ABSTRACT
In this paper, we give some applications of Nachbin’s Theorem (4) to approximation and interpolation in the the space of all k times continuously differentiable real functions on any open subset of the Euclidean space.
Keywords:
Nachbin’s theorem; approximation of differentiable functions; Stone-Weierstrass theorem; interpolation
RESUMO
Em 1949, Leopoldo Nachbin estabeleceu uma versão do Teorema de Stone-Weierstrass para funções diferenciáveis de classe Ck em abertos do espaço euclidiano. Neste trabalho, apresentamos algumas aplicações desse teorema relacionadas com aproximação e interpolação no espaço das funções de classe Ck munido da topologia compacto-aberta.
Palavras-chave:
Teorema de Nachbin; aproximação de funções diferenciáveis; Teorema de Stone-Weierstrass; interpolação
1 INTRODUCTION
Let Ω be an open subset of ℛ p and let k be a nonnegative integer. We denote by C k (Ω; ℛ) the algebra of all k times continuously differentiable real functions on Ω and consider the compact open topology of order , that is, the topology of uniform convergence for the functions and all their partial derivatives up to the order k on compact subsets of Ω.
For a multi-index of non-negative integers, let be the order of α, , and for let represents the corresponding linear partial differential operator acting on C k (Ω; ℛ).
The topology is generated by the semi-norms σ k,Γ given by
where Γ runs over all compact subsets of Ω. By Proposition 3, p. 8 55 L. Nachbin, ”Elements of Approximation Theory”. Van Nostrand, Princeton, NJ (1967), reprinted by Krieger, Huntington, NY (1976)., C k (Ω; ℛ) is a topological vector space with respect to this topology.
In 1949 Nachbin 44 L. Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété. Comptes Rendus de l’Académie des Sciences de Paris, 228 (1949), 1549-1551. established the following interesting characterization of dense subalgebras of the space C k (Ω; ℛ).
Theorem 1.(Nachbin) Let Ω be an open subset of ℛpand L be a subalgebra of Ck (Ω; ℛ). Then L is dense in C k (Ω; ℛ) if and only if the following conditions are satisfied:
-
given x, y ∈ Ω with x ≠ y, there exists f ∈ L such that f (x) ≠ f (y);
-
given x ∈ Ω, there exists f ∈ L such that f (x) ≠ 0;
-
given x ∈ Ω and u ∈ ℛ p with u ≠ 0 , there exists f ∈ L such that .
The proof of this result can be found in 33 J. Mujica, Subálgebras densas de funciones diferenciables. Cubo Matemática Educacional, 3 (2001), 121-128. and (44 L. Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété. Comptes Rendus de l’Académie des Sciences de Paris, 228 (1949), 1549-1551..
Our aim is to use Nachbin’s theorem to give a proof of a density theorem and a simultaneous interpolation and approximation theorem in the space C k (Ω; ℛ).
2 THE RESULTS
The Urysohn’s Lemma (22 M. Moskowitz & F. Paliogiannis, ”Functions of Several Real Variables”. World Scientific, Singapore (2011). p. 281) for differentiable functions is the main tool we employed in the next lemma.
Lemma 1.Let Ω be an open subset of ℛ p , w 1 ,..., w m distinct points in Ω, and y 1 , . . . , y m distinct real numbers. If L is a dense vector subspace of C k (Ω; ℛ), then there exists a function h ∈ L such that.
Proof. Let L be a dense linear subspace of C k (Ω; ℛ) and be a subset of Ω. Consider the following linear mapping
Notice that T is continuous.
For each w i ∈ S consider an open neighborhood U i ⊂ Ω of w i such that w j ∉Ui, for all j ≠ i, j ∈ {1,..., m}. It follows from the Urysohn’s Lemma for differentiable functions that there exists an infinitely differentiable function , such that and , if x ∉U i , in particular, . Let the restriction of the function Φi to the subset Ω and e i ∈ ℛ m the vector whose i th coordinate is equal to 1 and the others are equal to 0.
The linear mapping T is surjective since for any (c 1 ,..., c m ) ∈ ℛ m , we have
where . Moreover, T(L) is closed because it is a linear subspace of ℛ m . Then by density of L and continuity of T, it follows that
Therefore, for any there exists h ∈ L such that , that is, .
We give a proof of the following density result.
Theorem 2.Let V be an open subset of ℛp, L a dense subalgebra of Ck (V; ℛ), and v 1 , . . . , v n distinct points in V. Consider the open subset of ℛ p ,
and the subalgebra
Then, M is dense in Ck (Ω; ℛ).
Proof. Clearly M is a subalgebra of C k (Ω; ℛ). Let x, y be any distinct points in Ω. Consider the following subset
of V. By Lemma 1 there exists h ∈ L such that and for j = 1, . . . , n. Then, h|Ω ∈ M and satisfies Conditions (a) and (b) of Theorem 1.
