2 I. Pritsker and R. Varga Wesay that the triple èg; W;æè has the rational approximation property if, iè iiè for any f èzè which is analytic in G and

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1 Rational approximation with varying weights in the complex plane Igor E. Pritsker and Richard S. Varga Abstract. Given an open bounded set G in the complex plane and a weight function W èzè which is analytic and diæerent from zero in G, we consider the problem of locally uniform rational approximation of any function fèzè, which is analytic in G, by particular ray sequences of weighted rational functions of the form W m+n èzèr m;nèzè, where R m;nèzè =P mèzè=q nèzè; with deg P m ç m and deg Q n ç n: The main result of this paper is a necessary and suæcient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, and ænd explicit criteria for the possibility of weighted approximation in these cases. x1 Introduction and General Results. In this paper, we shall develop the ideas of ë11ë and apply them to the study of the approximation of analytic functions in an open set G by weighted rationals W m+n èzèr m;n èzè: Speciæcally, we examine triples of the form where èg; W; æè iè G is an open bounded set in the complex plane C, which can be represented as a ænite or countable union of disjoint simply connected domains, i.e., G = çë `=1 G` èwhere 1 ç ç ç1è, iiè W èzè, the weight function, is analytic in G with W èzè 6= 0 for any z 2 G, and iiiè æ satisæes 0 ç æ ç 1. 9 é= é; è1.1è è1.2è

2 2 I. Pritsker and R. Varga Wesay that the triple èg; W;æè has the rational approximation property if, iè iiè for any f èzè which is analytic in G and for any compact subset E of G, there exists a sequence of rational functions fr mi;n i èzèg 1 i=0, where R mi;n i èzè = P mi èzè=q ni èzè; with deg P mi ç m i and deg Q ni ç n i for all i ç 0, and where èm i + n i è!1as i!1;such that lim i!1 m i and m i + n i = æ; lim i!1 kf, W m i+n i R mi;n i k E =0, 9 é= é; è1.3è where all norms throughout this paper are the uniform èchebyshevè norms on the indicated sets. Given a triple èg; W;æè, as in è1.1è which satisæes the conditions of è1.2è, we state below our main result, Theorem 1, which gives acharacterization, in terms of potential theory, for the triple èg; W;æè to have the rational approximation property. Let MèEè be the set of all positive unit Borel measures on C which are supported on a compact set E, i.e., for any ç 2 MèEè, we haveçècè = 1 and suppç ç E. denotes the boundary of the set G, and the logarithmic potential of an arbitrary compactly supported signed measure ç is deæned èsee Tsuji ë18, p.53ëè by Z U ç èzè := log 1 jz, tj dçètè: è1.4è Theorem 1. A triple èg; W; æè, satisfying è1.2è, has the rational approximation property è1.3è if and only if there exist a signed measure çèg; W;æè=æç + èg; W;æè,è1, æèç, èg; W;æè; è1.5è with ç + èg; W;æè;ç, èg; W;æè 2 Mè@Gè; and a constant F èg; W;æè such that U çèg;w;æè èzè, log jw èzèj = F èg; W;æè; for any z 2 G: è1.6è Below, we state some consequences of Theorem 1, while in Section 2, we state applications of Theorem 1 in a number of speciæc cases.

3 Rational Approximation with Varying Weights 3 Remark 1: Results on weighted rational approximation of analytic functions in open sets with multiply connected components èas opposed, in è1.2iè, to unions of simply connected domainsè will be considered elsewhere. Remark 2: The condition in è1.2iiè that W èzè 6= 0 for all z 2 G cannot be dropped, for if W èz 0 è = 0 for some z 0 2 G k, where G = S ç `=1 G`, then the necessarily null sequence æ W m i+n i èz 0 èr mi;nièz 0 è æ 1 i=0 trivially fails to converge to any fèzè, analytic in G, with fèz 0 è 6= 0; whence, the rational approximation property fails. Corollary 1. A triple èg; W; æè, satisfying è1.2è, has the rational approximation property è1.3è if and only if è1.3è holds for fèzè ç 1; i.e., if and only if this single function is locally uniformly approximable on compact subsets of G by a corresponding sequence of the weighted rational functions. Remark 3: The function fèzè ç 1 in Corollary 1 can be replaced by any function which is analytic in G and not equal identically to 0 in G: Corollary 2. Given a triple èg; W;æè, which satisæes è1.2è with ç ænite, assume that there exist a constant F and a signed measure ç with supp ç and çèc è=2æ,1; è1.7è such that U ç èzè, log jw èzèj = F; for any z 2 G: è1.8è Then, the triple èg; W; æè has the rational approximation property è1.3è if and only if the signed measure ç can be decomposed as with ç + ;ç, 2Mè@Gè: ç = æç +, è1, æèç, ; è1.9è Furthermore, let a Jordan decomposition of the signed measure ç; satisfying è1.7è and è1.8è, be given by ç = ç +, ç, ; è1.10è where ç + and ç, are positive measures with supp ç + ; supp ç, and çèsupp ç + ë supp ç, è=0: è1.11è

