Annals of Mathematics

Annals of Mathematics Asymptotically Neutral Distributions of Electrons and Polynomial Approximation Author(s): Jacob Korevaar Source: The Annals of Mathematics, Second Series, Vol. 80, No. 3 (Nov., 1964), pp. 403-410 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970655. Accessed: 30/06/2011 08:40 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at. http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at. http://www.jstor.org/action/showpublisher?publishercode=annals.. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics. http://www.jstor.org

Asymptotically Neutral Distributions of Electrons and Polynomial Approximation* By JACOB KOREVAAR 1. Introduction and results Let D be a bounded simply connected region (connected open set) in the complex plane. We wish to approximate holomorphic functions in D by polynomials; the error must become uniformly small on every fixed closed subset of D. By Runge's theorem such approximation is always possible. However, we impose an additional condition: the zeros of the approximating polynomials must belong to a prescribed set P. Polynomials whose zeros lie in P will be referred to as F-polynomials. It will be assumed for simplicity that F and D are disjoint. P will be called a polynomial approximation set relative to D if and only if every zero free holomorphic function in D can be approximated by F-polynomials. It may be observed that, if F is a polynomial approximation set and J is a simple arc in D, then every continuous function on J can be uniformly approximated by F-polynomials. G. R. MacLane [4] has shown that every rectifiable Jordan curve is a polynomial approximation set relative to its interior. Recently M. D. Thompson [5] gave a new proof of this result, extending it to a different special class of Jordan curves. M.D. Thompson [5] and the author [3] have also considered certain unbounded sets F. In the present paper we determine all bounded sets F that can serve as polynomial approximation sets relative to D. The discussion will be based on the notion of "asymptotically neutral families". A family of finite sequences (1.1) Zn1, Zn2,... Zen (with n = no co) of points ("electrons") on F will be called asymptotically neutral relative to D if and only if (1.2) Z. - 0 as n = nj o, ql,kz uniformly on every closed subset of D. It may be noted that the sum in (1.2) represents the (complex conjugate of the) electrostatic field at z due to * Work supported by contract NONR 22511 at Stanford University.

404 JACOB KOREVAAR electrons at the points (1.1), assuming forces inversely proportional to the distance. MAIN THEOREM. Let D be a bounded simply connected region, F a bounded set disjoint from D. Then each of the following four statements implies the others: ( i ) F is a polynomial approximation set relative to D; (ii) for some point C E F, the function 1/(C - z) can be approximated by F-polynomials in D; (iii) F contains an asymptotically neutral family relative to D; (iv) clos F divides the plane, and D belongs to a bounded component of its complement. COROLLARY. Let D be an arbitrary bounded simply connected region. Then (a) if E is any given closed subset of D, it is possible to place a large number of electrons on the boundary OD in such a way that the resulting electrostatic field is uniformly weak on E; (/l) the boundary OD is a polynomial approximation set relative to D. The main theorem has an extension to the case of unbounded sets F which belong to a half-plane; the corollary remains valid for arbitrary bounded open sets D with connected complement. These extensions will be discussed in a separate paper. It is an interesting physical problem if part (a) of the corollary can be extended to the case of arbitrary, or at least fairly general, three-dimensional bounded regions D with connected complement. Indeed, suppose such a region D is the interior of a negatively charged hollow conductor SD. Experiments indicate that the charges are distributed over SD in such a way that there is no measurable electrostatic field in the interior D. Can this phenomenon be explained by a model where the charges consist entirely of free electrons? The author wishes to thank M. D. Thompson and S. Hellerstein who were in on the early stages of the work, and above all Professor Beurling whose detailed comments on the manuscript led to great simplifications; in particular, the present proof of Theorem 5.1 is due to him. 2. The implications (i) (ii) (iii) - (iv) That (i) implies (ii) is trivial. (ii) (iii). Suppose that for some point C E F the function 1/(C - z) can be approximated by F-polynomials, uniformly on every closed subset of D. Then, by logarithmic differentiation, the function - 1/(C - z) is the limit of finite sums of the form

