SAKAI, AKIRA

Osaka City University Repository

Sakai, A.
Osaka J. Math.
19 (1982), 571-575
UNIFORM APPROXIMATION BY ENTIRE FUNCTIONS
OF SEVERAL COMPLEX VARIABLES
Dedicated to Professor Yukinari Tόki on his 70th birthday
AKIRA SAKAI
(Received October 11, 1980)
Introduction. Let G be a holomorphically convex open subset of Cn and
T a closed subset of G. We say that T is totally real, if it is the zero set of a non-
negative C2 function p which is strictly plurisubharmonic on T. It is known
that a real C1 submanifold M is totally real if and only if it has no complex tan-
gents (cf. [3]). The problem of uniform approximation on totally real sub-
manifols was studied to a great extent by many authors (cf. Wells [9], Hϋrmander
and Wermer [4], Nirenberg and Wells [5], Harvey and Wells [2J, [3] and Nune-
macher [6]). The result of [6] states that if M is a totally real submanifold
then there exists a holomorphically convex open neighborhood B such that
every continuous function on M is uniformly approximated on M by functions
holomorphic in B. In [8], the author extended this result to the case of totally
real sets with C°° defining functions. (A totally real set is not necessarily a sub-
manifold. The approximation theorem for totally real sets contains one for
totally real analytic subvarieties which was conjectured by Wells [9].)
In this paper, we give a sufficient condition for T and G under which every
continuous function on T is uniformly approximated on T by functions holo-
morphic in G. The theorem we prove contains the following result which is
a straight generalization to higher dimensions of Carleman's theorem [1].
Every continuous function on R"y canonically imbedded in Cny is uniformly
approximated on Rn by entire functions on n complex variables.
We shall make use of the ZΛmethod due to Hϋrmander and Wermer [4]
and the swelling method similar to one used in [8J.
1. Statements. Let S be a closed subset of an open set U of Cn. We
denote by H(S) (or H(S, U)) the algebra of uniform limits of restrictions of
functions holomorphic in a neighborhood of S (or in Uy resp.).
We use an abbreviation L[uy ξ] for the Levi form of a C°° function u:
By an exhaustion function σ of G we mean a positive C°° strictly plurisubharmonic
572 A. SAKAI
function which maps properly G into R. We define a form
A[<r ξ] = ±L[σ2;ξ]
Theorem. Z^ί G be a holomorphtcally convex open subset of Cn and σ be an
exhaustion function of G. If T is the zero set of a nonnegatίve C°° function p on
G satisfying
(1) L[p;ζ]ϊ>cA[σ ,ζ\, feC",
for some constant c>0, then H(T, G)=C(T).
When G is CΛ, this is a uniform approximation theorem by entire func-
tions. In this case, we can choose σ(z)=\z\2+l as an exhaustion function
of Cn and we have \ξ\2<A[σ\ ξ]<2\ξ |2, feC*. Therefore, we obtain
Corollary 1. If T is the zero set of a nonnegative C°° function p on Cn satisfy-
ing
(2) L[p ξ}>c\ξ\\
with some constant c>0, then H(T, Cn)=C(T).
If we write JR*=={*; j^ ==0, 7=!, •••,»}, then p(i)=ΣljV>l 2 ^s a defining
3
function of Rn satisfying (2). Thus we obtain the following corollary.
Corollary 2. H(R\ C") = C(Rn) .
The proof of Theorem is based on the following lemma essentially due to
[4]. (For the proof, see Proposition 1 of [7].)
Lemma 1. Let 8 be a nonnegative function defined in an open set V in Cn.
Suppose K is a compact subset of V satisfying the following condition : There exists
a constant η >0 such that for every sufficiently small £>0, we can find a holomor-
phically convex open set Vz satisfying
{z: dίst (z, K)<6} C F
ε
c
If F is a a C°° function on V satisfying
\dF(z)\<cδ(z)»+1,
then F \
 κ
 belongs to H(K).
2. Construction of an exhaustion iK
m
} of G. Let σ and p be func-
UNIFORM APPROXIMATION BY ENTIRE FUNCTIONS 573
tions satsifying the assumption of the theorem. For every positive number r,
the open set G
r
={z^G
r
: σ(z)<r] is relatively compact in G.
Let λ be a C°° function: Λ-*[0, 1] such that \(ή=l(t<Q) and \(t)=0
(t>2). For every positive number m, we set
Then we have
: ξ] = -
m »z
m
with α=sup{|λ'|+2|λ"|+l}, since \
m
(z)=Q for z<=G\G2m.
We set PQ= p and p
m
= p—m\
m
 for m>l. Since we may assume that
L[p'y ξ]>2aA[σ', ξ], ξ^Cny multiplying p by a constant if necessary, we have
L[Pm;ξ]>aA[σ;ξ]y ξ^Cn.
For each nonnegative integer m, we define the compact set K
m
 = \
It is easy to show that K
m
dK
m+1 and \]Km=G.
3. Approximation on K
m
. In this section, we fix a nonnegative integer
m. We shall prove the following lemma.
Lemma 2. Iff is a C°° function, then f \
 Ko<=H(K0, G). I f / i s a C°° function
which is holomorphic in an open neighborhood of G2my w>0, then f\Km^H(Km> G).
Proof. Since p
m
 is strictly plurisubharmonic in G and since K
m