4 4 I. Pritsker and R. Varga Then, the triple èg; W; æè has the rational approximation property è1.3è if and only if ç + èc è ç æ: è1.12è If è1.8è, then è1.9è or è1.12è holds true for a signed measure ç satisfying è1.7è and çèg; W; æè=ç and F èg; W; æè=f: è1.13è The study of weighted rational approximation has recently been introduced in papers by Borwein and Chen ë1ë, Borwein, Rakhmanov and Saæ ë2ë, and Rakhmanov, Saæ, and Simeonov ë12ë. The last two papers deal with weighted rational approximation only on the real line. Certain special cases of the triples èg; W;æè, in the notation of è1.1è, were considered in the complex plane in ë1ë, but that research did not attack the general question of necessary and suæcient conditions for èg; W; æè to have the rational approximation property of è1.3è, as in Theorem 1. x2 Applications. Finding the signed measure çèg; W; æè of Theorem 1, or verifying its existence, is a nontrivial problem in general. Since U çèg;w;æè èzè is harmonic in C n supp çèg; W;æè and, since it can be shown from è1.6è, if log jw èzèj is continuous on G and if G is a ænite union of G`;` = 1;2; :::; `0, that U çèg;w;æè èzè isequal to log jw èzèj + F èg; W;æè on supp çèg; W;æè then U çèg;w;æè èzè can be found as the solution of the corresponding Dirichlet problems. The signed measure çèg; W; æè can be recovered from its potential, using the Fourier method described in Section IV.2 of Saæ and Totik ë13ë. However, we next consider a diæerent method, dealing with speciæc weight functions, which allows us to deduce ëexplicit" expressions for the signed measure çèg; W; æè of Theorem 1. For simplicity, we assume throughout this section that G is given as in è1.2iè, but with ç ænite. We denote the unbounded component ofcngbyæ. Let ç + and ç, be two positive Borel measures on C, with compact supports satisfying such that supp ç + ç C ng and supp ç, ç C ng; è2.1è ç + èc è=ç, ècè=1: è2.2è For real numbers æ ç 0 and æ ç 0, assume that the weight function W èzè, satisfying

5 Rational Approximation with Varying Weights 5 log jw èzèj =, ç ç æu ç+ èzè, æu ç, èzè =,U ç èzè; z 2 G; è2.3è with ç := æç +, æç, being a signed measure, is analytic in G. Then, we state, as an application of Theorem 1, our next result as Theorem 2. Given any pair of real numbers æ ç 0 and æ ç 0, given an open bounded set G = S ç `=1 G`, as in è1.2iè with ç ænite, and given the weight function W èzè of è2.3è, then the triple èg; W; æè has the rational approximation property è1.3è if and only if the signed measure can be decomposed as ç := è2æ, 1+æ,æè!è1;æ;æè, æ^ç + + æ^ç, è2.4è ç = æç +, è1, æèç, ; è2.5è where ç + ;ç, 2Mè@Gè: Here,!è1; æ; æè is the harmonic measure at 1 with respect to æ, and ^ç + and ^ç, are, respectively, the balayages of ç + and ç, from C ng to G. Furthermore, if ç of è2.4è satisæes è2.5è, then èsee Theorem 1è çèg; W; æè=ç: è2.6è We point out that the harmonic measure!è1; æ; æè èdeæned in Nevanlinna ë8ë or Tsuji ë18ëè is the same as the equilibrium distribution measure for G; in the sense of classical logarithmic potential theory ë18ë. For the notion of balayage of a measure, we refer the reader to Chapter IV of Landkof ë6ë or Section II.4 of Saæ and Totik ë13ë. In the following series of subsections, we consider various classical weight functions and we ænd their corresponding signed measures, associated with the weighted rational approximation problem in G, as given in Theorem 1. Incomplete Rationals. With N 0 and N denoting respectively the sets of nonnegative and positive integers, the incomplete polynomials of Lorentz ë7ë are a sequence of polynomials of the form n o1 z mèiè P nèiè èzè ; deg P nèiè ç nèiè; èmèiè;nèiè2 N 0 è; è2.7è i=0 where it is assumed that lim i!1 mèiè =: æ, where æé0 is a real number. The nèiè question of the possibility of the approximation of functions by incomplete

