington, NY (1976).
• [5] J.B. Prolla. "Approximation of Vector-Valued Functions". Mathematics Studies, 25, North-Holland, Amsterdam (1977).
• [6] J.B. Prolla. A generalized Bernstein approximation theorem. Math. 104 (1988), 317330.
• [7] J.B. Prolla & M.S. Kashimoto. Simultaneous approximation and interpolation in weighted spaces. Rendi. Circ. Mat. di Palermo, Serie II Tomo LI (2002), 485494.
• [8] W. Rudin. "Real and Complex Analysis". McGraw-Hill, Singapore, Third Edition (1987).

Publication Dates

• Publication in this collection
  04 Oct 2013
• Date of issue
  Aug 2013

History

• Received
  24 Aug 2012
• Accepted
  17 May 2013