Sets of Approximation and Interpolation in $\cx$ for manifold-valued maps

0.28 MB·English

by  Debraj Chakrabarti

7 SETS OF APPROXIMATION AND INTERPOLATION IN C FOR
0
0 MANIFOLD-VALUED MAPS
2

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DEBRAJCHAKRABARTI
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] 1. Introduction
V
C 1.1. Notation. This article is devoted to the study of approximation and interpolation of maps
. fromacompactK ⊂CintoacomplexmanifoldM,andinparticulartogivingexamplesofsetsK
h
t for which certaintypes of approximationand interpolationare possible. For brevity, we introduce
a
m three properties A1, A2 and A3 that compact subsets of the plane may possess. We first define
property A .
[ 2
ForacompactK ⊂C, letK◦ be the interiorofK. We willletA(K,M)denotethe continuous
1
v maps from K into the complex manifold M which are holomorphic on K◦. Let O(K,M) ⊂
5
A(K,M) denote the subspace of those maps f which extend to a holomorphic map from some
6
6 neighborhood Uf of K to M. We endow M with an arbitrary metric, which makes A(K,M)
1 and O(K,M) into metric spaces with the uniform metric on maps. We will say that a compact
0
K ⊂C has the property A , if, for every complex manifold M, every finite set P ⊂M, and every
7 2
0 ǫ>0, we can approximate any f ∈A(K,M) by a map f ∈O(K,M) such that dist (f,f )<ǫ,
ǫ M ǫ
/
h and for each p ∈ P, we have fǫ(p) = f(p). That is, fǫ is an uniform approximation to f which
t
a interpolates the values of f on P.
m We now define the property A , which will be the special case of A with M = C. More
1 2
: precisely, we will say that K ⊂ C has the property A , if, for any finite P ⊂ K, any function
v 2
i in A(K) := A(K,C) can be uniformly approximated by functions in O(K) := O(K,C) which
X
interpolate the values of f on P. Obviously for a set K, A ⇒A .
r 2 1
a
It is easy to see that a set K has property A iff O(K) is dense in A(K). Note the trivial
1
fact that there is a constant C > 0 such that for g ∈ C(K), if L (g) is the Lagrange polynomial
P
which interpolates the values of g on P, we have kL (g)k < Ckgk . If f˜∈O(K) be such that
P K K
f −f˜ < ǫ , then |f −f |<ǫ, and f(p)=f (p) for p∈P, where f =f˜+L (f −f˜).
C+1 ǫ ǫ ǫ P
(cid:12) The(cid:12)compact sets K ⊂C such that O(K) is dense in A(K) can be characterized by Vituˇskin’s
(cid:12) (cid:12)
t(cid:12)heorem(cid:12) ( [15]). A sufficient condition is that C\K has finitely many connected components.
Also,suchapproximationcanbe localized,inthesensethatO(K)isdenseinA(K)iffeverypoint
z ∈K has a neighborhood U in C such that O(K∩U ) is dense in A(K∩U ).
z z z
WenowdefinepropertyA . Letφ:M→Cbeaholomorphicsubmersionsuchthatφ(K)⊃M.
3
Let A (K,M) (resp. O (K,M) ) be the subspace of A(K,M) (resp. O(K,M)) consisting of
φ φ
sections of φ over K, i.e. maps s : K → M such that φ◦s = I . We will say that a compact
K
1
2 DEBRAJCHAKRABARTI
K ⊂ C has property A if for every complex manifold M and every holomorphic submersion φ
3
such that φ(M)⊃K, and every finite set P ⊂K, every section σ ∈A (K,M) can be uniformly
φ
approximated by sections in O (K,M) which interpolate the values of σ on P.
φ
We also note the following elementary fact:
Lemma 1. For a compact K ⊂C, A ⇒A .
