363
I ntn. J. 4ath. & Math. Sci.
Vol. 6 No. 2 (1983) 363-370
FUNCTIONS IN THE SPACE R2(E)
AT BOUNDARY POINTS OF THE INTERIOR
EDWIN WOLF
Department of Mathematics
Marshall University
Huntington, West Virginia 25701
(Received September 15, 1982)
ABSTRACT.
We
Let E be a compact subset of the complex plane C.
denote by R(E)
(the restrictions to E of)
the algebra consisting of
rational functions with poles off E.
For p a i, let
Lebesgue measure.
dimensional
Let m denote 2
RP(E)
be the closure of R(E)
in
p
L (E, dm)
point evaluation for
R2(E).
a limit point of the set S
Let x c 5E be a bounded
2.
In this paper we consider the case p
Suppose there is a
[yly
C > 0 such that x is
Int E, Dist(y,SE) > C ly
xl].
For those y e S sufficiently near x we prove statements about
f(x) for all f e R(E)
If(y)
KEY WORDS AND PHRASES. Rtional functions, compact set L p
spaces, bounded point
evaluation, adsible function.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
50A98, 46E99.
INTRODUCTION AND DEFINITIONS
Let E be a compact subset of the complex plane
.
We denote by
R(E) the algebra consisting of (the restrictions to E of) rational
functions with poles off E.
measure.
A point x
RP(E)
For p
i, let
Let m denote 2
RP(E)
dimensional Lebesgue
be the closure of R(E)
in"
LP(E,dm).
E is said to be a bounded point evaluation (BPE) for
if there is a constant F such that
1
If(x)
<
F.[ If(z)IPdm(z)]Pfor
E
all fcR(E).
E. WOLF
364
In [4] we studied the smoothness properties of functions in
RP(E),
(see
p > 2, at BPE’s.
Fernstrm
When p
and Polking
[2]
2, the situation is quite different
Fernstrm [i]).
and
In [5] we
2
showed that at certain BPE’s the functions in R (E) have the following
smoothness property: Let x c bE be both a BPE for R 2 (E) and the
vertex of a sector contained in Int E.
Let L be a line segment that
bisects the sector and has an end point at x.
> 0
Then for each
there is a 6 > 0 such that if
y c L and
IY- xl
If(y)
< 6,
f(x)
llfl12
<
R(E).
for all f
The
goal of this paper is to extend this result to certain cases where
there may not be a sector in Int E having vertex at x, but x is still
a limit point of Int E.
If x
E is a BPE for R
that f(x)
2(E)
there is a function g e
R(E).
fg dm for any f
L2(E)
such
Such a function g is called a
E
representing function for x.
A point x e E is a bounded point derivation (BPD) of order s for
R2(E)
f,s,
if the map f
(x), f
R(E), extends from R(E) to a bounded
2
linear functional on R (E).
Let A (x) denote the annulus
n
A (x)
n
[z 12 -n-2
<
z- x
2
<
-n+l
[z 12 -n-I
].
&
If x
z- x
&
2-n].
Let
0, we will denote A n (0)
by A n and A n (0) by A n
For an arbitrary set X c C we let
C2(X)
denote the Bessel capacity
of X which is defined using the Bessel kernel of order 1 (see
We say that
is a positive, non-
is an admissible function if
decreasing function defined on (0,), and r.(r)
and tends to zero when r
0
[3]).
-i
is nondecreasing
+
Using the techniques of [4] and [2] one can prove:
THEOREM 1.1.
Let s be a nonnegative integer and E a compact set.
Suppose that x is a BPE for
R2(E)
and
is admissible.
Then x is
FUNCTIONS IN THE SPACE
.
if and only if
2.
2
e L (E)
s.(I z-x[)
(2-n) -2C 2 (An(X)-E)
2 2n (s+l)
n=0
365
L 2 (E) such that
represented by a function g
g
(z-x)
R2(E)
<
THE MAIN RESULTS
Let x
zl
that E c
R2(E).
5E be a BPE for
< i].
We may assume that x
0 and
Suppose there is a positive constant C such that
0 is a limit point of the set S
L2(E)
We will construct a function g
which represents 0 for
R2(E)
and has support disjoint from S.
LEMMA 2.1.
R2(E).
Let 0 eE be a BPE for
Suppose there is a
positive constant C such that 0 is a limit point of the set
S
[YlY
g
2
L (E) such that:
(i)
(ii)
e Int E, Dist(y, SE) a
g represents 0 for
m((supp g) N S)
elyl].
Then there is a function
R2(E),
0,
k
(iii)
Ig[ 2
For all n a 2,
A
=[]
dm < F
For each i, i
[zl2-i-i
of the set A.
i
C
E.
Let Y
l
<
2k
C2(A2k+I
n
where F is a constant independent
PROOF.
2
+ 1
of n.
