Approximation of analytic functions with prescribed boundary conditions by circle packing maps

10 Pages·0.18 MB·English

by  Tomasz Dubejko

APPROXIMATION OF ANALYTIC
FUNCTIONS WITH PRESCRIBED BOUNDARY
CONDITIONS BY CIRCLE PACKING MAPS
5
9
9
1

n Tomasz Dubejko
a
J

5 Abstract. Weuserecentadvancesincirclepackingtheorytodevelopaconstructive
2
method for the approximation of an analytic function F : Ω → C by circle packing

] maps providing we have only been given Ω, |F′| Ω, and the set of critical points of
G F. This extends the earlier result of [CR] for F w(cid:12)ith no critical points.
(cid:12)
M
.
h
t
a
1 Introduction
m
[
In [CR] an inverse problem for circle packing and locally univalent function was

1 considered. It was shown that for a given bounded simply connected domain Ω and
v
a positive continuous function λ : ∂Ω (0, ) one can construct a sequence of
7
→ ∞
1 locally univalent hexagonal circle packings and associated with it a sequence f
n
2 { }
of circle packing maps such that f f uniformly on compacta of Ω as n ,
1 n → → ∞
0 wheref istheunique(uptosomestandardnormalization)locallyunivalentanalytic
5 function in Ω with f′ : Ω (0, ) being continuous and f′ λ.
9 | | → ∞ | | ∂Ω ≡
/ Here we are interested in a generalization of the above pro(cid:12)blem, that is in ap-
h (cid:12)
proximation of an analytic function F : Ω C via circle packing maps, where F
t
a →
has a finite set of critical points in Ω and F′ : Ω [0, ) is continuous with
m
| | → ∞
F′ λ on ∂Ω. We will prove that this can be achieved, roughly speaking, by tak-
:
v | | ≡
ing the circle packing map from the portion of a regular hexagonal circle packing of
i
X suitably small mesh that fills up Ω to a combinatorially equivalent branched circle
r packing whose branch set approximates the set of critical points of F and whose
a
boundary circles have their radii determined by λ and the mesh. The precise setup
is laid out in Section 3 and the main result is contained in Theorem 3.1.
Techniques we employ here to verify our approximation scheme are ideologically
quite different from the ones used in [CR]; the latter are closely linked with hexago-
nal combinatorics and with local univalence of functions considered, the former are
based on advances in the theory of circle packing. In fact, arguments presented in
this paper can be easily extended to other then hexagonal patterns (cf. [HR],[St]).
As the proof of Theorem 3.1 will heavily rely on results from the theory of circle
packing, we combine them, together with an introduction of basic terminology, in
1991 Mathematics Subject Classification. 30G25, 30E25, 52C15.
Key words and phrases. circle packing, discrete analytic maps, discrete conformal geometry.
TheauthorgratefullyacknowledgessupportoftheTennesseeScienceAllianceandtheNational
Science Foundation. Research at MSRI is supported in part by grant no.DMS-9022140.
Typeset by AMS-TEX
2 TOMASZ DUBEJKO
Section 2. The reader who would like to obtain more information about the subject
should see [BSt] or [D1].
The author would like to acknowledge that both figures in this paper have been
createdwiththehelpofCirclePack, asoftwarepackagedevelopedbyKenStephen-
son.
2 Circle Packing Overview
Let K be a simplicial 2-complex which is a triangulation of a simply connected
domain in the complex plane C and which has its orientation induced from the
orientation of the plane. Write K(0), intK(0), bdK(0), K(1), and K(2) for the sets
of vertices, interior vertices, boundary vertices, edges, and faces of K, respectively.
