Iern. J. Math. & Mah. Sci.
Vol. 6 No. 3 (1983)459-466
459
SOME REMARKS ON THE SPACE R2(E)
CLAES FERNSTRM
Department of Mathematics
Unive rs ity o f S to ckho im
SWEDEN
(Received on April 26, 1982 and in revised form on September 28, 1982)
ABSTRACT.
Let E be a compact subset of the complex plane.
We denote by R(E) the
algebra consisting of the rational functions with poles off E.
LP(E),
1 < p <
,
is denoted by
RP(E).
The closure of R(E) in
section 2 we introduce the notion of weak bounded point evaluation of order
identify the existence of a weak bounded point evaluation of order
R2(E)
necessary and sufficient condition for
E such that
R2(E)
2.
In this paper we consider the case p
L2(E).
R2(E)
and
> i, as a
We also construct a compact set
has an isolated bounded point evaluation.
the smoothness properties of functions in
,
B
In
In section 3 we examine
at those points which admit bounded
point evaluations.
KEY WORDS AND PHRASES. Rational funns, compact set, LP-spaces, bonded point
evaluation, weak bounded point evaluation, Bessel capacity.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
30A98.
INTRODUCTION.
Let E be a compact subset of the complex plane E.
LP(E)
For each p, i <_ p
be the linear space of all complex valued functions f for which
fl
<
p
,
let
is inte-
grable with the usual norm
dm(z)}
1/p
where m denotes the two dimensional
Lebesque measure.
Denote by R(E) the subspace of all rational functions having no
poles on E and let
RP(E)
be the closure of R(E) in
be a bounded point evaluation (BPE) for
RP(E),
LP(E).
A point z 0
E is said to
if there is a constant F such that
460
C.
dm(z)ll/P
In [I] Brennan showed that
RP(E)
is a BPE for
Fernstrm
FERNSTRM
for all f
LP(E),
RP(E)
R(E)
2, if and only if no point of E
p
2 (See
The theorem is not true for p
and Polking [3].)
Fernstrm [2]
or
In this paper we show that if the right hand side of (i)
is made slightly larger a corresponding theorem is true for p
We also show that
2.
this theorem is best possible.
If z
f(z
E is a BPE for
0
[ f(z)g(z)dm(z)
0)
RPkE
there is a function g
for all f e R(E).
LqkE
i_ + 1
q
P
i, such that
The function g is called a representing
E
function for z
0.
that a set A, A C
one when
Let B(z,) denote the ball with radius 8 and centre at z.
,
has full area density at z if
(z_z0)-s (Iz-z01)
-i
g
Lq (E),
RP(E),
2 < p, represented by g e
where s is a nonnegative integer and
decreasing function such that r qb(r) -1’ 0 when
e E
0
r’0.
Lq(E)
and
is a non-
Then for every e > 0 there is a
in E having full area density at z such that for every f e
0
R(E) and for all
0
If()
<_
tends to
tends to zero.
Suppose now that z 0 is a BPE for
set E
m(AB(z,))m(B(z,))
We say
-i
f(z
0
f’(Zo) (
l!
Zo Is (I
Zol)
I
z
f(S)(z0)(r
s!
I/p
0
if(z)
p
dm(z
E
)I
z
0
)sl
This theorem is due to Wolf
We shall show that the theorem of Wolf is not true for p
2.
[8].
We shall also
show that a slightly weaker result is true and that this result is best possible.
The main tool to show this is to construct a compact set E with exactly one bounded
point derivation for
order s for
RP(E)
A point z
if the map f
linear functional on
2.
R2(E).
0
f(S)(z 0),
E is a bounded point derivation (BPD) of
f
R(E), extends from R(E)
to a bounded
RP(E).
BPE’S AND APPROXIMATION IN THE MEAN BY RATIONAL FUNCTIONS.
