Rational interpolation with restricted poles

JOURNAL

OF APPROXIMATION

THEORY

7, 1-7 (1973)

Rational Interpolation

with Restricted

Poles*

THOMAS BAGBY Department

of Mathematics, Indiana U&e&y, Communicated

Bloomington,

Indiana 47401

by Joseph L. Walsh

Received January 5, 1970

1. SUMMARY OF RESULTS In this paper we discussinterpolation to an analytic function on a compact set in the extended complex plane P, by rational functions, whose poles lie in a disjoint compact set. To expedite the statement of theorems we agree that for the rest of the paper A and B will denote disjoint compact setsin P, and R will denote P - (A u B). An interpolation scheme {ani , bni} consists of a collection of points a,i and bni in P defined for every positive integer n and every positive integer i < n, with the requirement that ani # b,j for 1 < i, j < n. The associatedpotential function u,,,(z) is defined (unless t and z simultaneously lie in the set ia n1>.‘.,arm} or in the set {bnl ,..., b,,}) by the formula

%tW = -i i log lb, ani, t, bnilI , 2=1

where the brackets denote the cross-ratio. In some discussions the reference point t is fixed, and we simplify the notation by writing u, = ~4,~~ . Let (a,, , b,i} be an interpolation scheme, and f an analytic function defined on an open set containing the points a,, ,..., arm; we say that the (n - 1)-degree rational function r,, interpolates to f via the interpolation scheme provided r, coincides with f at the points a,, ,..., arm and has the poles b,, ,..., b, n-l , with the usual agreementsconcerning multiplicities. A central problem is to relate the interpolating rational functions and the associatedpotentials. An interpolation scheme{a,, , bmi} lies on A, B if all points a,, are in A and all points bni are in B. The case B = {co] corresponds to the problem of polynomial approximation. This problem has been studied by many authors [6, Chapter 71, and the following definitive result is due to Walsh [6, Sections 7.2-7.41. * This work was supported by NSF Grant GP 11158. 1 Copyright All rights

0 1973 by Academic Press, Inc. of reproduction in any form reserved.

2

BAGBY

THEOREM 1. Let A be a nonempty compactum in P such that R = P - A is connected and regular for the Dirichlet problem and contains B = (a~>. Let u denote the unique harmonic function in R which has constant boundary values at A and satisfies lim,,,(u(z) + log 1z 1) = 0; and let V = u(A). Let {ani , co} be an interpolation scheme on A, B, and u, = nn,m the associated potentials. Then the following statements are equivalent. (la)

Asn+

cc wehaveinf,u,+

(lb)

As n + a3 we have u, + u untformly on compacta in R.

(1~) If f is any analytic function {a,i , OO},then P, + f untformly on A.

V. on A, and P, interpolates to ,f via

We turn now to the analogous theory for rational approximation. One setting for the theory [5; 6, Chapter 81 involves the “dual” problems of approximating analytic functions on A by rational functions having poles on B, and approximating analytic functions on B by rational functions having poles on A. Using a very general topological setting, Walsh [6, Section 8.3, Corollary 21 showed that a condition analogous to (lb) is sufficient for the dual approximation property, and the present author [ 1, Theorem 21 showed that Walsh’s condition, and a condition analogous to (I a), are both necessary and sufficient. The purpose of the present paper is to discuss analogous results for the “one-sided” problem of rational interpolation [3; 6, Chapter 91, where we are required to approximate analytic functions on A but not analytic functions on B, using a topological setting of similar generality. We summarize our results in the following two theorems. The first is a very general description of the interrelationship between the behavior of the potentials u, and the power of the interpolation scheme {ani , b,J for approximating analytic functions. THEOREM 2. Let A and B be nonempty, disjoint compacta in P such that P - A is connected and regular for the Dirichlet problem. Let {ani , b,J be an interpolation scheme on A, B and u, the associated potentials with respect to a fixed reference point in R = $ - (A u B). Then the following statements are equivalent.