A collection of circles in C is said to be a circle packing for K if there is 1-to-1
P
correspondence between the elements of K(0) and (K(0) v C(v) ) such
P ∋ ↔ ∈ P
thatif u,v K(1) thenC(u)andC(v)areexternallytangentandif u,v,w K(2)
h i ∈ h i ∈
is a positively oriented triple then so is C(u),C(v),C(w) .
h i
Let S be a simplicial map S : K C which is first defined on K(0) by
P P
→
mapping v K(0) to the euclidean center of C(v) and then is extended to
∈ ∈ P
K(1) and K(2) via barycentric coordinates. If K(2) and v is a vertex of
△ ∈ △
then α (v, ) will denote the angle at the vertex S (v) in the euclidean triangle
P P
△
S ( ). We define the carrier of as carr( ) := S (K) and the angle sum of at
P P
△ P P P
a vertex v as Θ (v) := α (v, ). It follows from our definition of a circle
P △∈K(2) P △
packing that Θ (v) is aPpositive integer multiple of 2π if v is an interior vertex of
P
K. Moreover, if is univalent, i.e. the circles of have mutually disjoint interiors,
P P
then Θ (v) = 2π for each v intK(0). However, the converse is not true (see
P
∈
Fig 1 (b)). If v intK(0) and Θ (v) = 2πn, n 2, then v will be called a branch
P
∈ ≥
vertex (point) of of order (n 1) (multiplicity n); geometrically this means that
P −
the circles in associated with the neighboring vertices of v wrap n-times around
P
C(v). The listing of all the branch vertices of together with their orders will
P
be called the branch set of and denoted br( ). A circle packing without branch
P P
points will be called locally univalent.
So far we have talked about circle packings but we have not said anything about
their existence. Before we can state necessary and sufficient conditions for the
existence we need the following definition, in which N will denote the set of non-
negative integers.
Definition 2.1. Let K be a finite or infinite triangulation of a simply connected
domain in C. A set (v ,l ),(v ,l ),... (intK(0)) N, possibly finite, is called a
1 1 2 2
{ } ⊂ ×
branch structure for K if every simple closed edge-path Γ in K has at least 2ℓ(Γ)+3
n
edges, where ℓ(Γ) = δ (Γ)l and δ (Γ) is equal to 1 if v is enclosed by Γ and
i=1 i i i i
0 otherwise. P
The following theorem answers the existence question for circle packings (see
[D1],[D2]).
Theorem 2.2. Suppose K is a triangulation of a simply connected domain in C.
Let b ,...,b be interior vertices of K, and let k ,...,k be non-negative integers.
1 m 1 m
Then there exists a circle packing forK with branch set B = (b ,k ),...,(b ,k )
1 1 m m
if and only if B is a branch structure for K. (cid:8) (cid:9)
BOUNDARY PROBLEM AND CIRCLE PACKINGS 3
(a)
(c)
(b)
Figure 1. Different packings of the same complex:
(a)univalent, (b)locallyunivalent, (c)branched(and
its decomposition into univalent sheets)
Notice that when 0 = k = ... = k then B is a trivial case of a branch structure;
1 m
in fact the number of pairs (b ,k ) in B is irrelevant in this case. Therefore we
i i
identify the sets (b ,0),(b ,0),... with the empty set.
1 2
For the purpos(cid:8)es of this paper w(cid:9)e will need a stronger version of Theorem 2.2
whenKisfinitewhichwillbestatedshortly,butfirstwehavetointroduceafunction
whichisquitehandywhenworkingwithcirclepackings. If isacirclepackingforK
P
thenafunctionr : K(0) (0, )definedbyr (v) := the euclidean radius of C(v)
P P
→ ∞
is called the radius function of . Now a stronger version of Theorem 2.2 (see [D1]).
P
Theorem 2.3. Suppose K is a finite triangulation of a simply connected domain
in C. Let b ,...,b be interior vertices of K, and let k ,...,k be non-negative
1 m 1 m
integers. For any given ρ : bdK(0) (0, ) there exists a circle packing for K
→ ∞ P
with branch set B = (b ,k ),...,(b ,k ) and with r ρ if and only if
1 1 m m P bdK(0) ≡
B is a branch structu(cid:8)re for K. Moreover, (cid:9)is unique up(cid:12)to isometries of C.