6 6 I. Pritsker and R. Varga polynomials is closely connected to that of the approximation of functions by the weighted polynomials fz æn P n èzèg 1 n=0 ; deg P n ç n: è2.8è The approximation question for the incomplete polynomials of è2.7è was completely settled, by Saæ and Varga ë14ë and by v. Golitschek ë4ë, for the real interval ë0; 1ë èsee Totik ë17ë and Saæ and Totik ë13ë for the associated history and later developmentsè, and by the authors ë11ë in the complex plane. We consider now the analogous problem for incomplete rational functions in the complex plane. A special case of incomplete rational approximation in the complex plane was studied by Borwein and Chen in ë1ë. The latest such developments, on the real line, are in Borwein, Rakhmanov and Saæ ë2ë and Rakhmanov, Saæ and Simeonov ë12ë. Since the weight W èzè :=z æ in è2.8è is multiple-valued in C if æ =2 N, we then restrict ourselves to the slit domain S 1 := C nè,1; 0ë and the singlevalued branch ofwèzèins 1 satisfying W è1è=1. Thus, where æé0 is a real number. W èzè :=z æ ; z2s 1 := C nè,1; 0ë; è2.9è Theorem 3. Given an open set G, asinè1.2iè with ç ænite, such that G ç S 1, and given the weight function W èzè of è2.9è, then the triple èg; z æ ;æè has the rational approximation property è1.3è if and only if the signed measure ç =è2æ,1+æè!è1;æ;æè, æ!è0; æ; æè è2.10è can be decomposed as ç = æç +, è1, æèç, ; where ç + ;ç, 2 Mè@Gè: Here,!è1; æ; æè and!è0; æ; æè are the harmonic measures with respect to the unbounded component æ of C ng, respectively, at z = 1 and at z =0. In special cases where the geometric shape of G is given explicitly, it is possible to determine the explicit form of the signed measure in è2.10è. As a simple example, we consider below the special case of a disk and æ =1=2: Corollary 3. Given the disk D r èaè := æ z 2 C : jz, aj ér æ, where a 2 è0; +1è and where D r èaè ç S 1 = C nè,1; 0ë, i.e., r é a, and given the weight function of è2.9è, then the triple èd r èaè;z æ ;1=2è has the rational approximation property è1.3è if and only if

7 Rational Approximation with Varying Weights 7 è a; æ 2 è0; 1=2ë; r ç r max èa; æè := a sin ç è2.11è ; æ 2 è1=2; +1è: 4æ Furthermore, if è2.11è is satisæed, then the associated signed measure çèd r èaè;z æ ;1=2è èsee Theorem 1è is given by dçèd r èaè;z æ ;1=2è = æ ç ç1, a2, r 2 2çr jzj 2 ds; è2.12è where ds is the arclength measure on the circle jz, aj = r. Remark 4: Generally, it is possible to show that the triple èd r èaè; z æ ; æè, as in Corollary 3 but with any æ 2 ë0; 1ë, has the approximation property è1.3è if and only if ç r ç a ; æ + æ ç 1; au 0 ; æ + æé1; where u 0 2 è0; 1ë is the largest solution of the equation ç ç 1, 2æ, è2æ, 1+2æèu 2 è2æ, 1è arccos 2uè2æ, 1+æè " è2æ, 1+2æè p è 1,u +2æ arccos 2 2 p =æç; æè2æ, 1+æè in the interval è0; 1ë. Exponential Weight. Consider the weight function W èzè :=e,z ; z2c: è2.13è This section is devoted to the study of weighted rational approximation, with respect to the exponential weight of è2.13è, in disks centered at the origin and in certain domains, arising in connection with Padçe approximations of the exponential function. Our next result treats the case of disks. Theorem 4. Given D r è0è := fz 2 C : jzj érgand given the weight W èzè of è2.13è, then the triple èd r è0è;e,z ;æèhas the rational approximation property è1.3è if and only if r ç r max èæè; 0 ç æ ç 1; è2.14è

