doi: 10.5540/tema.2018.019.03.0465
Applications of Nachbin’s Theorem concerning
Dense Subalgebras of Differentiable Functions
M.S. KASHIMOTO
Received on December 26, 2017 / Accepted on April 26, 2018
ABSTRACT. In this paper, we give some applications of Nachbin’s Theorem [4] to approximation and interpolation in the the space of allktimes continuously differentiable real functions on any open subset of the Euclidean space.
Keywords: Nachbin’s theorem, approximation of differentiable functions, Stone-Weierstrass theorem, interpolation.
1 INTRODUCTION
LetΩbe an open subset ofRpand letkbe a nonnegative integer. We denote byCk(Ω;R)the
algebra of allktimes continuously differentiable real functions onΩand consider the compact
open topology of orderkτk
u,that is, the topology of uniform convergence for the functions and
all their partial derivatives up to the orderkon compact subsets ofΩ.
For a multi-indexα= (α1,· · ·,αp)∈N0pof non-negative integers, let|α|:=α1+· · ·+αpbe
the order of α,α! :=α1!· · ·αp!,and for |α| ≤p letDα :=∂|α|/∂xα11· · ·∂x
αp
p represents the
corresponding linear partial differential operator acting onCk(Ω;R).
The topologyτk
uis generated by the semi-normsσk,Γgiven by
σk,Γ(f) =
∑
|α|≤k 1
α!sup{|(D
αf)(x)|:x∈Γ} for all f∈Ck(Ω
;R),
whereΓruns over all compact subsets ofΩ.By Proposition 3, p. 8 [5],Ck(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 spaceCk(Ω;R).
*Corresponding author: Marcia Sayuri Kashimoto – E-mail: mskashim@gmail.com
Theorem 1.(Nachbin) LetΩbe an open subset ofRpand L be a subalgebra of Ck(Ω;R).Then
L is dense in Ck(Ω;R)if and only if the following conditions are satisfied:
(a) given x,y∈Ωwith x6=y,there exists f∈L such that f(x)6=f(y);
(b) given x∈Ω,there exists f ∈L such that f(x)6=0;
(c) given x∈Ωand u∈Rpwith u6=0,there exists f∈L such that ∂∂uf(x)6=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 spaceCk(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.LetΩbe an open subset ofRp,w1, . . . ,wmdistinct points inΩ,and y1, . . . ,ymdistinct
real numbers. If L is a dense vector subspace of Ck(Ω;R),then there exists a function h∈L such
that h(wj) =yj, j=1, . . . ,m.
Proof. LetLbe a dense linear subspace ofCk(Ω;R)andS={w1, . . . ,wm}be a subset ofΩ.
Consider the following linear mapping
T:Ck(Ω;R) → Rm
f 7→ (f(w1), . . . ,f(wm)).
Notice thatTis continuous.
For eachwi∈Sconsider an open neighborhoodUi⊂Ωofwisuch thatwj∈/Ui, for all j6=i, j∈
{1, ...,m}. It follows from the Urysohn’s Lemma for differentiable functions that there exists an
infinitely differentiable functionΦi:Rm→R,0≤Φi≤1, such thatΦi(wi) =1 andΦi(x) =0,
ifx∈/Ui, in particular,Φi(wj) =0, j6=i. Letφi=Φi|Ωthe restriction of the functionΦito the
subsetΩandei∈Rmthe vector whoseithcoordinate is equal to 1 and the others are equal to 0.
The linear mappingTis surjective since for any(c1, . . . ,cm)∈Rm,we have
(c1, . . . ,cm) = m
∑
i=1
ciei= m
∑
i=1
ci(φi(w1), . . . ,φi(wm)) = m
∑
i=1
ciT(φi) =T m
∑
i=1
ciφi !
,
where∑mi=1ciφi∈Ck(Ω;R).Moreover,T(L)is closed because it is a linear subspace ofRm.
Then by density ofLand continuity ofT, it follows that
Therefore, for any(y1, . . . ,ym)∈Rm there existsh∈Lsuch that T(h) = (y1, . . . ,ym),that is, (h(w1), . . . ,h(wm)) = (y1, . . . ,ym).
We give a proof of the following density result.
Theorem 2.Let V be an open subset ofRp,L a dense subalgebra of Ck(V;R),and v 1, . . . ,vn
distinct points in V.Consider the open subset ofRp,
Ω=V\ {v1, . . . ,vn}
and the subalgebra
M={f|Ω: f∈L,f(v1) =. . .=f(vn) =0}.
Then, M is dense in Ck(Ω;R).
Proof. ClearlyMis a subalgebra ofCk(Ω;R).Letx,ybe any distinct points inΩ.Consider the following subset
S={x,y,v1, . . . ,vn}
ofV.By Lemma 1 there existsh∈Lsuch thath(x) =1,h(y) =−1 andh(vj) =0 forj=1, . . . ,n.
Then,h|Ω∈Mand satisfies Conditions (a) and (b) of Theorem 1.
