I nternat. J. Math. & Mh Sci.
Vol. 4 No. 4 (1981)661-665
661
RATIONAL MODULES AND HIGHER ORDER
CAUCHY TRANSFORMS
JAMES LI-MING WANG
Department of Mathematics
The University of Alabama
University, Alabama 35486
(Received March 24, 1981)
ABSTRACT.
We apply the higher order Cauchy transforms to describe the closures of
rational modules with respect to the L
p
norms, the uniform norm and different
Lipschitz norms on a compact set in the plane.
KEY WORDS AND PHRASES.
Rational Module, Caucy
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
Transform
Primary OA 82, 46799
INTRODUCTION.
Let X be a compact subset of the complex
lane.
Let the module R(X)P
m
be the
m
space
{rO(z)+ rl(z) +
m}
r (z)
m
where each r. is a rational function with poles off X.
1
In recent years the problem of approximation by rational functions in the
areal mean has received great attention (see e.g.
problem of uniform approximation (see
[i], [3], [8])
[5], [12]), but
as well as the
little work has been devoted
to the corresponding problem of approximation by functions in a rational module
due to the lack of analyticity, perhaps.
The concept of rational modules arises
in a natural fashion when one attempts to study rational approximation in
Lipschitz norms.
In [i0],
O’Farrell studied the relation of the problems of
approximation by rational modules in different
Lipschitz norms, and in the uniform
norms, etc., to one another.
aut.hor
Not long ago the
proved in
[13]
that (X)P
1
662
J.L. WANG
LP(x)
is dense in
<_ p
for all 1
and R(X)P
<
2
is dense in
C(X) if X has no inte-
For an introductory survey of rational modules and all the standard
rlor.
notations, we refer the readers to the paper of O’Farrell
[i0].
However, the presence of an interior really complicates the situation.
LP(x)
be the closed subspace of
[4]
which consists of functions analytic in the
In thls note we employ a method of proof which goes back to
interior of X.
Brennan
LP(x)
Let
R(X)m
to describe the closures of
terms of the Cauchy transforms of functions in
LP(x) and
LP(x) for an
in
C(X) respectively in
arbitrary compact sub-
set X.
The author is indebted to J. Brennan for valuable conversations and
correspondences.
2
PRELIMINARIES
o
If X is a compact set, X will be its
The following notations will be used.
If V is any space of functions on X,
interior, X its topological boundary.
[V] p
will be the closure of V with respect to the
LP(x)
norm and
[V]u
the uniform
closure of V in C(X).
be a (finite Borel) measure on X.
Let
The Cauchy transform
is defined
by
(z)
for
gd/m,
We write V
d()
z
can be found in
Some basic properties for
write
I
[6].
If g is a function on X, we will
where dm denotes the two-dimenslonal Lebesgue measure.
iV
if
/
fay
0 for
iI
f in V and g
iV
if gdm
The following lemmas play important roles in this theory.
special case of the key lemma in
[i0], and lemma 2
i V.
Lemma i is a
is used by the author in
[13].
Throughout this paper, m will be a non-negatlve integer.
LEMMA i.
Let V be a measure on X.
Then V
i (X)Pm+1
if and only if
m
LEMMA 2.
If g e
LP(x),
then g is continuous when p > 2 and
is continuous
when i < p < 2.
If V is a space of functions on X, we put
regarded to be zero off X.
6
{f: f e V}.
Notice that f is
By induction, we can define the nth Cauchy transform:
663
RATIONAL MODULES AND CAUCHY TRANSFORMS
i
’,
,
on X.
nor V D
REMARK:
As a consequence of Lemma i, it is not hard to see that
[(x) + (x) +
[R(X)Pm] p
[(x) + (x) +
3.
=
In general, we don’t have V
2,3,4,
m
_l
+ (x)
+
(x)]p and [R(X)%] u
for all m > i, i < p <
u
MAIN RESULTS.
It is well known that
[Ii]
Sinanjan
for all i < p < 2
in the case of no interior, to Bers
[7]
and to Havin
LP(x)
a
[(X)]p
for I < p < 2.
[2]
This is due to
i (and open
for p
sets),
For p > 2, however, (X) is not always dense in
LP(x).
Some necessary and sufficient conditions are given by Brennan
Hedberg
[8],
[3]
etc., in terms of various capacities.
In this section, we will show that the higher order Cauchy transforms of (X)
and
LP(x)
a
always have the same closure in
Let X be a compact set.
THEOREM 3.