0,i,2,... consider all the intersections
Izl
2- i
with the bounded components of
be the closure of the union of these intersections.
Since Y. is compact, it can be covered by finit .ly many open discs of
radius
by B..
1
<C3-12 -i-l.
Let the union (finite) of these discs be denoted
The set B0 is bounded by finitely many closed Jordan curves
l
each of which is the union of finitely many circular arcs.
Each set
B. is contained in a set C0 bounded by finitely many closed Jordan
l
curves
Fij, J
1,2,..,n.l
such that if z belongs to any one of these
E. WOLF
366
2C3-12 -i-l.
curves, Dist (z,B i)
Now for each k, k
0,1,2,... choose a function
1
Co
lk
such that:
c
(1)
supp
(2)
I k(z)
1 for z
(3)
]% (z)
0 for z
.k
A2k+l
[z12-2k-2
J
2k+l
C.
i=o
lk (z)
x
where z
(5)
+
lk(Z)
kk+ l(z)
j2
# (Z)
-2k-3
2
2k-2
Izl
Fernstrm
lab 2k+2"
and Polking
[2] to
such that:
[z
k
dm(z)
<
k
Dist(z,E) <
].
F-2-2k(I-JSJ)C2(Ak-
E)
Here the constant F is independent
and k.
Since supp
set
C
(0,0), (0,1), and (1,0).
for
of
E
k
l for z near A
JD
(2)
[z
1 for z
> 0 we use a lemma of
obtain function
k(Z)
2k+l
+ ix and F and F are constants independent of k.
2
1
2
Given any
(l)
51k (z)
2k+l
[z
c
kk
].
Dist(z,E)
2k+l" lk lk on the
2k+l" k 1 on [JzJ 4 -I]
we have
A2k+l
Thus
o
[z JDist(z,E) < ].
h(z)
X(Z)
__i.
where h
k
lkh.
O
with
i near E.
X(z)
For each double index
there is a constant F
Set f
C
Choose X
(0,0)
such that
h.E #
2k+l
o
Ik
o
# 2k+l
h
k
(0,l)
Set
and (l,0)
FUNCTIONS IN THE SPACE
Since supp
c
lk
the above inequalities imply that
A2k+l
IDShk (z)
The subadditivity of C
2
367
R2(E)
<
F82(2k+l)(1+181)
and the convergence of
C2(A k
2
0
(see Theorem i.I) imply that the net
If
2
is bounded in L 1.
-,
,z,
z e E N
[zlDist(z, SE)
since f e L
2
I(E),
uz
a
for z c C
Clzl].
Set g
E and f(z)
5 f.
5z
There
2
is a subsequence that converges weakly to a function f c
which satisfi6sl f(z)
E)
Ll, lo c
0 for every
Then g e
and g is a representing function for 0.
L2(E)
The
proof of (iii) proceeds as in [5]
The above lemma can be used to prove the following theorem
in almost the same way that in
is used to prove
[5] Lemma 5.1
Theorem 5.1.
THEOREM 2.1.
2
Let 0 e 5E be a BPE for R (E).
Let C be a positive
constant such that 0 is a limit point of the set
S
[YlY
Int E, Dist(y,SE) > C
function for 0 and suppose that
admissible function.
y e S and
for all f
IYl
IYl]" Let g
-I
g(z) (Izl)
be a representing
2
L (E) where
is an
Then for any e > 0 there is a 6 > 0 such that if
< 6,
R(E).
Using this theorem and the methods in [4] one can prove:
COROLLARY 2.1.
hold.
Suppose that all the conditions of Theorem 2.1
Suppose, moreover, that s is a positive integer such that
g(z).z-S. (Iz )-i L2(E).
6 > 0 such that if y
If(y)
f(0)
for all f
f’(0)l.,
R(E).
Then for each c > 0 there exists a
S and
(y
0)
Yl
< 6,
.... f(S)0s.,
(y
0)sl
<
lY-01s(lYl)llfll2
E. WOLF
368
Finally, there is a corollary with weaker preconditions.
COROLLARY 2.2.
Let 0,g, and # be as in Theorem 2.1.
Suppose
there is a positive constant C such that 0 is a limit point of the
set
y
S
Int E, Dist(y, BE)
Y
C
Then for each e > 0 there is a 6 > 0 such that if y
If(y)
llf[12
f(0)
for all f
$
Yl]
(IYl
Yl
S and
< 6,
R(E).
The proof is similar to the proof of Theorem 2.1.
such
One uses the fact that there exists an admissible function
that
-IcL2(E)
g.#-l.
3.
EXAMP LES
We will construct a compact set E such that 0E, 0
EXAMPLE 1.
is a BPE for R
[Izl
&
i].
2(E),
and 0 is a limit point of Int
1,2,3,..., be the open disc centered on the
Let D i, i
positive real axis at 3
Let E
[J D0.
i
D
2
-i-3
Then since
i=l
(see
[3])., we
and having radius
r0
1
F(log
?)