Now let z ∈ Ω and u ∈ ℛ p , u ≠ 0. It follows from Lemma 1 that there exists g ∈ L such that and for j = 1, . . . , n. Hence, . If the Condition (c) of Theorem 1 is satisfied. Otherwise, notice that L is not a subset of
since L is a dense subalgebra of C k (V; ℛ) and B is a proper closed subalgebra of C k (V; ℛ). Thus, there exists ϕ ∈ L such that . Then, ϕg ∈ L and for j = 1, . . . , n, that is, . Moreover,
Thus, by Theorem 1, M is dense in Ck(Ω; ℛ).
For each positive integer l, 𝒫 l (ℛ p , ℛ) denotes the linear subspace of C k (ℛ p , ℛ) generated by the set of all functions of the form
where , the dual space of ℛ p . The elements of 𝒫 l (ℛ p , ℛ) are called the l - homogeneous continuous polynomials of finite type from ℛ p into ℛ . The subspace of C k (ℛ p , ℛ) consisting of all functions of the form
where , is denoted by 𝒫(ℛ p , ℛ). Its elements are called real continuous polynomials of finite type. The polarization formula shows that 𝒫(ℛ p , ℛ) is a subalgebra of C k (ℛ p , ℛ). Indeed, given ψ 1 and ψ 2 in (ℛ p )∗,
shows that ψ 1 ψ 2 ∈ 𝒫 2(ℛ p , ℛ), since ψ 1 + ψ 2 and ψ 1 - ψ 2 belong to (ℛ p )∗ .
Corollary 3.Let v1, . . . , vnbe distinct points in ℛp . Consider the open subset of ℛ p ,
and the subalgebra
Then, M is dense in Ck (Ω; ℛ).
Proof. First of all, we verify that the subalgebra P(ℛ p , ℛ) is dense in C k (ℛ p , ℛ). Given x, y ∈ ℛ p with x ≠ y, it follows from Hahn-Banach Theorem that there exists ψ ∈ (ℛ p )* such that ψ(x) ≠ ψ(y). Since , the Condition (a) of Theorem 1 is satis- fied. By definition, P(ℛ p , ℛ) contains all the constant functions. Now, let . Then, there exists . Let defined by . Since and for i ≠ j, it follows that
Therefore, by Theorem 1, P(ℛ p , ℛ) is dense in C k (ℛ p , ℛ) and the assertion follows from Theorem 2.
Motivated by an extended Stone-Weierstrass theorem (see Corollary 1.1 11 F. Deutsch, Simultaneous interpolation and approximation in linear topological spaces. SIAM J. Appl. Math., 14 (1966), 1180-1190.), we give a proof of a result concerning simultaneous interpolation and approximation in C k (Ω; ℛ). The tools are the Nachbin’s Theorem and the following result due to Deutsch.
Theorem 4. (Deutsch) Let Y be a dense vector subspace of the topological vector space Z and let T 1 ,..., T n be continuous linear functionals on Z. Then for each f ∈ Z and each neighborhood U of f there is y ∈ Y such that y ∈ U and .
Theorem 5.Let Ω be an open subset of ℛp, x1, ..., xndistinct elements of Ω and L a subalgebra of Ck (Ω; ℛ) that satisfies the following conditions,
-
given x, y ∈ Ω with x ≠ y, there exists f ∈ L such that f (x) ≠ f (y);
-
given x ∈ Ω, there exists f ∈ L such that f (x) ≠ 0;
-
given x ∈ Ω and u ∈ ℛ p with u ≠ 0, there exists f ∈ L such that.
Then, for each eq, and each neighborhood U of f there existssuch thatfor i = 1, . . . , n.
Proof. It follows from Theorem 1 that L is a dense subalgebra of the topological vector space C k (Ω; ℛ). Let . Notice that
is a continuous linear functional for each i = 1, · · · , n. Setting and Y = L, the conclusion follows from Theorem 4.
ACKNOWLEDGEMENTS
The author acknowledges the referees for the valuable comments and suggestions which improved the presentation of the paper.
REFERENCES
-
1F. Deutsch, Simultaneous interpolation and approximation in linear topological spaces. SIAM J. Appl. Math., 14 (1966), 1180-1190.
-
2M. Moskowitz & F. Paliogiannis, ”Functions of Several Real Variables”. World Scientific, Singapore (2011).
-
3J. Mujica, Subálgebras densas de funciones diferenciables. Cubo Matemática Educacional, 3 (2001), 121-128.
-
4L. Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété. Comptes Rendus de l’Académie des Sciences de Paris, 228 (1949), 1549-1551.
-
5L. Nachbin, ”Elements of Approximation Theory”. Van Nostrand, Princeton, NJ (1967), reprinted by Krieger, Huntington, NY (1976).
Publication Dates
-
Publication in this collection
Sep-Dec 2018
History
-
Received
26 Dec 2017 -
Accepted
26 Apr 2018