Now letz∈Ωandu∈Rp,u6=0. It follows from Lemma 1 that there existsg∈L such that
g(z) =1 and g(vj) =0 for j=1, . . . ,n. Hence, g|Ω ∈M. If ∂∂gu(z)6=0 the Condition (c) of
Theorem 1 is satisfied. Otherwise, notice thatLis not a subset of
B=
f∈Ck(V;R):∂f ∂u(z) =0
,
since Lis a dense subalgebra ofCk(V;R)andB is a proper closed subalgebra ofCk(V;R).
Thus, there existsφ∈Lsuch that∂ φ∂u(z)6=0.Then,φg∈Landφg(vj) =0 forj=1, . . . ,n, that
is,φg|Ω∈M.Moreover,
∂ φg
∂u (z) =
∂ φ
∂u(z)g(z) +φ(z)
∂g
∂u(z) =
∂ φ
∂u(z)g(z) =
∂ φ
∂u(z)6=0.
Thus, by Theorem 1,Mis dense inCk(Ω;R).
For each positive integerl,Pl(Rp,R)denotes the linear subspace ofCk(Rp;R)generated by
the set of all functions of the form
p(x) = [ψ(x)]l, x∈Rp,
where ψ ∈ (Rp)∗, the dual space of Rp. The elements of Pl(Rp,R) are called the l
-homogeneous continuous polynomials of finite type from Rp into R. The subspace of
p(x) =p0+ l
∑
j=1
pj(x), x∈Rp
wherep0∈R, pj∈Pj(Rp,R), j=1, . . . ,l,l∈N ,is denoted byP(Rp,R). Its elements
are called real continuous polynomials of finite type. The polarization formula shows that
P(Rp,R)is a subalgebra ofCk(Rp;R). Indeed, givenψ1andψ2in(Rp)∗,
ψ1(x)ψ2(x) = 1
4[(ψ1(x) +ψ2(x)) 2−(ψ
1(x)−ψ2(x))2]
shows thatψ1ψ2∈P2(Rp,R),sinceψ1+ψ2andψ1−ψ2belong to(Rp)∗.
Corollary 3.Let v1, . . . ,vnbe distinct points inRp.Consider the open subset ofRp,
Ω=Rp\ {v1, . . . ,vn}
and the subalgebra
M={f|Ω:f ∈P(Rp;R),f(v1) =. . .=f(vn) =0}.
Then, M is dense in Ck(Ω;R).
Proof. First of all, we verify that the subalgebra P(Rp;R) is dense inCk(Rp;R). Given
x,y∈Rpwithx6=y,it follows from Hahn-Banach Theorem that there existsψ∈(Rp)∗such that
ψ(x)6=ψ(y).Since(Rp)∗=P1(Rp;R)⊂P(Rp;R),the Condition (a) of Theorem 1 is
satis-fied. By definition,P(Rp;R)contains all the constant functions. Now, let06=u= (u
1, ...,up)∈
Rp.Then, there exists 06=u
j ∈R, j∈ {1, ...,p}. LetΠj:Rp→Rdefined by Πj(x) =xj,
x∈Rp.Since∂Πj
∂xj(x) =1 and
∂Πj
∂xi (x) =0 fori6= j,it follows that
∂Πj
∂u (x) =
p
∑
i=1
ui ∂Πj
∂xi
(x) =uj6=0.
Therefore, by Theorem 1, P(Rp;R) is dense inCk(Rp;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 inCk(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 T1, ...,Tnbe 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 Ti(y) =Ti(f),i=1, ...,n.
Theorem 5.LetΩbe an open subset ofRp,x
1, ...,xndistinct elements ofΩand L a subalgebra
(a) given x,y∈Ωwith x6=y,there exists f ∈L such that f(x)6=f(y);
(b) given x∈Ω,there exists f∈L such that f(x)6=0;
(c) given x∈Ωand u∈Rpwith u6=0,there exists f ∈L such that ∂f
∂u(x)6=0.
Then, for each f ∈Ck(Ω;R),and each neighborhood U of f there exists g∈L∩U such that
f(xi) =g(xi)for i=1, . . . ,n.
Proof. It follows from Theorem 1 thatLis a dense subalgebra of the topological vector space
Ck(Ω;R).LetS={x1, ...,xn} ⊂Ω. Notice that
Ti:Ck(Ω;R) → R
f 7→ f(xi)
is a continuous linear functional for each i=1,· · ·,n. Setting Z=Ck(Ω;R)andY =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.
RESUMO. Em 1949, Leopoldo Nachbin estabeleceu uma vers˜ao do Teorema de Stone-Weierstrass para func¸˜oes diferenci´aveis de classeCkem abertos do espac¸o euclidiano. Neste
trabalho, apresentamos algumas aplicac˜oes desse teorema relacionadas com aproximac¸˜ao e interpolac¸˜ao no espac¸o das func¸˜oes de classeCkmunido da topologia compacto-aberta.
Palavras-chave: Teorema de Nachbin, aproximac¸˜ao de func¸˜oes diferenci´aveis, Teorema de Stone-Weierstrass, interpolac¸˜ao.
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,Sub´algebras densas de funciones diferenciables. Cubo Matem´atica Educacional,3(2001), 121–128.
[4] L. Nachbin, Sur les alg`ebres denses de fonctions diff´erentiables sur une vari´et´e.Comptes Rendus de l’Acad´emie des Sciences de Paris,228(1949), 1549–1551.