3 2
Proof. Assume that K satisfies A , and let P and M have the same meaning as above, and let
3
f ∈A(K,M). ConsiderthecomplexmanifoldN =M×Candletφandπbetheprojectionsonto
C and M respectively. If F : K → N is defined by F(z) = (f(z),z), then clearly F ∈A (K,N)
φ
and since K has the property A , we can approximate F by maps G ∈ O (K,N) such that
3 φ
G(p)=F(p)=(f(p),p) for each p∈P. Then g =π◦G is in O(K,M) and an approximation to
f with g(p)=f(p) for each p∈P, which shows that K has property A . (cid:3)
2
(We may remark here that while holomorphic submersions φ : M → C always exist if M is
Stein ([7]), in general such M may be highly nontrivial, see e.g. [3].)
It is natural to ask the question: which are the compact sets in the plane which satisfy property
A or property A ? Given that approximation is local in nature, one can conjecture that using
2 3
patching arguments in the manifold M we may be able to show that every set with property A
1
has property A , or even A . We do not know if this program can be carried out. In this article,
2 3
we confine ourselves to to the much more modest goal of giving examples of sets K for which
properties A and A hold (see Theorems 1,2 and 3 below.)
2 3
1.2. Known results. Some resultshavebeen obtainedrecently regardingthe approximationand
interpolation of manifold-valued maps.
The following was proved in [2]. Define a Jordan domain to be an open set Ω ⋐ C, such that
the boundary ∂Ω has finitely many connected components, each of which is homeomorphic to a
circle. (We explicitly allow Ω to be multiply connected.)
Proposition 1. Let K =Ω, where Ω⋐C is a Jordan domain. For any complex manifold M, the
subspace O(K,M) is dense in A(K,M).
Thisispurelyastatementaboutapproximation,nointerpolationbeinginvolved. Thefollowing
result is contained in what was proved by Drinovec-Drnovˇsekand Forstneriˇc in [6], Theorem 5.1:
Proposition 2. Let Ω⋐C have C2 boundary, M be a complex manifold, and let f :Ω→M be a
map of class Cr (r ≥2) which is holomorphic in Ω. Given finitely many points z ,...,z ∈Ω, and
1 l
an integer k ∈N, there is a sequence of maps f ∈O(K,M) such that f agrees with f to order
ν ν
k at z for j =1,...,l and ν ∈N, and the sequence f converges to f in Cr(Ω) as ν →∞.
j ν
ThiscaninfactbeprovedwhenΩ⋐S,whereS isanon-compactRiemannsurface,andMisa
complexspace. Observethatthereisastrongerassumptiononthe smoothnessofthe mapthanin
property A , and also a stronger conclusion regardingapproximationand interpolation. However,
2
thesetofpointsofinterpolationisrestrictedtotheinteriorofΩ. Theauthorssubsequentlyproved
the following:
SETS OF APPROXIMATION AND INTERPOLATION 3
Proposition 3. If Ω⋐C is a domain with C2 boundary, M a complex manifold, and φ:M→C
a holomorphic submersion such that φ(M)⊃Ω, then O Ω,M is dense in A Ω,M .
φ φ
This is a (very) special case of [5], Theorem 5.1 rega(cid:0)rding(cid:1)the approximatio(cid:0)n of s(cid:1)ections of
submersions over strongly pseudoconvex sets in Stein manifolds, which may be thought of as a
far-reaching generalization of the Henkin-Ram´ırez-Kerzman approximation theorem for functions
on such domains ([9], Theorem 2.9.2. )
1.3. Sets with property A or A . We recallsome definitions from elementary point-set topol-
2 3
ogy. Let X be a topological space. For an integer n≥0, we say that dim(X)≤n if the following
holds : given any open cover A of X, there is a refinement B of A such that any point of X is
contained in at most n+1 of the sets in the cover B. If dim(X) ≤ n holds, but dim(X) ≤ n−1
does not, we say that the dimension dim(X) of X is n. The dimension is clearly a topological
invariantofX. Also,if dim(X)=n,then eachconnectedcomponentofX (ormoregenerally,any
closed subset) has dimension less than or equal to n.
Wesaythatatopologicalspaceislocally contractibleifthereisabasisofthetopologyconsisting
of contractible open sets. We can now state the following:
Theorem 1. Let K ⊂ C be a connected compact set, such that dim(K)=1, and suppose that K
is locally contractible. Then K has property A .