DISTRIBUTIONS OF ELECTRONS AND POLYNOMIAL APPROXIMATION 405 (2.1) 1_, with XjkE. The number vj of terms in the sum (2.1) must tend to infinity, for, otherwise, a suitable subsequence of the sums (2.1) would tend to a finite sum 1l/(zk - z) with Zk C cios F. It follows that the sequences C9 Cjl9...s 9 civi j = 1, 29... form an asymptotically neutral family relative to D. (iii) (iv). Suppose that F contains an asymptotically neutral family (1.1) relative to D, and let U denote the union of the points occurring in the sequences (1.1). LEMMA 2.1. Clos U divides the plane, and D belongs to a bounded component of its complement. PROOF. Let D' be the component of the complement of clos U which contains D, and suppose that D' is unbounded. Define 1 1 (2.2) f.(z) - E z n = n- )co. n klznk -Z The fn(z) are holomorphic in D', uniformly bounded on any given compact subset, and tend to zero in D. Hence by Vitali's theorem, fn(z) 0 every- - where in D', contradicting the fact that at any finite point z0 far from the bounded set U, the fn(z) are bounded away from zero. Thus D' must be bounded. 3. Asymptotically neutral families If clos F contains an asymptotically neutral family, so does P. Hence in discussing the implication (iv) y (iii) it may be assumed that F is closed. Suppose now that F divides the plane, and that the component D+ of complement F which contains D is bounded. It will be shown that F has a fairly well-behaved subset which carries an asymptotically neutral family. Let Do be the unbounded component of complement clos D+. Then Do is a simply connected region which contains the point at infinity. The boundary F- = OD- is part of OD+, hence part of F. It may be observed that if F is a Jordan curve, then D+ = interior F, Do = exterior F, and F- F. THEOREM 3.1. F- contains an asymptotically neutral family relative to D. The points of any such family are dense on F-. For every point C E c-9 the function 1/(C - z) can be approximated by F--polynomials in D. The nth roots of unity, n = 1, 2,..., form an asymptotically neutral family on the unit circumference. Indeed, in this case the sum in (1.2) is

406 JACOB KOREVAAR equal to the negative of the logarithmic derivative of I - 1, and it is easy to see that this quantity tends to 0 uniformly on any disc z p < 1. In the general case we introduce the one-to-one conformal mapping z = D(W) of the exterior I w > 1 of the unit circle in the extended w-plane onto the region Do in the extended z-plane, normalized by the conditions b(oo) = co, V(co) > 0. We extend the mapping to the boundary by setting 4)(ei0) - limr 1, J(rei0) wherever the limit exists; for the exceptional points e"9 (which form a set of vanishing outer logarithmic capacity [1]), we take.i(ei0) equal to one of the radial limit values. Then all points z = b(ei0) belong to F-; it will be shown that there exist asymptotically neutral families on F- consisting of images of suitably shifted roots of unity. Images of roots of unity have been used before, notably as interpolation points by Fejer [2]. That they should play a role in the present problem is entirely natural. Indeed, writing (b(ei0) = (0) one has.b(rei0) p(0) as r I 1 for almost all 0, and boundedly on (0, 2wc); it follows that 27r 1 do= 1 dw2 1 =0 0 P (0) - z lwl=r>l (w)- Z iw (o) - z for all z E D. Thus if 9(0) is sufficiently well-behaved, the special Riemann sums (3.1 27r An (3.1) n, P~'9(27rk/n) - z will be good approximations, for large n, to the value 0 of the integral. For the special case where F = F- is a rectifiable Jordan curve, M. D. Thompson [5] has shown that the sums (3.1) are so small in the interior of F that the points 9(27ck/n), k 1, *--, n, n 1, 2, *--, form an asymptotically neutral family. 4. Proof of Theorem 3.1 (the implication (iv) -- (iii)) We first prove a simple lemma on infinite series. LEMMA 4.1. Let a(n) > 0, E'na(n) I < co. Then (4.1) lim infn NO n2 Ea1a(in) = -. PROOF. Suppose that (4.1) is false. Then there are numbers r > 0 and A such that for every prime p? A,

DISTRIBUTIONS OF ELECTRONS AND POLYNOMIAL APPROXIMATION 407 p,= la(vp)? rjp. Summing over p one obtains (4.2) L p(n)a(n) > 2 A'/P =a where p(n) is the sum of the different prime factors? A of n. Now p(n) <n, hence (4.2) contradicts the convergence of 10na(n). This contradiction establishes (4.1). We are now ready for the principal lemma. LEMMA 4.2. Let Do and 9(0) be as in? 3, let D* = complement clos Do, and define (4.3) An(0 Z) 1 zed*. k~=1 (0( + 2Wrk/n) - z Then there is a sequence of integers nj co such that for almost all 0, *40(s Z) ) 0as j co, uniformly on every closed subset of D*. PROOF. Let d)(w) be the function which maps I w > 1 onto D- defined in? 3. For z E D*, the equation t 1 b(u) ) z,1,zw defines a one-to-one conformal mapping of I w 1 > 1 onto a bounded region R(z) in the t-plane; the point w co corresponds to t = 0. The image R(z) will be contained in the disc I t < 1/a(z), where J(z) denotes the distance between z and Do. Thus by the area formula Setting Jq7n I c (z) I2 < 1/J2(z) - (4.4) D* 1 cn(z) 122(z)da(z) = a(- ) it follows that the series 'iina(n) converges. We now observe that 1-1 I ch(z)e (( - z hence by Parseval's formula + (0, z) - n{jc(z)e-in0 + C2,(z)e 2inO +... } 2 r (4.5) do I *.(0, z) 12J2(z)da(z) =2n2 27 a(vn). h By Lemma 4.1, the right-hand side of (4.5) tends to zero for a sequence