Denote the Bessel kernel of order one by G where G is defined in terms of its Fourier
transform by
REMARKS ON THE SPACE R2 (E)
(Z)
L2()
For f
(i +
IZl 2)
-1/2
we define the potential
u
The Bessel capacity C
(z)
am
G(z-’r) f(’r)
for an arbitrary set X, XCI2, is defined by
2
din(z), where the nfmum
u f (z)
461
i for all z s X.
s
taken over all f
The set function C
2
L()
C2(X)
[nf
such that f(z)
0 and
is subadditive, increasing, translation
invariant and
C 2(B(z,6))
=
6
og
J
O
< i.
For further details about this capacity see Meyers [5].
The
BPD’s
-n-I
<
Theorem 2.1
can be described by the Besse! capacity.
Iz-z01
_<
2
-n
Let
An(Z 0)
denote the annulus
The following theorem is proved in [3]:
Then z is a BPD of order s for R2(E) if and
Let E be a compact set.
only if
n=
Definition
22n(s+l) C2(An(Z)
E) <
Set
log
L z (z)
iZ_ZoI
for
Z-ZoI
<
for
Z-Zol
>
0
1
e
A point z 0 e E is called a weak bounded point evaluation (w BPE) of
order $, S > 0
for
R2(E)
if there is a constant F such that
for all f e R(E)
We are now going to generalize theorem 2.1 in two directions.
Theorem 2.2
Let s be a nonnegative integer and E a compact set.
is a BPE for
R2(E)
represented by g e
L2(E) and
Suppose that z 0
that
is a positive,
nondecreasing function defined on (0,) such that r (r)
decreasing and tends to zero when r
function g e
L2(E)
g
such that
(z-zo)- ([ Z-Zo[)
e L 2 (E)
-)
0
+
Then z
0
-i
is non-
is represented
by
462
FERNSTRM
C.
if and only if
22n(s+l)
(2
-n -2
C
E)
2(An(z 0)
<
.
n=0
Then z is a w BPE of order
Lot E be a compact set.
Theorem 2.3
B
for R2(E) if and
only if
n
-g2 2n C2(A
n
(z) -E) <
n=l
The proofs of theorem 2.2 and theorem 2.3 are almost the same as the proof of theorem
2.1.
Wolf proved in [8] that the condition
We omit the proofs.
22n(s+l) (2-n) -2 C 2(An(z 0
E))
<
is necessary in theorem 2.2.
n=0
The compact sets E for which
L2(E)
can be described in terms of the
The following theorem is proved in Hedberg [4] and Polking [6].
Bessel Capacity.
Theorem 2.4
R2(E)
Let E be a compact set. Then the following are equivalent.
L 2(E).
(i) Re(E)
(ii) C 2(B(z,6)-E)
(iii) lim sup
C
2(B(z,6))
C2 (B(z,) -E) >
for all balls
B(z,6).
0 for all z.
If we combine theorem 2.3 and theorem 2.4 we get the following theorem.
Theorem 2.5
Let B > i and E be a compact set.
admits no w BPE of order
B
for
Then
L2(E)
R2(E)
if and only if E
R2(E).
Now we shall show that theorem 2.5 is not true for
B
< i.
We first need the following
theorem.
Theorem 2.6
There is a compact set E such that
E) < C 2(B(0,1/2))
(i) C 2(B(0,1/2)
(ii)
n
-I
n=l
22nC2(An(z
E)
for all z.
The proof is a modification of a proof in [2] or [3], where a weaker theorem is
Since we shall need the construction of E later, we give some details.
proved.
roof.
There are constants F
Fl(log
)
-i
<_
C2(B(z,6))
F
I and 2 such that
_<
F2(log )-i
for all 6, 6 <
60
<
I.
REMARKS ON THE SPACE R 2 (E)
Choose
,
Let A
0
C2 (B(0,1/2)).
<
log2n
Cover A
be the closed unit square with centre at the origin.
-n
with side 2
disc
i, such that
>
n
463
Bn(i)
exp(-c4n
Call the squares A
n
Bn
such that
log2n).
n
(i)
An
and
(i),
(i)
1,2,..., 4
i
n
In every A
0
with 4
(i)
n
n
squares
put an open
have the same centre and the radius of
Bn
(i)
is
Repeat the construction for all n, n > 2.