8 8 I. Pritsker and R. Varga where r max èæè is the unique positive solution, for r in the interval ëjæ, 1 j; +1è; of the following equation: 2 q r 2,, æ, 1 2æ 2,, æ, 1 2 æ ç arccos æ, ç 1 2 r = ç 2 è1, æè : è2.15è Moreover, if è2.14è holds, then the associated signed measure çèd r è0è, e,z ;æè is given by dçèd r è0è;e,z ;æè=è2æ,1,2rcos çè dç 2ç ; where dç is the angular measure on jzj = r and where z = re iç. è2.16è In particular, r max è1è = 1 2 èsee also Theorem 3.8 of ë10ëè, r max, 1 2æ = ç 4 and r max è0è = 1 2. We remark that the solution, r max èæè of è2.15è, can be veriæed to be symmetric about æ = 1, as a function of æ in the interval ë0; 1ë. 2 Next, we again consider the weight function W èzè :=e,z of è2.13è, but we now consider the triple èg æ ; e,z ; æè, where G æ,ageneralized Szegío domain, is deæned below. To begin, ærst assume that 0 éæé1. Then following ë15ë, the two conjugate complex numbers, deæned by z æ æ := exp fæ i arccosè2æ, 1èg ; è2.17è have modulus unity, and we consider the complex plane C slit along the two rays R æ := æ z 2 C : z = z + æ + iç or z =z, æ,iç; for ç ç 0 æ : è2.18è This is shown below in Figure 1. èfor readers who are familiar with ë15ë, the quantity ç := lim in that paper and æ of è1.3iè are related through i!1 n i m i æ = 1.è Next, the function 1+ç ^g æ èzè := p 1+z 2,2zè2æ, 1è è2.19è has z + æ and z, æ as branch points, which are the ænite extremities of R æ. On taking the principal branch for the square root, i.e., on setting ^g æ è0è = 1 and extending g æ analytically on the doubly slit domain C nr æ, then g æ is analytic and single-valued on C nr æ. It can also be veriæed that 1æz +^g æ èzè does not vanish on C nr æ. Next, we deæne the functions è1+z +^g æ èzèè 2æ and è1,z +^g æ èzèè 2è1,æè so that their values at z = 0 are respectively, 2 2æ and 2 2è1,æè, with remaining

9 Rational Approximation with Varying Weights 9 R γ + z γ z γ Rγ Figure 1. The set C nr æ. values determined by analytic continuation. These functions are also analytic and single-valued on C nr æ. With these deænitions, we then set ç ç 1,æ 1,æ 4æ ze^g æèzè æ w æ èzè := è0 éæé1è; è2.20è è1 + z +^g æ èzèè 2æ è1, z +^g æ èzèè 2è1,æè and it follows that w æ èzè is also analytic and single-valued on C nr æ. For the omitted cases æ = 0 and æ = 1, it can be veriæed that w 1 èzè = lim w æèzè æ!1 and w 0 èzè = lim w æèzè satisfy æ!0 ç w 1 èzè =ze 1,z for Re zé1; and è2.21è w 0 èzè =ze 1+z for Re zé,1: èagain, for those familiar with ë15ë, the function w æ èzè ofè2.20è is exactly the function w ç èzè in ë15, eq. è2.5èë.è It is known èsee ë15, Lemma 4.1ëè that w æ èzè is univalent injzjé1, for any æ with 0 ç æ ç 1, and this allows us to deæne the domain G æ := æ z 2 C : jw æ èzèj é 1 and jzj é 1 æ : è2.22è

10 10 I. Pritsker and R. Varga Its æ,isawell-deæned Jordan curve which lies interior to the unit disk, except for its points z æ æ of è2.17è. This is shown in Figure 2. We call G æ an extended Szegío domain, as the special case æ = 1 corresponds to a domain originally treated by Szegío in ë16ë in z γ Gγ 0 1 Figure 2. The set G æ. We now have all the necessary deænitions for the statement of our next result. - z γ Theorem 5. For any æ with 0 ç æ ç 1, let G æ be the domain of è2.22è, and let W èzè =e,z be the weight function of è2.13è. Then, the triple èg æ ;e,z ;æè has the rational approximation property è1.3è. To conclude this section, we note that, except for the ænal result of Theorem 5, all preceding results stated in Sections 1 and 2 are of the ëif and only if" type, i.e., these results are by deænition sharp. The result of Theorem 5, however, leaves open the possibility that for a given æ with 0 ç æ ç 1, there could be a larger domain H, with G æ ç H, such that the triple èh; e,z ;æè has the rational approximation property è1.3è, but we strongly doubt this. Also of general interest is the extension of the results of this paper to triples èg; W;æè of è1.1è, where one has the sharpened rational approximation property, that is, for any fèzè, analytic in G and continuous in G, there is a sequence of rational functions fr mi;n i g 1 i=0 satisfying è1.3iè, such that

