RATIONAL MODULES AND HIGHER ORDER CAUCHY TRANSFORMS

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.


INTRODUCTION.
Let X be a compact subset of the complex lane.Let the module R(X)P be the m m space {rO(z)+ rl(z) + r (z)m} m where each r. is a rational function with poles off X.
In recent years the problem of approximation by rational functions in the areal mean has received great attention (see e.g.[i], [3], [8]) as well as the problem of uniform approximation (see [5], [12]), but 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.Not long ago the aut.hor proved in [13] that (X)P 1 is dense in LP(x) for all 1 <_ p < and R(X)P 2 is dense in C(X) if X has no inte- rlor.For an introductory survey of rational modules and all the standard notations, we refer the readers to the paper of O'Farrell [i0].
However, the presence of an interior really complicates the situation.Let LP(x) be the closed subspace of LP(x) which consists of functions analytic in the interior of X.In thls note we employ a method of proof which goes back to Brennan [4] to describe the closures of R(X) in LP(x) and C(X) respectively in m terms of the Cauchy transforms of functions in LP(x) for an arbitrary compact sub- set X.
The author is indebted to J. Brennan for valuable conversations and correspondences.

PRELIMINARIES o
The following notations will be used.If X is a compact set, X will be its interior, X its topological boundary.If V is any space of functions on X, [V] will be the closure of V with respect to the LP(x) norm and [V] the uniform p u closure of V in C(X).
Let be a (finite Borel) measure on X.The Cauchy transform is defined by (z) I d() z Some basic properties for can be found in [6].If g is a function on X, we will write for gd/m, where dm denotes the two-dimenslonal Lebesgue measure.
We write V i V if / fay 0 for iI f in V and g i V if gdm i V.
The following lemmas play important roles in this theory.Lemma i is a special case of the key lemma in [i0], and lemma 2 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 If V is a space of functions on X, we put 6 {f: f e V}.Notice that f is regarded to be zero off X.By induction, we can define the nth Cauchy transform: i ' _l m 2,3,4, In general, we don't have V = , nor V D , on X.
REMARK: As a consequence of Lemma i, it is not hard to see that for all m > i, i < p < [(x) + (x) + + (x) u 3. MAIN RESULTS.

LP(x) for all i < p < 2 This is due to
It is well known that [(X)]p a Sinanjan [Ii] in the case of no interior, to Bers [2] for p i (and open sets), and to Havin [7] for I < p < 2. For p > 2, however, (X) is not always dense in LP(x).Some necessary and sufficient conditions are given by Brennan [3] Hedberg [8], etc., in terms of various capacities.
In this section, we will show that the higher order Cauchy transforms of (X) and LP(x) always have the same closure in LP(x) (modulo (X)), for all i < p < .a THEOREM 3. Let X be a compact set.Then [R(X)Pm] p [(X) + LP(x) + + LP(x)] for all m > 1 and I < p < .It is enough to show that [(X)PI] p a p 1 < p < since the arguments for m > I are similar.
c [(X) + LP(x) ^] by the remark in section 2. To prove the other inclusion, we let g be any function in Lq(x), i < q <_ , -i q-i p + 1 such that g (X)P I.We shall show that g LP(X)a Lemma i implies i (X) and therefore 0 off X. Also g is continuous by Lemma 2. It follows that g 0 everywhere on X.An argument similar to J. Brennan [4] using the theory of singular integrals and Schwartz lemma concludes that both and g vanishes almost everywhere on X. Hence it suffices to prove that f g dm 0 for all f in LP(x)'a Again, a similar construction in [4] gives o has support in X and, a sequence of functions On, n 1,2, such that each Pn writing [ for /x + i/y, -Pn + g in the Lq(x) norm.Thus for any f in LP(x) a 0 lira f f(-p dm fo f dm n X The following theorem shows that, among other things, every nth Cauchy trans- form of a function in LP(x) (n 2,...,m) belongs to the uniform closure of R(X) for all m > 2, 2 < p < .m m [(X) + THEOREM 4. Let X be a compact set.Then [R(X) ]u R(X)^+ ePa(X) + + LP(x) ]a u for all m _> 2 and 2 < p < .
q] for all PROOF This follows from heorem 3 and the fact that L oc i < q < 2 and any measure on X (see [6],p.37) We may consider the Cauchy transform as a linear map from C the space of c 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.
for all COROLLARY 6.Let X be a compact set.Then [R (X)L] i V]llp I where V is any space of functions on X such that [(x) + m for each fixed m and 0 < < 1 [v]+/-p [(x) _+/-p REMARK.L. I. Hedberg has pointed out to the author that the problem of approximation in L p by functions in R(X) 1 is closely related to the problem of approximation by harmonic functions (see [9]).