Set
n
4
n= 2
The subadditivity of C
(i)
n
i=l
now gives (i).
2
In order to prove (ii) it is enough to prove
F1
for all n, n
E) >
C2(A
32 4nlog n
2
(i)
Consider all B
n(i)
k(i)
We get 4
<_ k _<
n
discs with radius
exp(-4n+(n+Z) log2 (n+Z))
Call the discs
D (r)
n
n2-n+l
4
1,2
r
(2.1)
O
Bk(i._ An
such that
n
> n
0 < Z < n
2
n.
-1
3
Thus
2
F1
a4 n
2
jn
1
"=
j
log’q <-
Set Dn
t
D
r
C2(Dn
(r)
<
F2
4
jn
n
.=
1
j
logZj
(r)
n
Since the distances between the discs are large compared to their radii, it can be
shown that
C2(Dn) >_
Z C2(Dn(r))
if n is large.
r
(See theorem 2’ in [2] or theorem 2 in [3] for a proof.)
Thus if n is large,
C2(An (i).
which is (2.1)
E) >
C2(Dn)
2
F
n
84
jn
1
>---n
1
J
log2j
F1
16a4nlog
n
C.
464
Theorem 2.7
FERNSTRM
There is a compact set E such that
e 2(E)
(i)
R
2(E)
2
(ii) E has no w BPE of order one for R (E).
Proof
3.
The theorem follows immediately from theorem 2.3, 2.4, and 2.6.
BPE’S AND S!IOOTHNESS PROPERTIES OF FUNCTIONS IN
R2(E).
In this section we treat the theorem of Wolf mentioned in the introduction for the
2.
case p
Theorem 3.1
Let
r L
be a positive, nondecreasing function defined on (0,) such that
0(r)
(r)
-I
is nondecreasing and r L
Suppose that z 0 is a BPE for
(z-z 0)
-s
(Iz-z01)
-i
Zo,
-I
0 when r
0+
represented by g and
L (E), where s is a nonnegative integer.
g
> 0 there is a set E
such that for every f
(r-Zo)
i!
I
f()
E
0 in E, having full area
R(E) and every
f(s) (Zo)
f- (z0)
f(z0)
f()
(r)
2
Then for every $ > i and
density at
R2(E)
0(r)
s!
(-Zo)
E
T
0
s
12 Lz0 (z)
The proof of theorem 3.1 is only a minor modification of the proof of theorem 4.1 in
[8].
l.breover, there is a proof of theorem 3.1 for B
2 in Wolf [7].
We omit the
proof.
Remark.
E (the boundary of E) be both a BPE for
Let z
0
sector contained in Int
secotr.
Let s
E.
R2(E)
and the vertex of a
Let L be a line which passes through z 0 and bisects the
0 and let
be as in theorem 2.2.
For those y e L
E
that are
sufficiently near z 0 Wolf showed in [9] that
f(z)]2
If(y)-f(zo) _<
dm(z)l
1/2
for all f
Our next step is to prove that theorem 3.1 is not true for B
I.
R(E).
We first need a
theorem, which we think is interesting in itself.
Theorem 3.2
Then there is a compact set E such that
Let s be a nonnegative integer.
(i)
(ii)
n -I
E
22n
C2(An(Z)-E)
22n(s+l) C2(An(O)-E)
if z # 0.
<
Proof
.
R2(E)
REMARKS ON THE SPACE
465
Let B k
We shall modify the set constructed in the proof of theorem 2.6.
Let all B
denote the same discs as in that proof.
Jl
Choose
C2(AIj)
E
J>Jl
< 2
diam(Alj i
and
Suppose that we have chosen
An+2
Choose
2’
An+3
and
diam(An+I Jn+l
-1
Jn"
Let all
AIJI’
All
(k)
Bj
Anl
which intersect
,AnJn, be denoted by
Jn+l
C2
(An+l
< 2 -(n+l
j
< 2 -(n+3)
Let A0 be the closed unit square with centre at the origin.
k) such that B (k)
l<n< and
union of all
B
Anm
, 22n(s+l)
We have
C 2(An(0)
E) <_
0.