=\p
m
<
n{<r<2w+3}, K
m
 is (?G-convex and therefore we have H(Km)=H(Km,G).
It suffices to prove thatf\Km^H(Km).
Let -ψ be a C°° function satsfying ι|r=l in an open neighborhood of G2m and
Λ/r=0 in G\G2m+ι. We consider the function
If ^^G2wί, we have L[δw; 1] = ^ .: f], fΦO. If z^T\G2m then pw-p=0
and dp=Q. Hence we have
L[S
m
't f]^ψL[p; |]+(l-ψ)L[P; ?]|||-2>0, ^Φθ.
Therefore we can find an open neighborhood Ω
m
 of K^
w
 so that δ
m
 is strictly
plurisubharmonic in Ω
m
. There exists a constant -η > 0 such that 8
m
(z) <
97 dist(*, K
m
) and σ(.s:)<2m+3+77 dist(^, G2m+3). If we set δ(^)—max {0, 8m(z)}
574 A. SAKAI
and V9={z&Ωm: δm(z)<£η} nG2w+3+ε, then, for sufficiently small £>0, Vt
is holomorphically convex and satisfies
{*: dist (*, K
m
)<ε} C F
ε
c {z: S
We can now find a C°° extension Foff\
τ
 on G which satisfies
for an open neighborhood V of .ίΓ
m
 and for some positive constant c. The
way of construction of .F is the same as one in Lemma 6 of [7]. We note that,
i f/ is holomorphic in an open neighborhood U of G2w, then F is holomorphic
in U. By Lemma 1, we have f\Km=F\Km^H(Km)9 which proves the lemma.
4. Global approximation. Let / be an arbitrary function in C°°(G) and
let £ be any positive number. We shall construct a sequence {/
w
} of functions
holomorphic in G and satisfying
l/m-/m-ιl<2 m 1B on K
m
and
!/*-/!< |]2-v£ on ΓnG2m+3
We define the function/
ε
=lim/
m
. A standard argument shows that/
ε
 is holo-
morphic in G and that |/
ε
—/1 <£ on T.
The construction of {/
m
} is as follows. By Lemma 2, we can find a function
/o holomorphic in G such that
I/0-/K2-* on K0=TΓ\G3
Suppose fjy j=l> •• ,m—l are already defined. Let i/r be a C°° function:
G-»[0, 1] satisfying ι|r=l in an open neighborhood U of G2w and i/r=0 in
G\G2m+ι. Set^
r
=ψ'/
wί
_1+(l—Λ/T)/. Then ^ is holomorphic in U. By Lemma
2, we can find a function /
m
 holomorphic in G so that
-
1
ε on K
m
.
Since g=f
m
-\ in C7 and K
m
c:U9 we have
-
1f on *„.
Since |^-/|=^|/
m
-1-/|<^2-v£on ΓnG2 m + 1and since g=f on Γ\G2M+1,
1_ V = 1
we nave
-
1+Σ2-vθ on
This completes the proof of the theorem.
UNIFORM APPROXIMATION BY ENTIRE FUNCTIONS 575
REMARK 1. The question arises whether the same conclusion as Theorem
can be obtained under the condition that p is C°° strictly plurisubharmonic in
G. (There is a simple example of T such that every defining function of T is not
strictly 'plurisubharmonic in1. G and such that H(T, G)ΦC(Γ).) When T is
compact this condition is sufficient. This follows at once from Theorem 2 of
[7] and the fact that T is then (?G-convex. We do not know whether it is true
even when T is not assumed to be compact.
REMARK 2. It is reasonable to conjecture that the theorem will be valid
even when a defining function p of T is of class C2. In fact, when T is a sub-
manifold, C2-diίfrentiability of p is sufficient to derive the approximation by
functions holomorphic in a neighborhood of T (c.f. Harvey-Wells [2] and Nune-
macher [6]). The C°° differentiability assumption in this paper was necessary
because of the /Λmethod we employed.
References
[1] T. Carleman: Sur un Theorem de Weierstrass, Ark. Math. Ast. Fys. 20 (1927), 1-5.
[2] F.R. Harvey, and R.O. Wells Jr.: Holomorphic paproximation and hyperfunction
theory on a C1 totally real submanifold of a complex manifold. Math. Ann. 197 (1972),
287-318.
[3] F.R. Harvey and R.O. Wells Jr.: Zero sets of non-negative strictly plurisubharmonic
functions, Math. Ann. 201 (1973), 165-170.
[4] L. Hormander and J. Wermer: Uniform approximation on compact sets in Cn,
Math. Scand. 23 (1968), 5-21.
[5] R. Nirenberg and R.O. Wells Jr.: Approximation theory on differentiable submani-
folds of a complex manifold, Trans. Amer. Math. Soc. 142 (1969), 15-35.
[6] J. Nunemacher: Approximation on totally real submanifolds, Math. Ann. 224 (1976),
129-141.
[7] A. Sakai: Uniform approximation in several complex variables, Osaka J. Math. 15
(1978), 589-611.
[8] A. Sakai: Uniform approximation on totally real sets, Math. Ann. 253 (1980), 139-
144.
[9] R.O. Wells Jr.: Real-analytic subvarieties and holomorphic approximation. Math.
Ann. 179(1969), 130-141.
Department of Mathematics
Himeji Institute of Technology
Shosha 2167, Himeji,
671-22, Japan



Uniform approximation by entire functions of several complex variables

Sakai, Akira

Institutional Repositories DataBase (IRDB)

Osaka University Knowledge Archive

http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-01903-5.pdf

Uniform approximation by entire functions of several complex variables

Abstract

Similar works

Full text

Available Versions

Osaka City University Repository

Institutional Repositories DataBase (IRDB)

Osaka University Knowledge Archive