P (cid:12)
Suppose that and are circle packings for K. Then and will be called
P Q P Q
combinatorially equivalent (or shortly c-equivalent) with complex K. The map
4 TOMASZ DUBEJKO
F := S S −1 : carr( ) carr( ) will be called the circle packing map from
P,Q Q P
◦ P → Q
to (shortly, cp-map). The map F# : carr( ) (0, ) defined on the set
P Q P,Q P → ∞
of vertices of S (K) by F# (S (v)) = rQ(v) and then extended affinely to faces
P P,Q P rP(v)
of S (K) will be called the ratio function from to . The ratio function allows
P
P Q
one to measure “distortion” of circle packings or how distinct two packings are.
This is due to the following rather elementary fact, which gives, for example, the
uniqueness in Theorem 2.3.
Fact 2.4.
(1) If and are circle packings for K with identical branch sets (possibly
P Q
empty) then F# does not obtain its infimum or supremum in intK(0)
P,Q
unless it is constant.
(2) If and are circle packings for K such that br( ) = and br( ) =
P Q P ∅ Q 6 ∅
then F# does not attain its supremum in intK(0).
P,Q
We will now recall some results regarding circle packings and approximation
of analytic functions. Let H be the regular hexagonal 2-complex of mesh 2 with
vertices at 2k+l(1+√3i), k,l Z. Write H := 1H. Let be the univalent circle
∈ n n Pn
packing whose carrier is H , i.e. the regular hexagonal circle packing of circles of
n
radius 1. We notice that if B = (b ,k ),...,(b ,k ) is a branch structure for
n 1 1 m m
H , n 1, then necessarily k =(cid:8) = k = 1, i.e. (cid:9)the b ’s are simple branch
n 1 m i
≥ ···
points. Thus, when dealing with complexes H , we adopt a convention by writing
n
b ,...,b for (b ,1),...,(b ,1) . We now recall the definition of a discrete
1 m 1 m
{ } { }
complex polynomial.
Definition 2.5. A map f : C C is a discrete complex polynomial for H ,
n
→
n 1, with the branch set B = b ,...,b if the following are satisfied:
1 m
≥ { }
(i) there exists a circle packing for H with the branch set B such that f is
n
B
the cp-map from to ,
n
P B
(ii) f has a decomposition f = ϕ h, where h is a self-homeomorphism of C
◦
and ϕ is a complex polynomial.
If f is a discrete complex polynomial for H and is as in (i) then will be called
n
B B
the range packing of f.
The following theorem is a result of Corollary 4.9 and Lemma 5.2 from [D2].
Theorem 2.6.
(1) If B = b ,...,b is a branch structure for H , n 1, then there exists
1 m n
{ } ≥
a circle packing for H with the branch set B such that the cp-map f :
n
B
C C from to is a discrete complex polynomial.
n
→ P B
(2) There exists a constant κ 1 depending only on m such that if is a circle
≥ B
packing for H , n 1, and the cp-map from to has valence at most
n n
≥ P B
m+1 then rB(w) 1,κ for any neighboring vertices w,v, where r is the
rB(v) ∈ κ B
radius function of (cid:0). In p(cid:1)articular, any discrete complex polynomial for H ,
n
B
n 1, with the branch set containing at most m points is K-quasiregular,
≥
K = K(m).
The above theorem is a key factor in a proof of an approximation result for
discrete complex polynomials which will be stated shortly. However, first we
BOUNDARY PROBLEM AND CIRCLE PACKINGS 5
need to lay down some foundations. Let F be a classical complex polynomial
with the critical points x ,...,x of orders k ,...,k , respectively, i.e. B =
1 m 1 m
(x ,k ),...,(x ,k ) is the branch set of F. For each sufficiently large n let
1 1 m m
(cid:8)B = b (n),...,b ((cid:9)n), ,b (n),...,b (n) be a branch structure for H .