+
[(X) + LP(x)
a
since the arguments for m >
It is clear that
Then
for all m >
LP(x)]
a
p
[R(X)PI] p c
(modulo (X)), for all i < p <
[R(X)Pm] p
1 and I < p <
-i
+
q-i
1 such that g
^] p by
[(X) + LP(x)
a
Lemma i implies
Lemma 2.
J. Brennan
that both
that
fg
(X)PI.
i (X)
[4]
Lq(x),
0 off X.
0 everywhere on X.
i < q
<_
To
,
LP(X)a
Also g is continuous by
An argument similar to
using the theory of singular integrals and Schwartz lemma concludes
and g vanishes almost everywhere on X.
dm
for all
the remark in section 2.
We shall show that g
and therefore
It follows that g
p
I are similar.
prove the other inclusion, we let g be any function in
p
.
.
[(X) + LP(x)
a
[(X)PI] p
It is enough to show that
PROOF.
1 < p <
+
LP(x)
0 for all f in
a sequence of functions
writing [ for /x
On,
+ i/y,
0
LP(x)’a
Again, a similar construction in
n
1,2,
-Pn
+ g in the
lira
Hence it suffices to prove
f f(-p
such that each
n
Lq(x)
dm
dm
gives
o
has support in X and,
Thus for any f in
norm.
foX f
Pn
[4]
LP(x)
a
664
J.L. WANG
The following theorem shows that, among other things, every nth Cauchy trans-
form of a function in
R(X)m
LP(x)
(n
for all m > 2, 2 < p <
Let X be a compact set.
THEOREM 4.
R(X)^ +
.
2,...,m) belongs to the uniform closure of
ePa(X) +
+
Then
[R(X)
m]u .
LP(x)
]a u for all m _> 2 and 2
[(X) +
< p <
q]
L oc for all
This follows from heorem 3 and the fact that
PROOF
i < q < 2 and any measure
on X (see
[6],p.37)
We may consider the Cauchy transform as a linear map from C c
the space of
infinitely dlfferentiable functions on the complex plane with compact support,
into C
In
[I0], O’Farrell
obtains various pairs of norms so that the Cauchy
transform is bounded, and thus we have the following corollaries.
Let X be a compact set.
COROLLARY 5.
Then
for all
m >
2, 2 < p <
and 0 <
COROLLARY 6.
[(x) +
[v]+/-p
V]llp
Let X be a compact set.
Then
[R (X)L]
i
I where V is any space of functions on X such that
[(x)
REMARK.
< i.
m_+/-p
for each fixed m and 0 <
< 1
L. I. Hedberg has pointed out to the author that the problem of
approximation in L
p
by functions in
R(X)1
approximation by harmonic functions (see
is closely related to the problem of
[9]).
REFERENCES
i.
T. BAGBY, Quasi topologies and rational approximation, J. Functional
Analysis i0 (1972), 259-268.
2.
L. BERS, An approximation theorem, J. Analyse Math. 14 (1965), 1-4.
3.
J. BRENNAN, Invariant subspaces and rational approximation, J. Functional
Analysis 7 (1971), 285-310.
4.
J. BRENNAN, Invariant subspaces and weighted polynomial approximation,
Ark. mat. ii (1973), 167-189.
5.
T. W. GAMELIN, "Uniform algebras", Prentice Hall, N. J., 1969.
6.
J. GARNETT, "Analytic capacity and measure
No. 297, 1973.
7.
V. P. HAVIN, Approximation in the mean by analytic functions, Soviet Math.
Dokl. 9 (1968), 245-248.
Springer Lecture Notes,
RATIONAL MODULES AND CAUCHY TRANSFORMS
665
8.
L. I. HEDBERG, Approximations in the mean by analytic functions, Trans, Amer.
Math. Soc. 163 (1972), 157-171.
9.
L. I. HEDBERG, Two approximation problems in function spaces, Ark. Mat.
16 (1978), 51-81.
i0.
A. O’FARRELL, Annihilators of rational modules. J. Functional Analysis
19(1975) 373-389.
ii.
S. O. SINANJAN, The uniqueness property of analytic functions on closed sets
without interior points, Siblrsk. Mat. Z. 6 (1965), 1365-1381.
12.
A. G. VITUSHKIN, The analytic capacity of sets in problems of approximation
theory, Russian Math. Surveys, 22 (1967), 139-200.
13.
J. WANG, Approximation by rational modules on nowhere dense sets, Pac. J.
Math., 80 (1979), 293-295.