(2a) If a subsequence unk converges untformly on compacta in R, then the limit function u has a constant boundary value at A. (2b) If f is any analytic ,function on A, and r, interpolates to f via (ani , bni}, then rn + f untformly on A. We next state a theorem which has a form closer to the forms of Theorem I and a theorem given by Shen in a different setting [3; 6, Section 9.91. For this we need to introduce some more notation. If P - A is regular for the Dirichlet

INTERPOLATION

WITH

RESTRICTED

POLES

3

problem, and if b and t are distinct points in P - A, we letpb,t be the unique harmonic function in P - A - {b} which has a constant boundary value at A, vanishes at the point t, and satisfies lii p&z) - log 1z - b I is finite, if b # co, l=$p,,,(z) + log 1z I is finite, if b = co. We let V,,, denote the constant boundary value ofp,,, at A. Now if {ani, b,i} is an interpolation scheme on A, B, then for each n the potential function pn,t = (l/n) x:ipO,i, t is harmonic throughout all of P - A except for the points b,<(i < n) and has the constant boundary value W,,, = (l/n) xi Vbnirt at A. In particular, if A and B are bounded, we may think heuristically of pn+ as the logarithmic potential due to a unit positive charge on A which has settled into equilibrium under the influence of a fixed unit negative charge equally divided among the points b,i (i < n). In some discussions the reference point t is fixed, and we simplify the notation by writing p,, = pnSt and W, = W,,, . THEOREM 3. Let the hypotheses of Theorem 2 hold, and let the data u, , pn , W, be taken with respect to the samefixed referencepoint in R. Then thefollowing statementsare equivalent.

(3a) (3b) (3~) {ani , b,i},

As n + 03 we have inf, u, - W, + 0. As n + co we have u, - pn + 0 untformIy on compactain R. If f is any analytic function on A, and r, interpolates to f via then r, + f untformly on A.

Theorem 2 gives a general characterization, in terms of the associated potentials u, , of the interpolation schemes{a,i , b,i} which can be used to approximate every analytic function on A. On the other hand, if poles bni are prescribed, we may use them to compute the data pn and W,; then Theorem 3 gives a characterization, in terms of the associated potentials U n, of the interpolation points ani which can be used with these poles to approximate every analytic function on A. We will find it convenient to give a unified proof of these theorems in Section 3, following some preliminary results on measure-theoretic potential theory in Section 2. Under the hypothesesof Theorems 2 and 3, there always exist interpolation schemes{a,i , bni} for which the equivalent statements of the theorems hold true. In fact, if any poles bni (i < n) are preassigned, then there exist points ani (i < n) such that the equivalent statements of the theorems hold true. This may be proved by methods of Shen [3; 6, Chapter 91 and Walsh

4

BAGBY

[6, Chapter 91, who studied the rate of convergence of rational functions interpolating to analytic functions. Part of the emphasis of the present paper is in obtaining converse theorems under the weakest possible assumptions, and in this senseour work is complementary to that of Shen and Walsh. 2. POTENTIAL THEORY IN THE PRESENCEOF A FIXED CHARGE In the proofs of Theorems 2 and 3 we will use a generalized form of potential theory which we describe in the present section. We assume that the disjoint compacta A and B are bounded, and that P - A is regular for the Dirichlet problem. In this paper the word measurewill signify a signed Bore1 measure on P. If E C [FDis compact we let &2’(E) denote the set of all unit positive measures with support in E. If u is any measure with bounded support, we introduce the potential function

wherever it is defined. If {ani , bni} is an interpolation scheme on A, B, we use the notation CL,,for the unit positive measure on A with l/n units of mass at a,( (i < n), and fllafor the unit positive measureon B with l/n units of mass at bmi(i < n). Then we clearly have u,,, = U, -B . The following theorem describes mathema&&ly a charge p which has settled into equilibrium on A, in the presenceof a fixed charge --v on B. It is analogous to a well known result of Frostman [2; 4, Chapter III], and may be proved by the sametechniques. THEOREM A. Let v E J%!(B).Then there exists a unique measurep E d(A) for which the reduced energy integral

assumesits minimum value K. Moreover, the potential u,-, is continuous at all points of [FD- B and u,-,(z) = Kfor all z E A. We indicate the dependence of p and K on v in this theorem by writing TV= LZ’[v]and K = Z[v]. There is a close connection between these concepts and those introduced in the preceding section. For example, if v is the discrete measure consisting of l/n units of mass at each point /Ini (i < n), then = uZtVland W,,, = ,X[v]. P?Z.m

INTERPOLATION WITH RESTRICTED POLES

The following convergencetheorem will be neededlater. THEOREM B. Let {vn} be a sequence of measures in J&‘(B), and suppose that v, -j v in the weak-star topology. Then 9[vn] -+ 2?[v] in the weak-star topology and X[v,J -+ Z[v].

