Applications of Nachbin’s Theorem concerning Dense Subalgebras of Differentiable Functions

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 $k\begin{array}{c}\end{array}{\tau }_u^k$, 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 $\alpha =\left({\alpha }_1\mathrm{,...,}{\alpha }_p\right)\in {𝒩}_0^p$ of non-negative integers, let $\left|\alpha \right|:={\alpha }_1+\mathrm{...}+{\alpha }_p$ be the order of α, $\alpha !:=a_1!\mathrm{...}{\alpha }_p!$, and for $\left|\alpha \right|\le p$ let $D^{\alpha }:={\partial }^{\left|\alpha \right|}/\partial x_1^{{\alpha }_1}\mathrm{...}\partial x_p^{{\alpha }_p}$ represents the corresponding linear partial differential operator acting on C ^k (Ω; ℛ).

The topology ${\tau }_u^k$ is generated by the semi-norms σ ^k,Γ given by

${\sigma }_{k,\Gamma }\left(f\right)={\displaystyle \sum _{\left|\alpha \right|\le k}\frac{1}{\alpha !}}sup\left\{\left|\left(D^{\alpha }f\right)\left(x\right)\right|:x\in \Gamma \right\}\begin{array}{cc}& \end{array}\text{for all}f\in C^k\left(\Omega ;\mathcal{R}\right),$

where Γ runs over all compact subsets of Ω. By Proposition 3, p. 8 ⁵, C ^k (Ω; ℛ) is a topological vector space with respect to this topology.

In 1949 Nachbin ⁴ 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 ℛ^pand L be a subalgebra of C^k (Ω; ℛ). Then L is dense in C ^k (Ω; ℛ) if and only if the following conditions are satisfied:

1. given x, y ∈ Ω with x ≠ y, there exists f ∈ L such that f (x) ≠ f (y);

2. given x ∈ Ω, there exists f ∈ L such that f (x) ≠ 0;

3. given x ∈ Ω and u ∈ ℛ ^p with u ≠ 0 , there exists f ∈ L such that $\frac{\partial f}{\partial u}\left(x\right)\ne 0$.

The proof of this result can be found in ³ and ⁽⁴.

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 (² 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$h\left(w_j\right)=y_i,j=\mathrm{1,...,}m$.

Proof. Let L be a dense linear subspace of C ^k (Ω; ℛ) and $S=\left\{w_1\mathrm{,...,}{w}_m\right\}$ be a subset of Ω. Consider the following linear mapping

$\begin{array}{l}T:C^k\left(\Omega ;\mathcal{R}\right)\to {\mathcal{R}}^m\\ \begin{array}{ccc}& & \end{array}\begin{array}{c}\begin{array}{c}\end{array}\end{array}\begin{array}{c}\end{array}f\mapsto \left(f\left(w_1\right)\mathrm{,...,}f\left(w_m\right)\right).\end{array}$

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 ${\Phi }_i:{\mathcal{R}}^m\to R\mathrm{,0}\le {\Phi }_i\le 1$, such that ${\Phi }_i\left(w_i\right)=1$ and ${\Phi }_i\left(x\right)=0$, if x ∉U _i , in particular, ${\Phi }_i\left(w_j\right)=\mathrm{0,}j\ne i$. Let ${\varphi }_i={\Phi }_{i|\Omega }$ the restriction of the function Φ_i 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

$\left(c_1\mathrm{,...,}{c}_m\right)={\displaystyle \sum _{i=1}^m{c}_i{e}_i={\displaystyle \sum _{i=1}^m{c}_i\left({\varphi }_i\left(w_1\right)\mathrm{,...,}{\varphi }_i\left(w_m\right)\right)={\displaystyle \sum _{i=1}^m{c}_{i}T\left({\varphi }_i\right)=T\left({\displaystyle \sum _{i=1}^m{c}_i{\varphi }_i}\right),}}}$

where ${\displaystyle {\sum }_{i=1}^m{c}_i{\varphi }_i\in C^k\left(\Omega ;\mathcal{R}\right)}$. 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

${\mathcal{R}}^m=T\left(C^k\left(\Omega ;\mathcal{R}\right)\right)=T\left(\overline{L}\right)\subset \overline{T\left(L\right)}=T\left(L\right).$ (2.1)

Therefore, for any $\left(y_1\mathrm{,...,}{y}_m\right)\in {\mathcal{R}}^m$ there exists h ∈ L such that $T\left(h\right)=\left(y_1\mathrm{,...,}{y}_m\right)$, that is, $\left(h\left(w_1\right)\mathrm{,...,}h\left(w_m\right)\right)=\left(y_1\mathrm{,...,}{y}_m\right)$.

We give a proof of the following density result.

Theorem 2.Let V be an open subset of ℛ^p, L a dense subalgebra of C^k (V; ℛ), and v ₁ , . . . , v _n distinct points in V. Consider the open subset of ℛ ^p ,

$\Omega =V\backslash \left\{v_1\mathrm{,...,}{v}_n\right\}$

and the subalgebra

$M=\left\{f{|}_{\Omega }:f\in L,f\left(v_1\right)=\mathrm{...}=f\left(v_n\right)=0\right\}.$

Then, M is dense in C^k (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

$S=\left\{x,y,v_1\mathrm{,...,}{v}_n\right\}$

of V. By Lemma 1 there exists h ∈ L such that $h\left(x\right)=\mathrm{1,}h\left(y\right)=-1$ and $h\left(v_j\right)=0$ 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 $g\left(z\right)=1$ and $g\left(v_j\right)=0$ for j = 1, . . . , n. Hence, $g{|}_{\Omega }\in M$. If $\frac{\partial g}{\partial u}\left(z\right)\ne 0$ the Condition (c) of Theorem 1 is satisfied. Otherwise, notice that L is not a subset of