Funrthe{r,1s1uppose th1ek1sequen·c·e· Bm1 has thempkrmopert}y that lim b (n) = x fonr
{ n} n→∞ ij i
j = 1,...,k and each i, 1 i m. It is easy to see that such a sequence B
i n
≤ ≤ { }
can be constructed for all sufficiently large n. As a consequence of Theorem 5.3
and Remark 5.4 of [D2] we have
Theorem 2.7. For each sufficiently large n there exists a discrete complex polyno-
mial f for H with the branch set B such that the functions f and f# converge
n n n n n
uniformly on compacta of C to F and F′ , respectively.
| |
The uniform convergence of f# F′ in the last theorem follows from the
n
→ | |
following result that we will need later (Theorem 1 of [DSt]).
Theorem 2.8. Let Ω C be a bounded simple connected domain. Let
n
⊂ {Q }
and be sequences of circle packings such that for each n the packing is
n n
{Q } Q
univalent, carr( ) Ω, and and are c-equivalent. Moreover, suppose that
e n n n
Q ⊆ Q Q
the sets carr( ) exhaust Ω and that the supremum of radii of circles in goes
Qn e Qn
to 0 uniformly on compacta of Ω as n . In addition, assume that functions g
n
→ ∞
converge uniformly on compacta of Ω to an analytic function g : Ω C as n ,
→ → ∞
where g : carr( ) carr( ) is the cp-map from to . Then the ratio
n n n n n
Q → Q Q Q
functions g# converge uniformly on compacta of Ω to g′ .
n e | | e
3 The Main Result
In this section we will be concerned with the following problem. Let Ω C
⊂
be a Jordan domain. Let 0 Ω and ξ Ω, ξ > 0. Suppose λ : ∂Ω (0, )
∈ ∈ → ∞
is a continuous function and B = (x ,k ),...,(x ,k ) is a subset of Ω N.
1 1 m m
×
It is a well-known fact that there ex(cid:8)ists the unique analyt(cid:9)ic function F in Ω such
that F(0) = 0, F(ξ) > 0, the set of critical points of F is equal to B, and F′
| |
has a continuous extension to Ω with F′ λ on ∂Ω. We are interested in the
| | ≡
approximation of F using cp-maps having only been given Ω, λ, B, and ξ.
We will show that this can be achieved, roughly speaking, by taking the cp-map
from a portion of a regular hexagonal circle packing of suitable small mesh that
fills up Ω to a c-equivalent circle packing whose branch set approximates B and
whose radius function on the boundary is approximately equal to λ times the mesh
of the hexagonal packing. More precisely, write O for the maximal “complete”
n
subcomplex of H contained in Ω (i.e., a simply connected simplicial 2-complex
n
consisting of all faces of H whose closures are in Ω, together with their edges and
n
vertices). Denote the portion of associated with O , where is as in
n n n n
Q P P
Section 2. Suppose that for each sufficiently large n there exists a circle packing
c-equivalent to such that
n n
Q Q
e (1) if rn : O(n0) (0, ) is the radius function of n, v bdO(n0), and zv is a
→ ∞ Q ∈
point on ∂Ω closest to v, then r (v) = 1λ(z ),
n n v e
(2) if the set B = b (n),...,b (n), ,b (n),...,b (n) of distinct
n { 11 1k1 ··· m1 mkm }
vertices of O is the branch set of then the sequence B has the
n n n
Q { }
property that lim b (n) = x for j = 1,...,k and each i, 1 i m,
n→∞ ij i e i ≤ ≤
6 TOMASZ DUBEJKO
(3) if f : carr( ) carr( ) is the cp-map from to then f (0) = 0
n n n n n n
Q → Q Q Q
and f (ξ) > 0,
n e e
then will be called an approximating sequence of circle packings for F. We
n
{Q }
will prove the following theorem which is our main result.
e
Theorem 3.1. Let Ω C be a Jordan domain. Let 0 Ω and ξ Ω, ξ > 0.