A 0- (The
Set E
l<m<Jn)
-n
2
<
n=l
n=!
Let z
An+l(0)
3’’’’ so that their diameters are decreasing.
Jn+l so that
22(n+I)(I) E
>
< 2
-3
Jl
and which do not coincide with
i,
AI(0)
so that
22(s+l)
An+l
which intersect
so that their diameters are decreasing.
All, AI2, AI3,...
be denoted by
-.
k)j
If
is large all
(k)
Bj
(k)
B]
C
A(z)
differ
fro
Anm
< n <
and i < m < j
I n._
Now exactly as in proof of theorem 2.6 it follows
:.
Corollary 3.3
Proof
Remark
n
-I
22n C2(An(Z)-EI)
q.e.d.
2
There is a compact set E with exactly one BPD of order s for R (E).
Just combine theorem 3.2 and 2.1.
The situation for p # 2 is different.
all points z
Theorem 3.4
E, E compact, are
not
BPE for
In [i] Brennan showed that if almost
RP(E),
Let s be a nonnegative integer and
E admits no
BPE’s
2
for R (E).
be as in theorem 2.2.
Then there
is a compact set E such that
(i) z 0 is a BPE for R2(E).
(ii) There is a representing function g for z that satisfies
0
466
C.
FEPdqS
TRM
2
(Iz-z01)-i geL(E)
(z-z0)-s
(iii) For every
T
e
E,
and every positive integer n there is a
z0
T
function f e R(E) such that
f(s) (z0)
f, (z0)
f()-f(z0)- i (r-z0)
> n
Proof
If(z)
Lz0 (z)dm(z)
s’
(-z0)Sl
>
1 /2
Theorem 3.2 gives that there is a compact set E such that
n -I
22nC 2(An(z)
E)
z
z0
n=l
2n (s+l)
(2-n) -2 C 2(An(z0)
E)
<
n=l
Now theorem 2.1 gives (i) and theorem 2.2 gives (ii).
Moreover theorem 2.1 gives that
2
z 0 is a BPD of order s for R (E) and theorem 2.3 that
is not a w BPE of order i for
2(E)
R
This gives (iii).
REFERENCES
i.
Brennan, J. E.,
Analysis, 7
2.
Invariant Subspaces and Rational Approximation, J. Functional
(1971), 285-310.
Bounded Point Evaluations and Approximation in L p by Analytic
Functions, in "Spaces of Analytic Functions Kristiansand, Norway 1975"
65-68, Lecture Notes in Mathematics No 512, Springer-Verlag, Berlin
Fernstrm, C.,
196.
3.
FernstrSm,
C and Polking, J. C., Boundee Point Evaluations and Approximation in
by Solutions of Elliptic Partial Differential Equations. ’.J. Functional
Analysis, 28, 1-20 (1978).
LP
4.
Hedberg, L. I.,
Functions,
5.
Meyers, N. G., A Theory of Capacities for Potentials of Functions in Lebesgue
Classes, Math. Scand., 26 (1970), 255-292.
6.
Polking, J. C. Approximation in L p by Solutions of Elliptic Partial Differential
Equations, Amer. J. Math., 94 (1972), 1231-1244.
7.
Wolf, E.,
RP(x),
Non Linear Potentials and Approximation in the Mean by Analytic
Math. Z., 129 (1972) 299-319.
Bounded Point Evalutions and Smoothness Properties of Functions in
Doctoral Dissertation, Brown University, Providence, R.I., 1976.
8.
Wolf,.. Bounded Point Evaluations
R(X ), Tra___ns. Amer. Math. Soc.
9.
at Certain Boundary
Wolf, E., Smoothness Properties of Functions in
Points, Internat. J. Math. and Math. Sci., 2 (1979), 415-426.
and Smoothness Properties of Functions in
238 (1978), 71-88.
R2(X)