3
Examples of sets that satisfy the hypotheses are
(1) arcsinC,whereByanarcαinatopologicalspaceX wemeanacontinuousinjectivemap
α : [0,1] → X from the unit interval. The injectivity avoids pathologies like space filling
curves. By standard abuse of language, we will refer to the image α([0,1]) as the arc α.
(2) planar realizations of connected graphs i.e. finite simplicial complexes having only 0-
simplices (“vertices”) and 1-simplices (“edges”.) We can think of these as unions of arcs
in the plane, any two of which meet at at most one endpoint of each.
Let K ⊂C, then dim(K)≤1 iff K◦ =∅ ([1], p.133,Theorem 20.) However, it is possible for a set
K with K◦ = ∅, to be not locally contractible, and yet to satisfy A (recall that A is equivalent
1 1
to O(K) being dense in A(K).) An example is the “bouquet of circles” K = ∞ C , where C
n=1 n n
is the circle in the plane with center at 1 +i0, and radius 1. It can be shown that this K has
n n S
property A , yet Theorem 1 does not apply to it. (We don’t know if K has property A or A .)
1 3 2
For completeness, we record the following easy fact:
Lemma 2. Let K be a compact subset of C with dim(K)=0. Then K has property A .
3
We now go on to give examples of two-dimensional sets which have properties A or A . Let
2 3
k ≥ 0 be an integer, ∞ or ω. We introduce a class of sets in the plane denoted by C . We will
k
let C denote the class of compact sets K ⊂ C such that there is an integer N and domains Ω ,
k j
where j =1,...,N with the following properties:
• For k 6= 0, each Ω is a domain with Ck boundary, and for k = 0, each Ω is a Jordan
j j
domain.
4 DEBRAJCHAKRABARTI
• The Ω ’s are pairwise disjoint : Ω ∩Ω =∅ if i6=j.
j i j
• K =∪N Ω . We will refer to each Ω as a summand of K.
i=1 j j
• whenever i6=j, the set
P :=Ω ∩Ω =∂Ω ∩∂Ω
ij i j i j
is finite.
• If k 6=0, at each point of P , the boundaries ∂Ω and ∂Ω are tangent to each other.
ij i j
We can now provide a supply of sets which have property A :
2
Theorem 2. Each compact set K of class C has property A .
0 2
This can be thought of as a generalization of Proposition 1. The second result gives examples
of sets with property A :
3
Theorem 3. Each compact set K of class C has property A .
2 3
This again can be thought of as a generalizationof Proposition 2.
1.4. In §2 below, we prove a general result (Theorem 4 ) regarding approximation of sections of
submersionsoverasetK whichadmitsadecompositionK =K ∪K intoa“goodpair”(K ,K )
1 2 1 2
of compact sets. This is a direct generalization of the results in [2], §4.1 for maps into manifolds.
Proofsofallnew statements aregivenindetail,althoughthey aresimilartothe proofsinthe case
of maps. This is the tool for patching maps which is used to prove Theorems 1 and 3.
In §3, we give proofs of Lemma 2, Theorem 1, and Theorem 3. For the last two, the crucial
ingredientis three successiveapplicationsofTheorem4 to producea sectionwhichis holomorphic
in a neighborhood of the given compact K. For Theorem 3 we also require a result from [2]
(Theorem 5). In §4 we deduce Theorem 2 from Theorem 3. This requires a result from [11]
regarding conformal mappings continuous to the boundary.
Itshouldbe notedthatthe resultsprovedherehaveanalogsfortheapproximationandinterpo-
lationofAk maps,i.e., mapswhichareCk andholomorphicintheinterior. The proofsareexactly
the same. For notational simplicity we stick to the case of maps which are only continuous to the
boundary. Also,thisarticleislargelyselfcontained,withtheexceptionoftheproofsofTheorem5
and Proposition 5.
2. Approximation on Good Pairs
2.1. Some definitions. We will call a set K ⊂ C nicely contractible if there is a homotopy
c:[0,1]×K →K with the following properties:
(1) for each t, the map z 7→c(t,z) is in A(K,C),
(2) z 7→c(1,z) is the identity map on K.
(3) there is a z ∈K such that c(0,z)≡z .
0 0
(4) for t6=0, z 7→c(t,z) maps K◦ into K◦.