11 Rational Approximation with Varying Weights 11 lim æ f, W m i+n i æ R mi;n i =0: G i!1 For the essentially polynomial case of æ = 0 and W èzè :=e,z, this is treated in part in ë10, Theorem 3.2ë. Some general results in weighted polynomial approximation on compact sets are obtained in ë9ë. To our knowledge, general results on the sharpened rational approximation property have not as yet been treated in the literature. References ë1ë Borwein, P. B. and W. Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 è1995è, 85í106. ë2ë Borwein, P. B., E. A. Rakhmanov, and E. B. Saæ, Rational approximation with varying weights I, Constr. Approx. 12 è1996è, 223í240. ë3ë de Bruin, M. G., E. B. Saæ, and R. S. Varga, On the zeros of generalized Bessel polynomials, I and II, Nederl. Akad. Wetensch. Indag. Math. 23 è1981è, 1í25. ë4ë Golitschek, M. v., Approximation by incomplete polynomials, J. Approx. Theory 28 è1980è, 155í160. ë5ë Kuijlaars, A. B. J., The role of the endpoint in weighted polynomial approximation with varying weights, Constr. Approx. 12 è1996è, 287í 301. ë6ë Landkof, N. S., Foundations of Modern Potential Theory, Springer- Verlag, Berlin, ë7ë Lorentz, G. G., Approximation by incomplete polynomials èproblems and resultsè, in Padçe and Rational Approximations: Theory and Applications, E. B. Saæ and R. S. Varga èeds.è, Academic Press, New York, 1977, pp. 289í302. ë8ë Nevanlinna, R. Analytic Functions, Springer-Verlag, New York, ë9ë Pritsker, I. E., Polynomial approximation with varying weights on compact sets of the complex plane, Proc. Amer. Math. Soc., to appear. ë10ë Pritsker, I. E. and R. S. Varga, The Szegío curve, zero distribution and weighted approximation, Trans. Amer. Math. Soc. 349 è1997è, 4085í 4105.

12 12 I. Pritsker and R. Varga ë11ë Pritsker, I. E. and R. S. Varga, Weighted polynomial approximation in the complex plane, Constr. Approx., to appear. ë12ë Rakhmanov, E. A., E. B. Saæ and P. C. Simeonov, Rational approximation with varying weights II, preprint. ë13ë Saæ, E. B., and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, ë14ë Saæ, E. B., and R. S. Varga, On incomplete polynomials, Numerische Methoden der Approximationstheorie, L. Collatz, G. Meinardus, H. Werner, èeds.è, ISNM 42, Birkhíauser-Verlag, Basel, 1978, 281í298. ë15ë Saæ, E. B. and R. S. Varga, On the zeros and poles of Padçe approximants to e z : III, Numer. Math. 30 è1978è, 241í266. ë16ë Szegío, G., íuber eine Eigenschaft der Exponentialreihe, Sitzungsber. Berl. Math. Ges. 23 è1924è, 50í64. ë17ë Totik, V., Weighted Approximation with Varying Weights, Lecture Notes in Math., vol. 1569, Springer-Verlag, Heidelberg, ë18ë Tsuji, M., Potential Theory in Modern Function Theory, Maruzen, Tokyo, ë19ë Varga, R. S., and A. J. Carpenter, Asymptotics for zeros and poles of normalized Padçe approximants to e z, Numer. Math. 68 è1994è, 169í185. ë20ë Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain, Colloquium Publications, vol. 20, Amer. Math. Soc., Providence, ë21ë Yosida, K., Functional Analysis, Springer-Verlag, Berlin, Igor E. Pritsker Dept. of Math. Case Western Reserve Univ., Cleveland, Ohio èu.s.a.è iep@po.cwru.edu Richard S.Varga Inst. for Computational Mathematics Dept. of Math. & Computer Science Kent State University, Kent, Ohio èu.s.a.è varga@mcs.kent.edu

2 I. Pritsker and R. Varga Wesay that the triple (G W ) has the rational approximation property if, for any f(z) which is analytic in G and for any co

2 I. Pritsker and R. Varga Wesay that the triple (G W ) has the rational approximation property if, for any f(z) which is analytic in G and for any co Rational approximation with varying weights in the complex plane Igor E. Pritsker and Richard S. Varga Abstract. Given an open bounded set G in the complex plane and a weight function W (z) which is analytic