Proof. We use the notation pn = Y[v,], p = .Y[v], K, = X[vn] and K = X[v]. Suppose we have a sequenceof indices n for which pL,converges in the weak-star topology and K, converges: pn -+ y and K, + C. We first

observe that aKz < 1 u,-,, dcL, and hence C <

s

u,-, dp = K.

On the other hand, we have for each z E A the estimate C = lim u~~-~~(z)= lim inf

s

log &

d&z - vn)(f>

b s log , z ; t , 4y - v)(t)

and hence K < j uY+ dy < C. Thus K = C, and since $ uY-”dy = K we have y = p. This proves Theorem B.

3. PROOF OF THEOREMS 2 AND 3 Since these theorems are invariant under Mobius transformations, we may assume that A and B are bounded sets and that the data u, , pn , W, are taken with the reference point at infinity. (3a) Z- (3b). If (3b) fails, we can find a sequenceof indices n such that - pn is bounded away from zero at some point of R. We can find a UT2 subsequenceof these indices, by the theory of normal families, such that U, - pn converges to a harmonic function g, uniformly on compact subsets of P - A. If E > 0 is arbitrary, we see from (3a) that U, - pn > --E in P - A for n sufficiently large. It follows that g 3 0 in P - A. However, g( co) = 0, sinceeach of the functions u, - p,, vanishes at co, and we conclude that g is identically zero in P - A. This is a contradiction.

BAGBY

(3b) 3 (2a). This follows from Theorem B. (2a) * (3~). If (3~) fails, we can find a sequenceof indices n such that sup, 1f - rc j is uniformly bounded away from zero. We can find a subsequence of these indices such that 01,+ 01and flla + /I in the weak-star topology. From (2a) we see that u,-,(z) + C as z E R approaches A. Since u,-, is subharmonic in P - A, we have sup, u,.+ = C, < C. Now if C, < C, < C, < C and if yj = {z E R : u,-, = C,}, then we have the following explicit formula [6, Section 8.1, Eq. (4)] for z E y3 :

Thus, if E > 0 is arbitrary, we have for large n s;p If-

r,l d sup IfY3

r,I

where L(yZ) denotesthe length of y2 . In particular we have supa 1f - r, / -+ 0, which is a contradiction. (3~) * (3a). We begin with the observation that for each n the function u?l- pn is harmonic in P - A and vanishes at co; we conclude that infA u, < W, . If (3a) fails, we can find a sequenceof indices n such that inf,, u, - W, is bounded away from zero. We can find a subsequenceof these indices for which ~ll~-+ 01and j$, --t p in the weak-star topology, and thus by Theorem B, pn converges to p = 9[fl] in the weak-star topology, and W, converges to W = .X[/3]. Let E > 0 be arbitrary. Since ZA,-~- u,-~ is harmonic in P - A and has the value 0 at co, we can find a point t E R - {co} such that u,-,(t) > W - E, and hence u,(t) > W - E for IZ sufficiently large. We now apply (3~) to the function f(z) = l/(t - z). We have the formula [6, Section 8.1, Eq. (3)]

1 rIi w - E.

INTERPOLATION WITH RESTRICTED POLES

7

Since E was arbitrary, we have lip inf u,(z) = W, a contradiction. REFERENCES 1. T. BAGBY, On interpolation by rational functions, Duke Math. J. 36 (1969), 95-104. 2. 0. FROSTMAN, Potentiel d’tquilibre et capacite des ensembles, Comm. Sem. Math. Lund

3 (1935), l-118. 3. Y. SHEN, On interpolation and approximation by rational functions with preassigned poles, J. Chinese Math. Sot. 1 (1936), 154-173. 4. M. TSUJI, “Potential Theory in Modern Function Theory,” Maruzen, Tokyo, 1959. 5. J. L. WALSH, A duality in interpolation to analytic functions by rational functions, Proc. Nat. Acud. Sci. U.S.A. 19 (1933), 1049-1053. 6. J. L. WALSH, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Sot. Coil. Publ. 20 (1935).