SETS OF APPROXIMATION AND INTERPOLATION 5
Of course, convex sets are nicely contractible, as are strongly star-shaped sets (these are sets K
such that the maps c(t,z) may be taken as dilations with stretch factor t and center z ∈ K.)
0
Another important class of nicely contractible sets are arcs. The property of nicely contractible
setswhichisusedintheproofofLemma3(andthereforeTheorem4isthefollowing: ifK isnicely
contractible and M is a connected complex manifold, then A(K,M) is in fact contractible. We
can take the contraction to be the map φ from A(K,M) to itself given by φ (f)(z)=f(c(t,z)).
t t
LetK andK becompactsubsetsofC. Wewillsaythat(K ,K )isagood pairifthefollowing
1 2 1 2
hold:
(1) K \K ∩K \K =∅.
1 2 2 1
(2) K := K ∩K has finitely many connected components, each of which is nicely con-
1,2 1 2
tractible.
Let M be a complex manifold of complex-dimension n, and φ : M → C be a submersion. We
will say that an open set U ⊂ M is φ-adapted, if U is biholomorphic to an open set in Cn, and
there are holomorphic coordinates (z ,··· ,z ) on U such that with respect to these coordinates
1 n
(and the standard coordinate on C), the map φ takes the form (z ,··· ,z ) 7→ z . (This is of
1 n n
course the same as saying z =φ| .)
n U
2.2. The main result. We introduce the following notation and definitions:
(1) Let K =K ∪K , where (K ,K ) is a good pair.
1 2 1 2
(2) Let M be a complex manifold and φ : M → C be a holomorphic submersion such that
φ(M)⊃K.
(3) Let B ⊂ C be compact and such that B∩K = ∅, and each function g ∈ A(K ) can be
1 2
approximated uniformly on K by functions in A(K ∪B) ;i.e., if g ∈ A(K ) and ǫ > 0,
2 2 2
then there is a g ∈A(K ∪B) such that |g−g |<η on K .
η 2 η 2
(4) Let P be a finite subset of K.
We now state the following result regarding the approximation of sections of φ over K:
Theorem 4. With K ,K ,M,φ, B, and P as above, let s ∈ A (K,M) be a section of φ such
1 2 φ
that each s(K ) has a φ-adapted neighborhood in M for j = 1,2. Then, given η > 0, there is an
j
s ∈ A (K,M) such that dist(s,s ) < η on K, s extends as a holomorphic section of φ to a
η φ η η
neighborhood B of K ∩B, and for each p∈P, we have s (p)=s(p). Moreover, s (K ∪B ) has
η 2 η η 2 η
a φ-adapted neighborhood in M.
Theproofof4willbereducedtothesolutionofanon-linearpatchingprobleminEuclideanspace
bytheuseofcoordinateinneighborhoodsofs(K ). Wenowdescribetheingredientsrequiredinthe
j
proof. Themostimportantisthe followingversion(resemblingthatA.Douadyin[4],pp.47-48)of
H. Cartan’s lemma on holomorphic matrices.
Lemma 3. Let G be a complex connected Lie Group, and let (K ,K ) be a good pair, and let
1 2
g ∈A(K ,G). Then, for j =1,2 there are g ∈A(K ,G) such that g =g ·g on K .
1,2 j j 2 1 1,2
6 DEBRAJCHAKRABARTI
For a proof, see [2], Lemma 4.4, where it is assumed that G=GL (C), and eachcomponent of
n
the intersection is star shaped, but the proof is valid in general.
If P ⊂ C is a finite set, we will let AP(K,Cn) denote the closed subspace of the Banach space
A(K,Cn) consisting of those functions which vanish at each p ∈ K ∩P. We will require the
following version of the solution of the additive Cousin problem continuous to the boundary (the
case with P =∅ is in fact used in the proof of Lemma 3 ):
Lemma 4. Let K ,K be a compact subsets of the plane such that K \K ∩K \K =∅, and let
1 2 1 2 2 1
P be a finite subset of the plane. There exist bounded linear maps T :AP(K ,C)→AP(K ,C)
j 1,2 j
such that for any function f in AP(K ,C) we have on K ,
1,2 1,2
(1) T f +T f =f,
1 2
Proof. We reduce the problem to a ∂ equation in the standard way. Let χ be a smooth cutoff
which is 1 near K \K and 0 near K \K . Let λ:=f.∂χ, so that λ∈A(K ,C). Let
1 2 2 1 ∂z 1,2
1 λ(ζ)
Λ (z)= dζ∧dζ
f
2πi ζ−z
ZK1,2
Let q be the Lagrange interpolation polynomial with the property that q (p)=Λ (p) for p∈P.