⊂ ∈ ∈
Suppose λ : ∂Ω (0, )is a continuous functionand B = (x ,k ),...,(x ,k )
1 1 m m
→ ∞
is a subset of Ω N. Let F be the unique analytic function i(cid:8)n Ω such that F(0) = 0(cid:9),
×
F(ξ) > 0, the set of critical points of F is equal to B, and F′ : Ω [0, ) is
| | → ∞
continuous and F′ λ. Then there exists an approximating sequence of
| | ∂Ω ≡ {Qn}
circle packings for F(cid:12). Moreover, if the maps f are defined as in (3) above then f
(cid:12) n e n
and f# converge uniformly on compacta of Ω to F and F′ , respectively.
n
| |
n n
Q Q
e
.x
.0
fn .0
Ω
Figure 2. The circle packing map in an
approximation sequence: λ 1.4, B = (x,1)
≡ { }
Before we give a proof of the above theorem we want to make two remarks.
Remark 3.2. We observe that our construction of an approximating sequence
n
{Q }
is based exclusively on Ω, λ, B, and ξ, hence it gives a constructive method for the
e
approximation of F.
Remark 3.3. If B = then we obtain the result of [CR].
∅
To proof Theorem 3.1 we will need the following
Lemma 3.4. Let Ω and F be as in Theorem 3.1. Then there exists a sequence of
polynomials χ such that χ F and χ′ F′ uniformly in Ω as n .
n n n
{ } → | | → | | → ∞
Proof. First we need a formula for F. Let u be the harmonic function in Ω with
u = logλ. Denote by v the harmonic conjugate of u in Ω. Write τ : Ω D for
∂Ω
| →
the Riemann mapping with τ(0) = 0 and τ(ξ) > 0. Then the function F is given
by
z
F(z) = B(τ(η))eu(η)+iv(η)dη,
Z
0
where B(z) = c m z−τ(xi) ki and c is a suitable unimodular constant (such
Qi=1(cid:0)1−τ(xi)z(cid:1)
that F(ξ) > 0).
BOUNDARY PROBLEM AND CIRCLE PACKINGS 7
Let Ω be a sequence of Jordan domains such that Ω Ω Ω , n 1, and
n n+1 n
{ } ⊂ ⊂ ≥
the boundary of Ω converges to ∂Ω in the sense of Fr´echet (cf. [LV, p.27],[H],[W]).
n
Write τ : Ω D for the Riemann mapping with τ (0) = 0 and τ (ξ) > 0. Then
n n n n
→
τ τ uniformly in Ω. Define F : Ω C by
n n n
→ →
z
F (z) = B(τ (η))eu(τ−1◦τn(η))+iv(τ−1◦τn(η))dη.
n n
Z
0
Then F F uniformly in Ω. Moreover, as F′(z) = B(τ (z)) eu(τ−1◦τn(η)) for
n n n
→ | | | |
z Ω, and B and u are continuous in D and Ω, respectively, we also get F′ F′
n
∈ | | → | |
uniformly in Ω. By Runge’s theorem for each n there exists a polynomial χ such
n
that sup F (z) χ (z) < 1 and sup F′(z) χ′ (z) < 1. The last implies
the asserzt∈ioΩn|onf the−lemnma.| (cid:3)n z∈Ω| n − n | n
We are now ready to show Theorem 3.1.
Proof of Theorem 3.1. We first need to verify the existence of an approximating
sequence ofcirclepackingsforthefunction F. ThisfollowseasilyfromTheorem2.3,
Definition 2.1, and the geometry of simplicial complexes O . In fact, Definition 2.1
n
andthegeometryofsimplicialcomplexesO areonlyneededtoobtainthecondition
n
(2) in the definition of an approximating sequence. The condition (3) of that
definition is achieved immediately by applying translations and/or rotations, if
required, to already constructed packings.