408 JACOB KOREVAAR of integers n tending to infinity. Thus for a suitable subsequence {nj}, I * +(A z) 2J2(z)da(z) -, 0 as j -> co for almost all 0. It follows that, for any one of these 0, and for z belonging to the closed subset of D* where J(z)? 2s > 0, uniformly as j >co. 0(0, Z) = I 2 72 I <-Z I <e 2 (? A, n )da(c) 0 PROOF OF THEOREM 3.1. Lemma 4.2 proves the existence of asymptotically neutral families on F- relative to D c D*. We next show that the union U of the points occurring in any asymptotically neutral family (1.1) on F- is dense on W-. Consider a small disc A with center C F-. Such a disc contains points of D+ as well as points of Do. Now by Lemma 2.1, it is impossible to go from D to infinity without crossing clos U. Thus A must contain infinitely many points of U, and hence C eclos U. We can then write C = lim Zflh, where h = h(n) and n o co through a suitable sequence. Thus, using the same n, - z irni ~kh (n) ZEnk - Z uniformly on every closed subset of D. Integrating and exponentiating, we obtain the final statement of the theorem. 5. The implication (iv) (i) If clos F is a polynomial approximation set, then so is F; hence, we may again assume that F is closed. Suppose now that F divides the plane, and that D belongs to a bounded component of its complement. Let F- be the subset of F defined in? 3. We will use Theorem 3.1 to prove that every zero free holomorphic function f(z) in D can be approximated by F--polynomials. By the Borel-Caratheodory inequality, it is sufficiento show that every harmonic function log I f (z) I can be approximated by sums of the form (5.1) X + n>1log IZk - z with X real, ZkGP-. The maximum principle, finally, shows that it is sufficiento consider approximation on rectifiable Jordan curves in D. Thus the desired result will follow from THEOREM 5.1. For any rectifiable Jordan curve y in D, let C(z) denote the Banach space of all real-valued continuous functions on -y under the maximum norm, and let A(z) be the subset of all uniform limits on -i of

DISTRIBUTIONS OF ELECTRONS AND POLYNOMIAL APPROXIMATION 409 sums (5.1). Then A(z) = C-(). PROOF. It is not clear a priori that A(z) is a vector space. In order to apply the usual method of testing by continuous linear functionals, we therefore begin by looking for a large subset of A(z) which is a subspace of CQz). By Theorem 3.1, -log I z I c A(-Y) for every C e r-. It follows that for every integer v and every pair of points 4', 4" E (5.2) >(log I C"-z I-log IC'-z l) E A(i). At each point C of the connected set F-, there is at least one "tangential" direction s with the following property: there are sequences {JC},{"} of points on F- such that C X -",, and the direct-ion of the vector from ' to C" converges to s. For such sequences, log - z - log I - z - log I C - z I uniformly for z C a. Thus by (5.2), alog I - z e A('y) as for every tangential direction s at C c F- and every real number X. We let A0(,Y) c A(') be the closed subspace of C(z) generated by the functions (a/as) log I - z I and the constant 1, and will prove that A,(y) = C(7). Suppose on the contrary that A,(-) # C(z). Then there exists a real measure dpa # 0 on y such that log -zi df(z) -0 = d(z) = 0 for all C e F- and corresponding tangential directions s. Now consider the logarithmic potential u(g) = 5 log - z I dp(z) The function u(c) is harmonic outside y and vanishes at infinity. At each point C e F-, we have Ou/Os = 0 for every tangential direction s. We will show that u(c) = constant on F-. Since d~i? 0, the function u(c) cannot be constant everywhere outside y (or it would vanish and then di = 0). Thus the holomorphic function Ou/8t - itu/&r2 can have only a finite number of zeros C, on F-. Let C be a point of F- different from the X. Since grad u(c) / 0, the set F- has a unique tangential direction s at C (we, of course, identify opposite directions). This tangential direction, furthermore, must

410 JACOB KOREVAAR depend continuously on C. Thus for the part of F- in a small disc about C, all small chords have approximately the same direction. It follows that F- coincides locally with a smooth Jordan arc, and that u() = constant locally on F-. Since F- is connected and u(c) is continuous on F-, u(c) = constant on F-. We conclude that u(c) = constant outside a; hence, u(c) = 0, dse- 0, and Ao(7) = A(Y) =-C7) UNIVERSITY OF WISCONSIN REFERENCES 1. A. BEURLING, Ensembles exceptionnels, Acta Math., 72 (1939), 1-13. 2. L. FEJPR, Interpolation und konforme Abbildung, Gdttinger Nachrichten, 1918, 319-331. 3. J. KOREVAAR, Approximation by polynomials whose zeros lie in a given set (Survey). Tagung uber Approximationstheorie, Oberwolfach, 1963, to appear. 4. G. R. MACLANE, Polynomials with zeros on a rectifiable Jordan curve, Duke Math. J., 16 (1949), 461-477. 5. M. D. THOMPSON, Approximation by polynomials whose zeros lie on a curve, Ph. D. Thesis, University of Wisconsin, 1962. To appear in Duke Math. J. 1964. (Received November 19, 1963) (Revised January 3, 1964)