f f f
Observe that both Λ and q are linear in f and continuous in the sup norm when restricted to
f f
compact sets.
We can now define (assuming (1−χ).f =0 where χ=1 even if f is not defined):
(T f)(z)=(1−χ(z)).f(z)+Λ (z)−q (z)
1 f f
and (assuming χ.f =0 where χ=0 even if f is not defined):
(T f)(z)=χ(z).f(z)−Λ (z)+q (z),
2 f f
Since T f is clearly holomorphic (resp. continuous) where f is, the result follows. (cid:3)
j
We will use the following standard result regarding Banach spaces, which can be proved by
iteration (see [10] pp. 397-98):
Lemma 5. In a metric space X, let B (p,r) denote the open ball in X of radius r centered at p.
X
Let E and F be Banach Spaces and let Φ:B (p,r)→F be a C1 map. Suppose there is a constant
E
C >0 such that:
• for each h ∈ B (p,r), the linear operator Φ′(h) : E → F is surjective and the equation
E
Φ′(h)u=g can be solved for u in E for all g in F with the estimate kuk ≤Ckgk .
E F
• for any h and h in B (p,r) we have kΦ′(h )−Φ′(h )k≤ 1 .
1 2 E 1 2 2C
Then,
r
Φ(B (p,r))⊃B Φ(p), .
E F
2C
(cid:16) (cid:17)
We willusethe threelemmas aboveto giveaproofofthe followingresult(the maincomponent
ofwhichis due to Rosay,[12], andcomments in[13]) regardingthe solutionofa non-linearCousin
problem (see Lemma 4.5 of [2].) It is simply a translation of Theorem 4 to coordinates.
SETS OF APPROXIMATION AND INTERPOLATION 7
Lemma6. Letω beanopensubsetofCn andletF:ω →Cn beaholomorphic immersion. Assume
that F preserves the last coordinate, i.e., F is of the form F(z ,··· ,z ) = (F(z ,··· ,z ),z ) for
1 n 1 n n
some map F :ω →Cn−1.
Let(K ,K )denoteagood pairof compact subsetsofC, let P beafinitesubsetof K =K ∪K ,
1 2 1 2
and suppose for each p∈K \K we are given a point q(p)∈Cn−1. Let u ∈A(K ,Cn) be such
2 1 1 1
that
• u (K )⊂ω, and
1 1,2
• u is of the form u (z)=(t (z),z) where t :K →Cn−1
1 1 1 1 1
Given any ǫ>0, there exists δ >0 such that if u ∈A(K ,Cn) be such that
2 2
• ku −F◦u k<δ on K ∩K ,
2 1 2 2
• for p∈P ∩(K ), we have u (p)=F(u (p)),
1,2 2 1
• for p∈K \K we have u (p)=(q(p),p)∈Cn, and
2 1 2
• u (z)=(t (z),z), where t :K →Cn−1
2 2 2 2
then for j =1,2 there exist v ∈AP(K ,Cn) such that
j j
• kv k<ǫ,
j
• u +v =F(u +v ), and
2 2 1 1
• the last coordinate function of each of v and v is 0.
1 2
We now give a proof of Lemma 6.
Proof. In order to apply Lemma 5 we choose the Banach spaces E, F and the map Φ as follows:
• For j = 1,2, let B be the closed subspace of the Banach space AP(K ,Cn) consisting of
j j
thosemapswhoselastcoordinatefunctionis0,i.e.,B =AP(K ,Cn)⊕0. WenowletE the
j j
BanachspaceB ⊕B ,whichweendowwiththenormk·k :=max k·k ,k·k .