Wewillnow show thatif isanapproximatingsequence ofcirclepackingsfor
n
{Q }
F then the maps f : carr( ) carr( ) and f# : carr( ) (0, ) converge
n en n n n
Q → Q Q → ∞
uniformly on compacta of Ω to F and F′ , respectively. Let G be a complex
e| |
polynomial with the branch set B such that G(0) = 0 and G(ξ) = 1. If B is the
n
branch set of then it is not hard to see that B O(0) H(0) is a branch
n n n n
Q ⊂ ⊂
structure for H for all sufficiently large n. From Theorem 2.7 there exists, for
en
each sufficiently large n, a discrete complex polynomial g for H with the branch
n n
set B such that g G and g# G′ uniformly on compacta of C as n .
n n n
→ → | | → ∞
Write R : H(0) (0, ) for the radius function of the range packing of g .
n n n n
→ ∞ U
Since G′ > 0, g# G′ , and g#(v) = R (v)/(1), there exists
| |(cid:12)∂Ω n(cid:12)∂On −n−→−∞→ | |(cid:12)∂Ω n n n
σ 1 suc(cid:12)h that 1 (cid:12)R σ (cid:12)for all n. Denote by the portion of
≥ nσ ≤ n bdO(n0) ≤ n Un Un
associated with the subcom(cid:12)(cid:12)plex O of H . Recall that r : Oe(0) (0, ) is the
n n n n
→ ∞
radius function of . Since and are packings for O with the branch sets
n n n n
Q U Q
B , Fact 2.4 implies that rn ( ) : O(0) (0, ) attains its maximum and minimum
n e Rn ·e n →e ∞
in bdO(0). Hence
n
r
(3.1) 1 minλ n σmaxλ
σ ∂Ω ≤ Rn O(0) ≤ ∂Ω
n
(cid:12)
(cid:12)
because 1 min λ rn σmax λ. Now it follows from Theorem 2.6
σ ∂Ω ≤ Rn bdO(n0) ≤ ∂Ω
(applied to g ) and (3.1) (cid:12)that there exists a constant κ 1 such that
n (cid:12)
≥
r (w)
(3.2) 1 n κ for any n and any neighboring vertices v,w O(0).
κ ≤ r (v) ≤ ∈ n
n
8 TOMASZ DUBEJKO
The last conclusion and the construction of the maps f imply that f is a family
n n
{ }
of K-quasiregular mappings for some K 1. Moreover, since Ω is bounded and
≥
by Fact 2.4 the ratio functions f# are uniformly bounded by max λ, it follows
n ∂Ω
that the maps f are uniformly bounded. Hence f is a normal family. By
n n
{ }
taking a subsequence if necessary, assume that f f uniformly on compacta of
n
→
Ω as n . We will first show that the limit function f is not constant. If f
→ ∞
were constant then, by Theorem 2.8, functions f# would converge uniformly on
n
compacta of Ω to the constant zero-function. But this would contradict (3.1) for
sufficiently large n because rn = fn# and G′ > 0 in Ω x ,...,x .
(cid:0)Rn(cid:1)(cid:12)O(n0) (cid:0)gn#(cid:1)(cid:12)O(n0) | | \{ 1 m}
Hence f is non-constant K-quasiregular mapping in Ω.