C2(B(x
r))
&
exp(-22ii2).
,r
&
r
o
< 1
have
22nC2(A n
Z
n=l
Let D
E.
E)
n=l
22nC(D n)
<
F-
..
!_ <
n 1 n
Thus 0 is a BPE for
R2(E).
Dist(y,E) a C
intersects the positive real axis in a sequence
If C is a positive constant sufficiently
1
Int E,
will do), the se.t [YlY
small (any positive number <
IYl]
of disjoint intervals
EXAMPLE 2.
an’bn ]
such that
bn
0.
Next we construct a compact set E which is like
Example 1 in that 0 is a limit point of Int E and a BPE for
In this example, however, there exists no sequence
that
,f(Yn
f(0)
<
Ilfll
2,,
for all f
R(E) if
[yn]
.lYnl
R2(E).
c Int E such
< 6.
We will use
FUNCTIONS IN THE SPACE
important parts of
positive constant
r, r
r
<
o
< i.
R2(E)
369
Fernstrm’s construction in [i].
1 -i
such that C2(B(z,r)) < F(log ?)
Choose
,=
for all
a 1 such that
1
F.
c n=l
Let F be a
nlog
C2 (B(0
<
2n’
1/2))
Cover A with
be the closed unit square with center at 0
o
n
(i)
4
In
1 2
i
squares of side 2-n. Call the squares A n
(i)
have
and
that
put an open disc
every set A n
2
n
Let
the same center, and the radius of B n (i)is exp(-4 n log n)
i
1,2,3,... be an open disc centered on the positive real
D
i
i- 1
exp
< 2
and ro
&
axis such that D 1 c [zl 21
(i)
where the summation is
B
1,2,3,..., let G n
For each n, n
n
i
(i)
n
N ( Di
4 and B n
over those indices i such that 1 < i
1
Let A
o
.qn
Bn(i)
Bn(i)such
Set E
J
Ao
1
n=2
Gn.
Then
R2(EI
(-22ii2).
-i]
zl
An(i)
.
has no BPE’s in 5E 1 as is shown
in [i].
Now replace a suitable number of the discs
(i)
(i)c
to obtain a compact set E 2 such that 0 is the
G
j=2 J’
(see [1]).
only boundary point of E 2 that is a BPE for
B
n
B
n
R2(E2
If y e Int E 2, let norm(y)
i.
denote the norm of "evaluation at y" as a linear functional on
This can be done so that Int E
2
D
R2(E2
Then if
[yk]
c
Di,
and
Yk
on D. would be a BPE for
; otherwise some point
Di’ nrm(Yk)
R 2(E 2)
For each i choose an open disc D.1 c D.1 such that D.1 and D.1
are concentric and such that if y
2
space R (E 2
D’i
is greater than i.
D.
l
D’i’
then norm(y) for the
E. WOLF
370
Now let E
E
D
2
i=l
i
The radii of the D 1 are so small that 0 is also a BPE for
R2(E).
[yn]
Let
norm(y n)
be any sequence in Int E such that
norm of "evaluation at
on
Yn
> 0 is there a 6 > 0 such that if
lynl
R2(E).
< 6,
0.
Yn
Let
Then for no
f(0)lllfIl2
f(y n)
R(E).
for all f
EXAMPLE 3.
Let
be an admissible function.
set E in the same way that the set E
Obtain a compact
was obtained in Example 2 so
2
that:
(i)
is centered at 3
D.
1
(3. 2 -i-2
r.
1
22n
(2)
n=0
n=0
Let
Yi
3-i. (lyil) lyi
ClYilfor
all i.
3
-i-2
2 -i-2
and has radius
-i-2
#(2-n)-2C2(An(0)
22nc 2 (A n (x)
(3)
2
2
E)
-E) <
for x
,
and
0
x
D
i"
rl,Dist(Yi,E) a
that Dist(Yi,SE)
to the sequence [yi]
Then by the choice of
But there is no C > 0 such
Hence Corollary 2.2 applies
but Theorem 2.2 does not.
REFERENCES
2
I.
Fernstr6m,
2.
Fernstrm,
3.
Meyers, N. G., A theory of capacities for potentials of functions
Some remarks on the space R (E)
C.
University of Stockholm 1982.
C. and Polking, J. C., Bounded point evaluations and
p
approximation in L by solutions of elliptic partial
differential equations. J. Functional Analysis, 28, 1-20
(1978).
in Lebesgue classes, Math. Scand., 26
4.
5.
Math. Reports,
(1970)
255-292.
Wolf, E., Bounded point evaluations and smoothness properties of
functions in RP(x), Trans. Amer. Math. Soc. 238 (1978),
71-88.
2
Wolf, E., Smoothness properties of functions in R (X) at certain
boundary points, Internat. J. Math. and Math Sci. 2 (1979)
415-426.