1 2 E A(K1,Cn) A(K2,Cn)
• Let F be the Banach space AP(K ,Cn−1). (cid:16) (cid:17)
1,2
• Let π : Cn → Cn−1 denote the projection on the first n − 1 coordinates, and let the
open subset U of E be given by {(w ,w ) : (u + w )(K ) ⊂ ω}. (Observe that
1 2 1 1 1,2
{0}× B(K ,Cn) ⊂ U.) Let P = (P ,...,P ) be an n-tuple of polynomials such that
2 1 n
P (z) ≡ z, for p ∈ K ∩P, we have P(p) = F(u (p)), and for p ∈ K \K we have
n 1,2 1 2 1
(P (p),...,P (p))=q(p).
1 n−1
Let the map Φ:U →F be given by
Φ(w ,w ):=π◦ (P +w )| −F◦((u +w )| ) .
1 2 2 K1,2 1 1 K1,2
(Observe that, since F preserv(cid:2)es the last coordinate, the last coord(cid:3)inate function of (P +
w )| −F◦((u +w )| ) is actually 0. So the precomposition with π simply drops a
2 K1,2 1 1 K1,2
coordinate which is identically 0.)
A computation shows that Φ′(w ,w ) is the bounded linear map from E to F given by
1 2
(v ,v )7→π◦ v | −F′((u +w )| )(v | ) .
1 2 2 K1,2 1 1 K1,2 1 K1,2
(cid:2) (cid:3)
8 DEBRAJCHAKRABARTI
Observethat w plays no role whatsoeverin this expression,and therefore Φ′(w ,w )∈BL(E,F)
2 1 2
is in fact a smooth function of w alone, and we will henceforth denote it by Φ′(w ,∗).
1 1
Let G⊂GL (C) be the complex Lie subgroup of matrices of the form
n
A b
,
0 1 !
where A ∈ GL (C), b is an 1×(n−1) column vector, and 0 is the zero row vector of size
n−1
(n−1)×1. It is easy to see that G is connected.
We construct a right inverse to Φ′(u ,∗) . Let γ = F′ ◦ u | . Since F preserves the last
1 1 K1,2
coordinate, γ ∈ A(K ,G), and thanks to Lemma 3 above, we may write γ = γ ·γ , where
1,2 (cid:0) (cid:1) 2 1
γ ∈ A(K ,G). (We henceforth suppress the restriction signs.) Let j : F → A(K ,Cn) be the
j j 1,2
continuous inclusion induced by the map Cn−1 →Cn given by (z ,··· ,z )7→(z ,··· ,z ,0).
1 n−1 1 n−1
For g ∈F, let
S(g)=(−γ−1T (γ−1j(g)),γ T (γ−1j(g))),
1 1 2 2 2 2
where the T are as in equation 1 above (and extended to Cn componentwise). Observe that
j
−γ−1T (γ−1j(g)) and γ T (γ−1j(g)) vanish at each p ∈ P and they belong to B and B respec-
1 1 2 2 2 2 1 2
tively. It is easy to see that S : F → E is a bounded linear operator, and a computation shows
that Φ′(u ,∗)◦S =I . Choose θ >0 so small so that if w ∈B (u ,θ),
1 F 1 F 1
(1) kΦ′(w ,∗)−Φ′(u ,∗)k < 1 (possible by continuity), and
1 1 op 8kSk
(2) the equation Φ′(w ,∗)u = g can be solved with the estimate kuk ≤ 2kSkkgk, (possible
1
from the fact that small perturbations of a surjective linear operator is still surjective.)
Consequently, if ǫ < θ and u˜ ∈ B(K ,Cn), for the ball B ((0,u˜ ),ǫ) the hypothesis of Lemma 5
2 2 E 2
are verified with C =2kSk. We have therefore,
ǫ
Φ(B ((0,u˜ ),ǫ)) ⊃ B Φ(0,u˜ ),
E 2 F 2
2C
(cid:16) (cid:17) ǫ
= B P +u˜ −F(u ), .