(cid:12) (cid:12)
Wewillnowverifythatf isactuallyanalytic. Sincef isboundedK-quasiregular
n
mapping, by the Stoilow’s theorem ([LV]), it has a decomposition f = ϕ ψ ,
n n n
◦
where ψ : O D is a K-quasiconformal homeomorphism normalized by ψ (0) =
n n n
→
0 and ψ (ξ) > 0, and ϕ : D C is an analytic function with ϕ (0) = 0 and
n n n
→
ϕ (ψ (ξ)) > 0. By extracting a subsequence from f if necessary, we can assume
n n n
{ }
that ϕ ϕ and ψ ψ uniformly on compacta of Ω and D, respectively, where
n n
→ →
ϕ : Ω D is a K-quasiconformal homeomorphism, ψ : D C is analytic, and
→ →
f = ϕ ψ. Inparticular, ϕandψ arenot constant and thebranch set ofϕisequalto
◦
(ψ(x ),k ),...,(ψ(x ),k ) . Letz Ω x ,...,x . Fromtheequicontinuity
1 1 m m 0 1 m
∈ \{ }
o(cid:8)f the normal family ψ and(cid:9)its uniform convergence on compacta to the function
n
{ }
ψ we get that there exist ǫ > 0 and n such that f is 1-to-1 for n > n ,
0 n B(z ,ǫ) 0
(cid:12) 0
whereB(z ,ǫ) = z z < ǫ . Let (z )betheport(cid:12)ionof associatedwiththe
0 0 n 0 n
{| − | } Q Q
largest complete hexagonal generation of O around z contained in B(z ,ǫ). Then
e n 0 e 0
(z ) is a sequence of univalent hexagonal circle packings with their number
n 0
{Q }
of generations around z going to as n . From the Hexagonal Packing
e 0 ∞ → ∞
Lemma of [RS] we conclude that the quasiconformal distortion of the maps f at
n
z goes to 0 as n . Hence the quasiconformal distortion of the mappings ψ
0 n
→ ∞
at z goes to 0 as n . Thus ψ is 1-quasiconformal in Ω x ,...,x . Since
0 1 m
→ ∞ \{ }
ψ is a homeomorphism of Ω, the last implies that ψ is conformal in Ω. Therefore
f = ϕ ψ is analytic in Ω with the branch set B.
◦
To complete our proof we need to show that f = F. To achieve this we have
to prove that f′ has a continuous extension to ∂Ω and is equal to λ there. We
| |
observe first that since f is analytic in Ω, by Theorem 2.8, f# f′ uniformly
n
→ | |
on compacta of Ω as n . Let χ be a sequence of polynomials given by
n
→ ∞ { }
Lemma 3.4. Since F′ > 0 and F has a finite branch set in Ω, we can assume
| | ∂Ω
(cid:12)
that for each n the restriction of the branch set of χ to Ω is equal to the branch set
(cid:12) n
B of F. Take δ > 0 such that F′(z) > 0 for z Ωδ := z Ω : dist(z,∂Ω) δ .
| | ∈ { ∈ ≤ }
Given ǫ > 0 let N(ǫ) be such that
χ′ (z) F′(z) < ǫ, z Ω, n N(ǫ) and
n
| |−| | ∈ ≥
(3.4) (cid:12) (cid:12)
(cid:12) |χ′n(z)| 1 (cid:12)< ǫ, z Ωδ, n N(ǫ).
|F′(z)| − ∈ ≥
(cid:12) (cid:12)
(cid:12) (cid:12)
From the geometry of the complexes H , Theorem 2.7, and the fact that B is a
n n
branch structure for O , it follows that there exists a sequence of discrete polyno-
n
mials p such that p is a discrete polynomial for H , br(p ) Ω = B , and
n n n n n
{ } ∩
BOUNDARY PROBLEM AND CIRCLE PACKINGS 9
p χ and p# χ′ uniformly on compacta of C as n . Now (3.4)
n → N(ǫ) n → | N(ǫ)| → ∞
implies that there is N′(ǫ), N′(ǫ) N(ǫ), such that
≥
(3.5) p#n 1 < 2ǫ, z Ωδ, n N′(ǫ).