$B=\left\{f\in C^k\left(V;\mathcal{R}\right):\frac{\partial f}{\partial u}\left(z\right)=0\right\},$

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 $\frac{\partial \varphi }{\partial u}\left(z\right)\ne 0$. Then, ϕg ∈ L and $\varphi g\left(v_j\right)=0$ for j = 1, . . . , n, that is, $\varphi g{|}_{\Omega }\in M$. Moreover,

$\frac{\partial \varphi g}{\partial u}\left(z\right)=\frac{\partial \varphi }{\partial u}\left(z\right)g\left(z\right)+\varphi \left(z\right)\frac{\partial g}{\partial u}\left(z\right)=\frac{\partial \varphi }{\partial u}\left(z\right)g\left(z\right)=\frac{\partial \varphi }{\partial u}\left(z\right)\ne 0.$

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

$p\left(x\right)={\left[\psi \left(x\right)\right]}^l,\begin{array}{cc}& \end{array}x\in {\mathcal{R}}^p,$

where $\psi \in \left({\mathcal{R}}^p\right)*$, 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

$p\left(x\right)=p_0+{\displaystyle \sum _{j=1}^l{p}_j\left(x\right),\begin{array}{cc}& \end{array}}x\in {\mathcal{R}}^p$

where $p_0\in \mathcal{R},p_j\in {\mathcal{P}}^j\left({\mathcal{R}}^p,\mathcal{R}\right),j=\mathrm{1,...,}l,l\in \mathcal{N}$, 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 )^∗,

${\psi }_1\left(x\right){\psi }_2\left(x\right)=\frac{1}{4}\left[{\left({\psi }_1\left(x\right)+{\psi }_2\left(x\right)\right)}^2-{\left({\psi }_1\left(x\right)-{\psi }_2\left(x\right)\right)}^2\right]$

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, . . . , v_nbe distinct points in ℛ^p . Consider the open subset of ℛ ^p ,

$\Omega ={\mathcal{R}}^p\backslash \left\{v_1\mathrm{,...,}{v}_n\right\}$

and the subalgebra

$M=\left\{f{|}_{\Omega }:f\in \mathcal{P}\left({\mathcal{R}}^p;\mathcal{R}\right),f\left(v_1\right)=\mathrm{...}=f\left(v_n\right)=0\right\}.$

Then, M is dense in C^k (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 $\left({\mathcal{R}}^p\right)*=P^1\left({\mathcal{R}}^p;\mathcal{R}\right)\subset P\left({\mathcal{R}}^p;\mathcal{R}\right)$, the Condition (a) of Theorem 1 is satis- fied. By definition, P(ℛ ^p , ℛ) contains all the constant functions. Now, let $0\ne u=\left(u_1\mathrm{,...,}{u}_p\right)\in {\mathcal{R}}^p$. Then, there exists $0\ne u_j\in \mathcal{R},j\in \left\{\mathrm{1,...,}p\right\}$. Let ${\Pi }_j:{\mathcal{R}}^p\to \mathcal{R}$ defined by ${\Pi }_j\left(x\right)=x_j,x\in {\mathcal{R}}^p$. Since $\frac{\partial {\Pi }_j}{\partial x_j}\left(x\right)=1$ and $\frac{\partial {\Pi }_j}{\partial x_i}\left(x\right)=0$ for i ≠ j, it follows that

$\frac{\partial {\Pi }_j}{\partial u}\left(x\right)={\displaystyle \sum _{i=1}^p{u}_i}\frac{\partial {\Pi }_j}{\partial x_i}\left(x\right)=u_j\ne 0.$

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 ¹), 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 $T_i\left(y\right)=T_i\left(f\right),i=\mathrm{1,...,}n$ .

Theorem 5.Let Ω be an open subset of ℛ^p, x₁, ..., x_ndistinct elements of Ω and L a subalgebra of C^k (Ω; ℛ) that satisfies the following conditions,

1. given x, y ∈ Ω with x ≠ y, there exists f ∈ L such that f (x) ≠ f (y);

2. given x ∈ Ω, there exists f ∈ L such that f (x) ≠ 0;

3. given x ∈ Ω and u ∈ ℛ ^p with u ≠ 0, there exists f ∈ L such that$\frac{\partial f}{\partial u}\left(x\right)\ne 0$.

Then, for each eq$f\in C^k\left(\Omega ;\mathcal{R}\right)$, and each neighborhood U of f there exists$g\in L\cap U$such that$f\left(x_i\right)=g\left(x_i\right)$for 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 $S=\left\{x_1\mathrm{,...,}{x}_n\right\}\subset \Omega$. Notice that

$\begin{array}{l}{T}_i:C^k\left(\Omega ;\mathcal{R}\right)\to \mathcal{R}\\ \begin{array}{ccc}& & \end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}f\mapsto f\left(x_i\right)\end{array}$

is a continuous linear functional for each i = 1, · · · , n. Setting $Z=C^k\left(\Omega ;\mathcal{R}\right)$ 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

• ¹
  F. Deutsch, Simultaneous interpolation and approximation in linear topological spaces. SIAM J. Appl. Math., 14 (1966), 1180-1190.
• ²
  M. Moskowitz & F. Paliogiannis, ”Functions of Several Real Variables”. World Scientific, Singapore (2011).
• ³
  J. Mujica, Subálgebras densas de funciones diferenciables. Cubo Matemática Educacional, 3 (2001), 121-128.
• ⁴
  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.
• ⁵
  L. 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