F 2 1
2C
(cid:16) (cid:17)
So, if kP +u˜ −F(u )k < ǫ , we have 0 ∈ Φ(B ((0,u˜ ),ǫ)). This is exactly the conclusion
2 1 4C E 2
required, since any u such that u (p)=P(p) for each p∈K ∩P can be written as u =P +u˜
2 2 2 2 2
for some u˜ ∈B(K ,Cn). (cid:3)
2 2
We will now prove Theorem 4.
Proof. We omit the restriction signs on maps for notational clarity. For j = 1,2 let V be a φ-
j
adapted neighborhood of s(K ), and let F : V → Cn be a coordinate system such that F (z) =
j j j j
(F (z),φ(z)). Let F=F ◦F−1 be the associated transition function.Then F is a biholomorphism
j 2 1
fromtheopensetω =F (V ∩V )ontotheopensetF (V ∩V ),andFpreservesthelastcoordinate,
1 1 2 2 2 1
i.e., F is of the form F(z ,··· ,z )=(F(z ,··· ,z ),z ) for some map F :ω →Cn−1. Any section
1 n 1 n n
ofφ overK is representedinthe coordinatesystemF by a mapofthe formt :K →Cn, where
j j j j
t (z)=(t˜(z),z), with t˜ a map from K to Cn−1. Also, for j =1,2, given maps t :K →Cn of
j j j j j j
SETS OF APPROXIMATION AND INTERPOLATION 9
the form t = (t˜,z), they glue together to form a section over K ∪K (i.e. there is a section λ
j j 1 2
of φ over K ∪K such that t =F ◦λ) iff t =F◦t .
1 2 j j 2 1
Since B ∩K = ∅ by hypothesis, the pair of compact sets (K ,K ∪B) is good. Let ǫ > 0,
1 1 2
and let u = F ◦s. Observe that u (K ) ⊂ ω, and u is of the form u (z) = (t (z),z), where
1 1 1 1,2 1 1 1
t :K →Cn−1. Then Lemma 6 gives a δ >0 correspondingto u , the good pair (K ,K ∪B), ω
1 1 1 1 2
and F.
Let w = F ◦(s| ), then w ∈ A(K ,Cn), and is of the form w (z) = (w˜ (z),z). Thanks
2 2 K2 2 2 2 2
to the hypothesis regarding uniform approximation of functions in A(K ,C) by functions in
2
A(K ∪B,C),wecanfindau ∈A(K ∪B,Cn)ofthesameformasw suchthatku −F(u )k<
2 2 2 2 2 1
δ on K (Since F(u ) = w on K .) Further we may assume that for p ∈ P ∩K , we have
1,2 1 2 1,2 2
u (p)=w (p) and that the last coordinate function of u is z.
2 2 2
Then, by Proposition 6, there is a v ∈ AP(K ,Cn) and a v ∈ AP(K ∪B,Cn), such that
1 1 2 2
kv k<ǫ and u +v =F(u +v ), and the last coordinate functions of the v are 0’s. Hence the
j 2 2 1 1 j
maps u +v and u +v glue together to form a section of φ (which we call s˜ given by
1 1 2 2 ǫ
F−1(u +v ) on K
s˜ := 1 1 1 1
ǫ ( F−21(u2+v2) on K2and nearK2∩B
Clearly, s˜ is in A (K ∪K ,M), and extends to a holomorphic map near K ∩B. Moreover,
ǫ φ 1 2 2
dist(s˜ ,s) = O(ǫ). By construction, we have s (p) = s(p). Therefore, given η > 0, we can find s
ǫ ǫ η
with required properties. (cid:3)
3. Sets with property A
3
In this section we prove Lemma 2, and Theorems 1 and 3. In each we show that a certain set
K has property A . We will let M be a complex manifold, φ:M→C a holomorphic submersion
3
such that φ(M) ⊃ K, P a finite subset of K, and s a section of φ, s ∈ A (K,M). We let ǫ > 0,
φ
and want to show that there is an s ∈ O (K,M) such that dist(s,s ) < ǫ and s (p) = s(p) for
ǫ φ ǫ ǫ
each p∈P.
3.1. Proof of Lemma 2. If K is not finite, it can be written as a disjoint union of finitely many
singletons and a closed subset C homeomorphic to the Cantor middle-third set. (See e.g., [1], pp.