|F′(z)| − ∈ ≥
(cid:12) (cid:12)
(cid:12) (cid:12)
Write for the image packing of p . Denote by the portion of associated
n n n n
W W W
with the subcomplex O of H , and ̺ the radius function of . Since
n n n f n n
W W
and are circle packings for O with identical branch sets, Fact 2.4 shows that
n n f f
Q
̺rnn(·)e: O(n0) → (0,∞) has its maximum and minimum in bdO(n0). Hence (cid:0)fpn#n#(cid:1)(cid:12)O(n0) :
O(0) (0, )has itsmaximum and minimum inbdO(0). Fromthe last conclu(cid:12)sion,
n n
→ ∞
the construction of (i.e., f# F′ λ), the fact that
Qn n ∂(carr(Qn)) ≈ | | ∂(carr(Qn)) ≈
carr( n) Ωδ for laerge n, and ((cid:12)(cid:12)3.5), it follows th(cid:12)(cid:12)at there exists N′′(ǫ), N′′(ǫ)
Q ⊂ ≥
N′(ǫ), such that
(3.6) p#n (v) 1 < 3ǫ, v O(0), n N′′(ǫ).
(cid:12)f# − (cid:12) ∈ n ≥
(cid:12) n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We observe that f# max λ by Fact 2.4(2). Hence, since p# and f# have
n O(n0) ≤ ∂Ω n n
been defined as simp(cid:12)licial extensions of p# and f# , respectively, we obtain
(cid:12) n O(0) n O(0)
n n
from (3.6) (cid:12) (cid:12)
(cid:12) (cid:12)
(3.7) p#(z) f#(z) < 3ǫmaxλ, z carr( ), n N′′(ǫ).
n n n
− ∂Ω ∈ Q ≥
(cid:12) (cid:12)
(cid:12) (cid:12)
Recall that f# f′ and p# χ′ uniformly on compacta of Ω. Thus, by
n → | | n → | N(ǫ)|
letting n in (3.7), one gets
→ ∞
χ′ f′(z) < 3ǫmaxλ, z Ω.
(cid:12)| N(ǫ)|−| |(cid:12) ∂Ω ∈
(cid:12) (cid:12)
As ǫ was arbitrary, (3.4) implies that F′ f′ in Ω. In particular, f′ has a
| | ≡ | | | |
continuous extension to ∂Ω and is equal to F′ λ there. Hence f F and
the proof is complete. (cid:3) | |(cid:12)∂Ω ≡ ≡
(cid:12)
Concluding remarks. 1) Noticethat Theorem 3.1gives, inessence, a methodfor the
approximation of the integral of an analytic function on compacta of its domain.
2) Since the sequence of maps f# converges uniformly on compacta of Ω to
n
{ }
F′ , logf# converges uniformly on compacta of Ω x ,...,x to the harmonic
n 1 m
| | { } \{ }
function u in Ω x ,...,x which has isolated singularities at x of degree k ,
1 m i i
\ { }
i = 1,...,m, and which satisfies boundary condition u = logλ. We also note
∂Ω
|
that functions f# are easier to construct then functions f ; the latter require the
n n
radius functions of ’s and the centers of circles in ’s while the former require
n n
Q Q
only radius functions of ’s.
e n e
Q
3) Arguments presented so far in this paper can easily be extended to other
e
combinatorial patterns as follows. Let be a univalent circle packing whose carrier
O
is C (e.g. the “ball-bearing” packing – see [BSt] Figure 2(b)). Denote the geometric
complex of by L. Assume that L is of bounded degree, i.e. there is a uniform
O
10 TOMASZ DUBEJKO
bound on the number of neighboring vertices of each vertex in L. Further, suppose
has the property that for each ǫ > 0 there is n such that all circles of 1 ,
O ǫ nO
n n , contained in z < 1 have their radii at most ǫ. Then, one can generalize
ǫ
≥ {| | }
Definition 2.5 and Theorem 2.6 to discrete complex polynomials for L (= 1L) as
n n
it was done in [D2]. Moreover, one can also obtain Theorem 2.7 for complexes L
n
following the steps of the proof of Theorem 5.3 in [D2] and using Theorem 2.2 of
[HR]. It is now a matter of replacing H by L to define approximation sequences
n n
based on complex L and to verify Theorem 3.1 for such sequences.
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Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300
Mathematical Sciences Research Institute, Berkeley, CA 94720
E-mail address: [email protected]