108-109.) In particular, C\K is connected, and K has property A .
1
Observe that s ∈ A (K,M) is simply a continuous section of φ over K. We can cover K by
φ
finitely many sets {U } such that eachs(U ) has a φ-adaptedneighborhoodin M. We can choose
j j
arefinement{V }ofthiscover,suchthattheV ’sarepairwisedisjoint. ObservethatK :=K∩V
j j j j
has property A . Now, with respect to the φ-adaptedcoordinate system arounds(K ), the map s
1 j
has a representation on K of the form s(z)=(s˜(z),z), where s˜is continuous and takes values in
j
Cn−1. Approximating s˜by a holomorphic map in a neighborhood of K our result follows.
j
3.2. Proof of Theorem 1. We first introduce some combinatorial preliminaries. Let vw be an
edge in a graph Γ. We can construct a new graph Γ′ with one more vertex by “splitting the edge
10 DEBRAJCHAKRABARTI
vw.” More formally, if V is the vertex set of Γ, and E its edge set, the vertex set of Γ′ is V ∪{u}
(where u6∈V), and the edge set is (E\{vw})∪{vu,uw}.
Recall that a graph is n-colorable, if there is an assignment of n colors to its vertices so that
no two adjacent vertices have the same color. By a classical result of K¨onig (see [8], Theorem
12.1), the condition that a graph is 2-colorable is that it does not have a circuit (closed non-self
intersecting path) of odd length. We now have the following:
Lemma 7. Given any finite graph G, there are edges e ,...,e such that the graph G′ obtained
1 k
after successively splitting these edges is 2-colorable.
Proof. Let V be the vector space over the field Z/2Z with basis the set of edges of G. Given a
circuit C in G, traversing the edges x ,x ,...,x , associate with it an element c of V given by
1 2 m
c = x +x +···+x . Let Z be the subspace of V spanned by all c for each circuit C. Pick a
1 2 m
basis c ,...,c . Since any circuit can be written as a linear combination of the c ’s it is sufficient
1 k j
to split any edge of those c which have an odd number of summands. Moreover, from the fact
j
that the c are linearly independent over Z/2Z it follows that each c has an edge not contained
j j
in any c , k 6=j. The result follows. (cid:3)
k
Now we can prove Theorem 1. Note that as in the proof of Lemma 1 above, K◦ = ∅, so s is
simply a continuous section of φ over K.
Forz ∈K,there isaneighborhoodU ofz inK suchthats(U )hasaφ-adaptedneighborhood
z z
in M. Select a finite subcover F of {U } , and let G be a refinement of F which has only
z z∈K
double intersections (this is possible since dim(K) = 1, and K is connected.) We can write
G ={V ,...,V }.
1 M
We associate to G a graph N by taking the nerve: the vertices V and V are joined by an
G i j
edge in N iff V ∩V 6= ∅. Let V and W be open sets in G such that V ∩W 6= ∅. We define a
G i j
new open cover S (G) of K in the following way. Replace the sets V and W by V′, U and W′,
VW
where V′ ⊂V, W′ ⊂W, and U is a neighborhoodof V ∩W suchthat V′∪U∪W′ =V ∪W, and
V ∩U 6=∅,U∩W 6=∅butV ∩W =∅. IfU ischosensmallenough,then(G\{V,W})∪{V′,U,W′}
is againanopencoverof K with only double intersections. This is our G′ =S (G). At the level
VW
of nerves NG′ is obtained by splitting the edge VW in the graph NG, as in Lemma 7. Thanks
to Lemma 7 we can obtain after finitely many rounds of edge-splitting a new cover H, with only
double intersections such that the corresponding nerve N is 2-colorable. It also follows that for
H
each U ∈H, the set s(U) has a φ-adapted neighborhood in M. Further, by shrinking the U’s we
can assume that whenever an intersection U ∩U′ of two sets in H is non-empty, it is contractible.
Let us call the colors used in coloring N red and blue. We can now define
H
K = U, and K = U.
1 2
UU[∈rHed UU[∈blHue
It is easy to verify that (K ,K ) is a good pair, and for j = 1,2 the set s(K ) has a φ-adapted
1 2 j
neighborhood in M.