Numerische Mathematik

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1 Numer. Math. (1997) 76: Numerische Mathematik c Springer-Verlag 1997 Electronic Edition Approximation of conformal mappings of annular regions N. Papamichael 1, I.E. Pritsker 2,, E.B. Saff 3,, N.S. Stylianopoulos 4 1 Department of Mathematics Statistics, University of Cyprus, P.O. Box 537, Nicosia, Cyprus; nickp@pythagoras.mas.ucy.ac.cy 2 Department of Mathematics Computer Science, Kent State University, Kent, OH 44240, USA; pritsker@mcs.kent.edu 3 Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, 4202 East Fowler Ave., Tampa, FL 33620, USA; esaff@math.usf.edu 4 Department of Mathematics Statistics, University of Cyprus, P.O. Box 537, Nicosia, Cyprus; nikos@pythagoras.mas.ucy.ac.cy Received February 2, 1996 Summary. In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal mapping f of an annular region A onto a circular annulus. In particular, we consider the case where f has an analytic extension in compl(a), for this case, we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform convergence. We also estimate the rate of uniform convergence in the case where the annular region A has piecewise analytic boundary without cusps. In both cases we also give the corresponding rates for the approximations to the conformal module of A. Mathematics Subject Classification (1991): 30C30, 65E05, 41A20 1 Introduction Let A be an annular region of the complex plane bounded by two piecewise analytic Jordan curves Γ e Γ i, where Γ i is interior to Γ e, denote the exterior of Γ e by Ω (so that Ω) the interior of Γ i by G. Also, assume that 0 G, let z 0 be a fixed point on Γ i denote by f the conformal mapping of A onto a circular annulus E := {w :1< w <M}, so that f (z 0 ) = 1. Here the outer radius M of E is uniquely determined by A is called the conformal module of A. Research supported, in part, by a University of Cyprus research grant Research done in partial fulfillment of Ph.D. degree at the University of South Florida Research supported, in part, by NSF grant DMS Correspondence to: N. Papamichael page 489 of Numer. Math. (1997) 76:

2 490 N. Papamichael et al. The work of this paper is concerned with certain aspects of the convergence theory of an orthonormalization method (ONM) which was studied by Papamichael et al [8], [9], [10], [11] [12], for the purpose of approximating the conformal mapping f the conformal module M. The details of the basic ONM are as follows: Let L 2 (A) be the Hilbert space of all square integrable functions that are analytic in A, with inner product (1.1) (u,v):= u(z)v(z)dσ z A (where dσ z denotes 2-dimensional Lebesgue measure) corresponding norm u 2 := (u, u) 1 2. Also let L S 2 (A) denote the set L S 2(A) :={u:u L 2 (A) u has a single-valued indefinite integral}. The set L S 2 (A) is, in fact, a closed subspace of L 2(A), therefore, is itself a Hilbert space with inner product (1.1). Next, let the functions g H be defined respectively by (1.2) g(z) :=logf(z) log z (1.3) H (z) :=g (z)= f (z) f(z) 1 z, so that g is a branch analytic single-valued in A, H L S 2 (A), (1.4) f (z) =ze g(z). With these notations, it is well known ([3, pp ], [10, p. 686] [11, p. 101]) that the conformal module M of A is related to H by means of {( ) } 1 log z (1.5) M = exp dz H 2 2 /2π. i A z Let R m,n be the set of all rational functions of the form (1.6) R m,n (z) := n a k z k. k= m k/= 1 Then, the basic ONM consists of the following steps: (i) Orthonormalize the set of monomials {z k } n k= m, k /= 1, by means of the Gram-Schmidt process in order to obtain the orthonormal set {η k } n k= m, k /= 1. (ii) Approximate the function H by the Fourier sum (1.7) R m,n(z) := n (H,η k )η k = k= m k/= 1 n akz k, k= m k/= 1 where the coefficients (H,η k ) hence the coefficients ak can be computed, without the explicit knowledge of H, with the help of the formula page 490 of Numer. Math. (1997) 76:

3 Approximation of conformal mappings of annular regions 491 (1.8) (η k, H )=i η k log z dz; A see e.g. [3, pp ] [11, p. 101]. (The Fourier sum Rm,n is of course the best approximant to H in L S 2 (A) out of the subspace R m,n; i.e., (1.9) H Rm,n 2 = inf H R m,n 2.) R m,n R m,n (iii) Using (1.3) (1.5), approximate the mapping function f the conformal module M respectively by f m,n (z) := z ( z (1.10) exp z 0 (1.11) {( 1 M m,n := exp i A z 0 ) Rm,n(ζ)dζ ) } log z dz R z m,n 2 2 /2π. The results of many numerical experiments ([8], [9], [10], [11] [12]) suggest that (at least in the diagonal case m = n) uniform convergence of f m,n to f holds on A, under some mild assumptions on the geometry of A. On the other h, there is no adequate theory available to justify the ONM except for the case when f is analytic on the boundary of A (see [3, p. 250] [12, p. 654]). Even in the latter case, the available theory concerns diagonal approximants with equal number of positive negative powers of z, although it is reasonable to expect that this diagonal selection of powers need not always be the best choice. In this paper we seek to overcome some of the above shortcomings in the theory of the ONM. In particular, we consider the case where the function H can be extended analytically in compl(a), for this case, we derive optimal ray sequences of approximants to H. More specifically, we derive (in terms of the analytic continuation properties of H ) the asymptotically optimal proportion of positive negative powers of z in the best approximations R m,n to H.We also study the case where f is not analytic on A fill partially the gap in the theory of the ONM, by showing that uniform convergence on A holds if A is bounded by piecewise analytic curves without cusps. Finally, we consider the convergence of the ONM approximants to the conformal module M present results that provide theoretical justification for certain experimental observations concerning the quality of these approximants. The organization of the paper is as follows: In Sect. 2, we state (without proofs) our main results. In Sect. 3, we describe an adaptive version of the ONM, based on the choice of optimal ray sequences of approximants, also present the results of some numerical experiments illustrating the application of this adaptive method. Finally, in Sect. 4 we present the proofs of the results of Sect. 2. Here, our analysis is based mainly on methods of proof used in earlier papers ([1], [5], [6] [7]) in connection with the problem of approximating, by means of Bieberbach polynomials, the conformal mapping of a simply-connected domain onto a disk. page 491 of Numer. Math. (1997) 76:

4 492 N. Papamichael et al. 2 Main results 2.1 Analytic case optimal ray sequences Let φ be the conformal mapping of G := Int(Γ i ) onto the unit disk D := {w : w < 1}, normalized by φ(0) = 0 φ (0) > 0, let Φ denote the conformal mapping of the unbounded domain Ω := Ext(Γ e ) onto the exterior of the unit circle D := {w : w > 1}, normalized by Φ( ) = lim z Φ(z)/z > 0. Next, assume that the function H of (1.3) is analytic on A suppose that its nearest singularities in G in Ω are situated on the level curves L ri := {z : φ(z) = r i, 0 < r i < 1} L re := {z : Φ(z) = r e, 1 < r e < }, respectively. In other words, we assume that the function H is analytic in the annular region (2.1) A H := Ext(L ri ) Int(L re ), bounded by the curves L ri L re, has singularities on each of these curves. We now consider ray sequences {R m(j ),n(j ) } j =1 of rational functions of the form (1.6) such that (2.2) m(j ), n(j), as, the following limit exists (2.3) lim m(j ) =: α, 0 α 1. m(j )+n(j) We also assume that there exists a constant c > 0 such that (2.4) m(j +1) m(j) <c n(j +1) n(j) <c, for all j =1, 2,.... The result of the following theorem shows that the optimal proportion of positive negative powers of z in the best approximation Rm,n to H can be described asymptotically in terms of the location of the nearest singularities of H. Theorem 2.1. Let H be analytic on A let Rm,n be the best L S 2 (A) approximation to H out of R m,n. Then, a ray sequence {Rm(j ),n(j ) } j =1, satisfying (2.2) (2.4), is optimal in the sense of geometric convergence rate if only if (2.5) In this case, (2.6) lim lim sup f f m,n 1/(m+n) m m + n = = lim sup log r e log r e log r i =: α. H Rm,n 1/(m+n) = r α 1 e = r α i. Furthermore, an optimal ray sequence {R m,n} the corresponding sequence of approximants {f m,n } to the mapping function f converge locally uniformly in the annular region A H. page 492 of Numer. Math. (1997) 76:

5 Approximation of conformal mappings of annular regions 493 In the above in what follows denotes the uniform norm on A, i.e. for any function h analytic in A bounded on A, h := sup h(z). z A Also, the statement that the rate of geometric convergence in (2.6) is optimal means that for any ray sequence {R m(j ),n(j ) } j =1 satisfying (2.2) (2.4), lim sup H R m,n 1/(m+n) ri α. The following result gives the corresponding estimate of the error in the ONM approximation to the conformal module M of A. Theorem 2.2. Let {M m(j ),n(j ) } j =1 be a sequence of approximants to M corresponding to an optimal ray sequence {Rm(j ),n(j ) } j =1. Then, this sequence of approximants tends monotonically (with m + n) to M from above, (2.7) lim sup{m m,n M } 1/(m+n) = r 2(α 1) e = ri 2α. 2.2 Piecewise analytic boundary Here we assume that the boundary components Γ i Γ e of A are piecewise analytic Jordan curves without cusps. This means that each of Γ i Γ e consist of a finite number of analytic arcs, where two adjacent arcs meet at a corner form there an interior (with respect to A) angle νπ, with 0 <ν<2. Thus, in each case, the exterior (with respect to A) angle is λπ with ν + λ =2. Our next result extends Theorem 1 of [7] (concerning the convergence of Bieberbach polynomials for approximating the conformal maps of simplyconnected regions) to the mapping of annular regions. Theorem 2.3. Suppose that the boundary components Γ i Γ e of A are piecewise analytic without cusps let λ i π λ e π (0 <λ i,λ e < 2) be respectively the smallest exterior (with respect to A) angles among all joints on Γ i Γ e. Suppose, in addition, that m, n N satisfy (2.8) 0 < c 1 m n c 2 <, for some fixed numbers c 1 c 2. Then, with f m,n defined as in (1.10), there exists a constant K, depending only on c 1,c 2 the geometry of A, such that (2.9) f f m,n K log max(m, n) min(m γi, n γe ), with γ i = λ i /(2 λ i ) γ e = λ e /(2 λ e ). Remark. Because of (2.8), the result of the theorem is equivalent to the estimate page 493 of Numer. Math. (1997) 76:

6 494 N. Papamichael et al. (2.10) ( ) ( ) log n log m f f m,n = O = O, n γ where γ := min(γ i,γ e ). We note that the exponent γ cannot be increased in general. This is so because if γ>min(γ i,γ e ), then the converse theorem argument of [14] would imply that f Lip β for some β>(2 min(λ i,λ e )) 1 would contradict the known boundary behavior of f (see Sect. 4, Lemma 4.4). It follows that Theorem 2.3 is sharp up to the logarithmic factor in the numerator of (2.9). Theorem 2.4. Suppose that the annular region A satisfies the conditions of Theorem 2.3. Then there exists a constant K, depending only on the geometry of A, such that (2.11) log max(m, n) 0 M m,n M K, m, n =2, 3,..., min(m 2γi, n 2γe ) where γ i = λ i /(2 λ i ) γ e = λ e /(2 λ e ). We end this section by noting that the results of Theorems confirm the experimental observation ([9, p. 47], [10, Sect. 5] [11, Sect. 4]) that the ONM approximations to the conformal module M are more accurate than the corresponding approximations to the mapping function f. m γ 3 Adaptive ONM numerical examples The results of Theorem 2.1 suggest a process for balancing the boundary errors (3.1) max f (z) f m,n (z) z Γ i max f (z) f m,n (z). z Γ e In this section we show how this idea can be used in order to develop an adaptive version of the classical ONM, in cases when the function H (z) :=logf(z) log z can be extended analytically to an open set containing A. We also present two numerical examples illustrating the application of the adaptive ONM the convergence results of Sect We let E (m,n) i E e (m,n) denote estimates of the maximum error in modulus in the approximation f m,n to f on the boundary curves Γ i Γ e respectively. These are obtained, as indicated in [10], [11] [12], by means of (3.2) (3.3) E (m,n) i E (m,n) e := max 1 f m,n (zj) i j := max M m,n f m,n (zj e ), j where {zj i e } {zj } are two sets of boundary test points on Γ i Γ e respectively. (We expect (3.3) to be a reasonable estimate of the actual error in modulus on Γ e because, as we remarked at the end of Sect. 2, the approximations {M m,n } to M are more accurate than the corresponding approximations {f m,n } to f ; see Theorems also Lemma 4.3 in Sect. 4.) Then, the adaptive ONM algorithm can be described as follows: page 494 of Numer. Math. (1997) 76:

7 Approximation of conformal mappings of annular regions Set m := m min n := n min, where m min n min denote respectively the minimum number of negative positive powers of z to be used in the basis set. 2. Compute the ONM approximations f m,n to f M m,n to M, using the basis set (3.4) {z k } n k= m, k /= Compute the error estimates E (m,n) i 4. If E (m,n) i < E e (m,n) then: E (m,n) e, by means of (3.2) (3.3). Introduce into the basis set (3.4) the next positive power of z, i.e. set n := n +1. Otherwise: Introduce into the basis set (3.4) the next negative power of z, i.e. set m := m Check the termination criterion either terminate the process or go to Step 2. Regarding Step 5, the procedure is terminated when after a certain pair (m, n), due to the numerical instability of the Gram-Schmidt process, the maximum error max(e (m,n) i, E e (m,n) ) does not decrease any further. In presenting the numerical results we make use of the following notations: ζ i ζ e : These denote the locations of the nearest singularities of H in G Ω respectively. r i r e : As in Sect. 2, these denote the level indices of the nearest singularities of the function H, i.e. r i := φ(ζ i ) r e := Φ(ζ e ). (The values of r i r e are obtained by using either the exact mapping functions φ or Φ, when these are available, or are computed in approximate form by using the conformal mapping package BKMPACK of Warby [17].) α : This denotes the optimal ratio given by (2.5), i.e. (3.5) α := log r e log r e log r i. E (m,n) f : This denotes the larger of the two error estimates (3.2) (3.3), i.e. (3.6) E (m,n) f = max(e (m,n) i, E e (m,n) ). E (m,n) M : This provides an estimate of the error in the approximation M m,n to the conformal module M by means of (3.7) E (m,n) M = M m, n M m,n, where m + n is the largest value of the sum of indices m + n used in the adaptive ONM algorithm. (That is, this estimate is computed by using, instead of the exact value of M, the most accurate approximation to M that we have available.) page 495 of Numer. Math. (1997) 76:

8 496 N. Papamichael et al. δ f δ M : These are respectively estimates of the orders of the errors (2.6) (2.7). They are determined as follows: Let (m 1, n 1 ) (m 2, n 2 )betwo pairs of indices such that (3.8) m 1 m 1 + n 1 m 2 m 2 + n 2 α, (mj,nj ) (mj,nj ) let E f E M ; j =1,2, denote the error estimates (3.6) (3.7) corresponding to the pairs (m j, n j ); j =1,2, respectively. Then, we assume that E (m,n) f C 1 r δf α (m+n) i E (m,n) M C 2 r δm α (m+n) (3.9) i, seek to estimate δ f δ M by means of the formulae [ ] 1 log(e (m 2,n 2) f /E (m1,n1) f ) (3.10) δ f = α (m 2 + n 2 m 1 n 1 ) log r i (3.11) δ M = [ 1 α (m 2 + n 2 m 1 n 1 ) log(e (m2,n2) M log r i /E (m1,n1) M ) see (2.6) (2.7). (That is, from the theory we expect to obtain values δ f 1 δ M 2.) ] ; Example 3.1. Square with circular hole. Let A be the annular region bounded internally by the circle externally by the square Γ i := {z : z =0.8} Γ e := {z : z = ±1+iy, y 1} {z :z =x±i, x 1}. Then, the nearest singularities of H occur at the points ζ i =0.4 ζ e =1.6; see [11, p. 95]. Also, the mapping function φ is known trivially, i.e. φ(z) =1.25z. Therefore, r i =0.5. Finally, the value of r e is found in approximate form by using BKMPACK for the approximation of the mapping function Φ. In this way we find that (correct to six decimal places) r e is given by r e = Hence, from (3.5), α is given (correct to two decimal places) by α =0.33. page 496 of Numer. Math. (1997) 76:

9 Approximation of conformal mappings of annular regions 497 Example 3.2. Ellipse with circular hole. Let A be the annular region bounded internally by the circle externally by the ellipse Γ i := {z : z =2/3} Γ e := {z : z = x +iy, (x+1) 2 /4+y 2 =1}. In this case, the nearest singularities of H occur at the points ζ i = i ζ e =4/(9ζ i ), where ζ i is correct to all figures quoted; see [11, p. 100]. Also, the mapping functions φ Φ are known exactly, i.e. φ(z) =1.5z Φ(z) = z+1 ( ) 1/2 3 { (z+1) 2 }; see e.g. [2, p. 225]. In this way we find that (correct to six decimal places) r i r e are given by r i = r e = Hence, from (3.5), α is given (correct to two decimal places) by α =0.30. The results for Examples are given in Tables 1, 2 3, 4, respectively. All these results were obtained by using the adaptive ONM algorithm with starting pairs of indices (m min, n min )=(9,7) for Example 3.1, (m min, n min )=(4,4) for Example 3.2. Tables 1 3 contain, in each case, the values of the pairs (m, n) that were chosen by the adaptive process at each step of the algorithm, the corresponding values of the ratio m/(m + n) the values of the boundary errors E (m,n) i. These results illustrate how the algorithm seeks to balance the magnitude E (m,n) e of the two errors (by introducing the appropriate power of z into the basis set) show that, for the optimal values of (m, n) chosen by the algorithm, the ratios m/(m + n) are (as predicted by Theorem 2.1) close to the value of α. The domain of Example 3.1 has 4-fold rotational symmetry about the origin. Because of this the corresponding approximations (1.7) take the following form: R m,n(z) := k= m(4)n k/= 1 a k z k, with m := m(j )=4j+1, n := n(j )=4j 1; j =1, 2,.... page 497 of Numer. Math. (1997) 76:

10 498 N. Papamichael et al. Table 1. Square with circular hole: Steps of the adaptive algorithm (m, n) m/(m + n) E (m,n) i E e (m,n) ( 9, 7) e e-03 ( 9,11) e e-04 ( 9,15) e e-04 ( 9,19) e e-05 ( 9,23) e e-05 (13,23) e e-06 (13,27) e e-06 (13,31) e e-06 (17,31) e e-07 (17,35) e e-07 (17,39) e e-07 (21,39) e e-08 (21,43) e e-09 (21,47) e e-09 (25,47) e e-10 (25,51) e e-10 (25,55) e e-10 (29,55) e e-11 (29,59) e e-11 (29,63) e e-11 (33,63) e e-12 Tables 2 4 contain the ONM approximations M m,n, for some of the optimal pairs (m, n) of Tables 1 3, together with the corresponding estimates δ f δ M that measure respectively the orders of the errors in the approximations f m,n to f M m,n to M. These estimates were computed from (3.10) (3.11) by keeping the pair (m 1, n 1 ) fixed, as the initial pair in each table, taking recursively (m 2, n 2 ) to be each subsequent pair shown in the table. As expected (see (2.6) (2.7)), the values of δ f δ M are, in all cases, close to 1 2, respectively. Table 2. Square with circular hole: Orders of the errors E (m,n) f (m, n) M m,n δ f δ M ( 9,19) ( 9,23) (13,23) (13,27) (13,31) (17,31) (17,35) (17,39) (21,39) E (m,n) M page 498 of Numer. Math. (1997) 76:

11 Approximation of conformal mappings of annular regions 499 Table 3. Ellipse with circular hole: Steps of the adaptive algorithm (m, n) m/(m + n) E (m,n) i E e (m,n) ( 4, 4) e e-02 ( 4, 5) e e-02 ( 4, 6) e e-02 ( 4, 7) e e-03 ( 4, 8) e e-03 ( 4, 9) e e-03 ( 4,10) e e-03 ( 5,10) e e-03 ( 6,10) e e-03 ( 6,11) e e-03 ( 6,12) e e-03 ( 6,13) e e-03 ( 7,13) e e-03 ( 7,14) e e-04 ( 7,15) e e-04 ( 7,16) e e-04 ( 7,17) e e-04 ( 8,17) e e-04 ( 9,17) e e-04 ( 9,18) e e-05 ( 9,19) e e-05 ( 9,20) e e-05 ( 9,21) e e-05 (10,21) e e-05 (10,22) e e-05 (10,23) e e-05 4 Proofs 4.1 Proofs of Theorems For the proofs of our main results we shall make use of several lemmas. The first of these is an analogue, for rational functions, of the well-known Bernstein-Walsh lemma for polynomials. Lemma 4.1. For any rational function of the form (4.1) n r m,n (z) = a k z k k= m we have that (4.2) r m,n (z) r m,n Ω Φ(z) n, z Ω, r m,n (z) r m,n G (4.3) φ(z) m, z G, where Ω G denote the uniform norms on Ω G respectively. page 499 of Numer. Math. (1997) 76:

12 500 N. Papamichael et al. Table 4. Ellipse with circular hole: Orders of the errors E (m,n) f (m, n) M m,n δ f δ M ( 4, 9) ( 4,10) ( 6,12) ( 6,13) ( 7,14) ( 7,15) ( 7,16) ( 7,17) ( 8,17) ( 9,18) ( 9,19) ( 9,20) ( 9,21) (10,21) (10,22) (10,23) E (m,n) M Proof. The function r m,n (z)/[φ(z)] n is analytic in Ω continuous on Ω. Thus, by the maximum modulus principle, r m,n (z) Φ(z) n r m,n Φ n Ω = r m,n Ω, z Ω, from which (4.2) follows. Similarly, the function r m,n (z)[φ(z)] m is analytic in G continuous on G. Hence, r m,n (z) φ(z) m r m,n φ m G = r m,n G, z G, from which (4.3) follows. Lemma 4.2. For any rational function of the form (4.1) there exists a constant K, depending only on the geometry of A, such that H (r m,n a 1 /z) 2 K H r m,n 2. Proof. Let Γ := {z : f (z) = r}, where r (1, M ) is fixed. Then, since the indefinite integral of H is single-valued in A, {H (t) r m,n (t)}dt = 2πia 1. Γ On the other h by [4, p. 4] {H (t) r m,n (t)}dt Γ max m,n(t) t Γ Γ Γ πdist(γ, A) H r m,n 2. page 500 of Numer. Math. (1997) 76:

13 Approximation of conformal mappings of annular regions 501 Thus, H (r m,n a 1 /z) 2 H r m,n 2 + a 1 1/z 2 K H r m,n 2. Proof of Theorem 2.1. Our first task is to show that (4.4) lim sup H Rm,n 1/(m+n) max ( re α 1, ri α for any ray sequence satisfying (2.2) (2.4). We do this as follows: Suppose that lim sup H Rm,n 1/(m+n) < max ( re α 1, r α ) (4.5) i, assume that the max in (4.5) is equal to ri α, so that (4.6) lim sup H Rm,n 1/m < r i. Next, suppose that the value of lim sup in (4.6) is equal to q < r i let ε>0be such that q + ε<r i. Then, by making use of Lemma 4.1 ( Inequalities (2.4)), we find that for z L q+ε := {z : φ(z) = q + ε} Rm(j +1),n(j +1)(z) Rm(j ),n(j )(z) j =1 j =1 Rm(j +1),n(j +1) Rm(j max(m(j ),m(j +1)) ),n(j ) (q + ε) { H R m(j ),n(j ) + H Rm(j } +1),n(j +1) (q + ε) max(m(j ),m(j +1)) j =1 K 1 K 2 j =1 j =1 ( q + ε ) min(m(j ),m(j +1)) (q + ε) max(m(j ),m(j +1)) 2 ( q + ε ) m(j ) 2 <. q + ε This shows that the series { R (4.7) m(j +1),n(j +1) (z) Rm(j ),n(j )(z) } + Rm(1),n(1)(z) j =1 converges uniformly on L q+ε. Since (by (4.6)) the series converges uniformly to H on A, it follows that it converges in the region between Γ i L q+ε.in other words, we have shown that (4.7) converges to an analytic continuation of H through the level curve L ri, which is a contradiction. A similar argument can be used to establish a contradiction in the case when the max in (4.5) is ) page 501 of Numer. Math. (1997) 76:

14 502 N. Papamichael et al. equal to re α 1. This proves (4.4). (Observe that the right h side of (4.4) takes its minimal value when α is given by (2.5), i.e. when α satisfies the equation re α 1 = ri α ). We shall now show that (2.6) holds for a ray sequence defined by (2.5). For this, we first note that the function H can be expressed as H (z) =H e (z)+h i (z), z A H, where H e is analytic inside L re H i is analytic outside L ri (including the point at ) H i ( ) = 0. Also, by the results of Walsh [16, pp ], there exist two sequences of polynomials {P n (z)} n=1, where deg P n n, {Q m (t)} m=1, where deg Q m m, such that (4.8) lim sup H e P n 1/n Ω = 1, n r e ( ) 1 (4.9) lim sup H i Q m 1/m G m z = r i. Further, from Lemma 4.2, ( lim sup H Rm,n 1/(m+n) 2 lim sup 1 (P H n + Q m z { ( ) } lim sup H e P n + 1 1/(m+n) H i Q m z (4.10) r α 1 e = r α i, )) 1/(m+n) where, for the last step, we made use of (4.8), (4.9) (2.5). Next, by using Lemma 4.1 the estimate (cf. [4, p. 4]) H (z) R m,n(z) 1 πdist(z, A) H R m,n 2, we can show (with the help of series (4.7)) that lim sup H Rm,n 1/(m+n) which, in view of (4.4) (4.10), gives that z A, lim sup H Rm,n 1/(m+n) 2 2 (4.11) lim sup H Rm,n 1/(m+n) = lim sup H Rm,n 1/(m+n) 2 = r α 1 e = ri α. In order to show that an optimal ray sequence {Rm,n} j =1 converges to H locally uniformly in A H, we consider again the series (4.7) repeat the argument of the first part of our proof for z lying respectively on two level curves L ri+ε := {z : φ(z) = r i + ε} L re ε := {z : Φ(z) = r e ε}, where ε>0 is sufficiently small. The local uniform convergence of {Rm,n} j =1 to H in A H is then established by letting ε 0 in the two resulting estimates. page 502 of Numer. Math. (1997) 76:

15 Approximation of conformal mappings of annular regions 503 To complete the proof we recall that H is analytic single-valued in the annular region A H that f (z) = z ( z ) (4.12) exp H (ζ)dζ, z 0 where the integration is carried over a rectifiable path γ z lying in A H (see (1.2), (1.3) (1.4)). Since for z belonging to any compact subset of A H there is an upper bound for the length of γ z, it follows from (4.12) (1.10) that ( f (z) f m,n (z) = z z ) ( z ) z 0 exp H (ζ)dζ exp Rm,n(ζ)dζ z 0 z 0 z z { C 1 z 0 H (ζ) R m,n (ζ) } dζ z 0 z C 1 γ z max H (ζ) Rm,n(ζ). ζ γ z z 0 This shows that {f m,n } j =1 also converges locally uniformly to f in A H. Thus, by choosing arbitrary z A γ z A, we obtain from the previous estimate (4.11) that lim sup f f m,n 1/(m+n) ri α. To prove that equality holds, we recall that g(z) g(z 0 )= H(ζ)dζ, z A H, z 0 let z g m,n (z) := Rm,n(ζ)dζ, z A H, m, n =2, 3,..., z 0 observe that for z A H z z { C 2 H (ζ) R m,n (ζ) } dζ f(z) f m,n(z). This gives z 0 z 0 z lim sup g g(z 0 ) g m,n 1/(m+n) Finally, if we assume that z 0 lim sup f f m,n 1/(m+n). lim sup f f m,n 1/(m+n) < ri α, then by applying the same argument as in the first part of this proof to the sequence of rationals {g m,n } j =1 (instead of {R m,n} j =1 ) we can show that this sequence converges to g(z) g(z 0 ) in a region that contains A H in its interior. That is we can show that g, hence H, are analytic on A H which is a contradiction. page 503 of Numer. Math. (1997) 76:

16 504 N. Papamichael et al. Lemma 4.3. For the ONM approximation M m,n to the conformal module M we have that (4.13) 0 M m,n M H Rm,n 2 2, where a b means that there exist two constants K 1 K 2 (that depend only on the geometry of the annular region A) such that bk 1 a bk 2. Proof. One can readily see from (1.5) (1.11) that M m,n M H 2 2 R m,n 2 2. The result follows because, since R m,n is the Fourier sum of H, H 2 2 R m,n 2 2 = H R m,n 2 2. Proof of Theorem 2.2. The sequence of approximations {M m,n } to M, given by (1.11), decrease monotonically with m + n. This is so because, from (1.7), R m,n 2 2 = n (H,η k ) 2. k= m k/= 1 Therefore, the result of Theorem 2.2 follows immediately from Lemma 4.3 (4.11). 4.2 Proofs of Theorems We shall follow essentially the methods of proof used by Andrievskii [1] Gaier [6], in connection with the use of Bieberbach polynomials for the conformal mapping of simply-connected domains. In particular, the proof of Theorem 2.3 involves the following three steps: (i) estimating the L 2 -error in the approximation Rm,n to H, using the extremal property (1.9); (ii) relating the norms r m,n r m,n 2, where r m,n is a rational approximant of the form (4.1); (iii) estimating the uniform error in the approximation f m,n to f, thus proving (2.9). First, however, we present a lemma concerning the boundary behavior of the mapping function f its derivatives. This extends a well-known result of Warschawski [18] (for the conformal mapping of simply-connected domains) to the mapping of annular regions. The lemma follows immediately from Warschawski s result through the localization principle, i.e. by considering a small simply-connected neighborhood of a boundary point. Lemma 4.4. With the hypotheses of Theorem 2.3, suppose that two analytic arcs, of either Γ i or Γ e, meet at a point ζ form there a corner of interior (with respect to A) angle νπ, 0 <ν 2. Then, all the limits (4.14) lim (f (z) f (ζ))/(z ζ) 1 ν = Λ0,Λ 0 /=0, z ζ page 504 of Numer. Math. (1997) 76:

17 Approximation of conformal mappings of annular regions 505 (4.15) lim (z z ζ ζ)n 1 ν f (n) (z) =Λ n, n=1, 2,...,Λ 1 /=0, exist for unrestricted approach z ζ, z A. Similarly, for the inverse mapping τ := f 1, all the limits (4.16) lim (τ(w) ζ)/(w f w f (ζ) (ζ))ν = Λ 0,Λ 0/=0, (4.17) lim (w f w f (ζ) (ζ))n ν τ (n) (w) =Λ n, n=1, 2,...,Λ 1/=0, exist for unrestricted approach w f (ζ), w E. (I) The L 2 -error in the approximation to H Lemma 4.5. Under the hypotheses of Theorem 2.3, there exists a constant K, depending only on the geometry of A, such that H Rm,n { log max(m, n)}1/2 (4.18) 2 K, m, n =2, 3,..., min(m γi, n γe ) where γ i = λ i /(2 λ i ) γ e = λ e /(2 λ e ). Proof. The proof consists of the following three main steps: (i) reducing the problem of estimating the L 2 -error of the approximation to H to that of estimating the uniform error of an approximation to a modified function; (ii) estimating this uniform error by making use of results from the theory of uniform approximations; (iii) establishing the result of the lemma by making use of the extremal property of R m,n. (i) Let h denote the function (4.19) h(z) :=(H(z)+1/z)ω(z), where ω(z) := l (z z j ), j=1 the product extends over all corner points z j A, j =1, 2,..., l. Also, let ν j π,0<ν j <2, j =1, 2,..., l, be respectively the inner angles at the points z j. Then, (1.3) (4.15) imply that for z A, near z j, j =1, 2,..., l, h(z) =O( z z j 1 ν j ), this in turn implies that h is continuous on A. Consider the set of rational functions of the form n (4.20) r m,n (z) = a k z k, k= m page 505 of Numer. Math. (1997) 76:

18 506 N. Papamichael et al. let rm,n be the best uniform approximant to h on A out of this set. Next, for n l, let p l 1 be the Lagrange polynomial interpolating rm,n at the points {z j } l j =1, i.e. p l 1 (z) := l j=1 ω(z) ω (z j )(z z j ) r m,n(z j ), define the function r m,n l by means of the equation (4.21) r m,n(z) p l 1 (z) =ω(z) r m,n l (z). Our next task is to consider using the function r m,n l 1/z in order to approximate H in the L 2 (A) norm. For this we first note that, since h(z j )=0, j=1, 2,..., l, the degree of p l 1 is fixed, (4.22) p l 1 (z) max 1 j l r m,n(z j ) l j =1 ω(z) ω (z j ) z z j C (B) h r m,n, z B, where B C is an arbitrary compact set. Next, we choose a positive number δ m,n < 1 small enough so that each disk S j = {z : z z j δ m,n }, j = 1, 2,...,l,is contained in the annular region A m,n bounded by the level curves L 1+1/n := {z : Φ(z) =1+1/n} L 1 1/m := {z : φ(z) =1 1/m}. (For this it is sufficient to take δ m,n = o(min(m 2, n 2 )), m, n =1, 2,... ; see [15, p. 181].) Since rm,n(z) p l 1 (z) is uniformly bounded on A, by a constant independent of m n, it follows (from Lemma 4.1) that the same is true for A m,n. Thus, by the maximum modulus principle, r m,n l(z) ω(z) z z j C 1, z S j, δ m,n therefore, (4.23) r m,n l (z) C 2 δ m,n, z l S j. Let s j = {z : z z j δ 2 m,n}, j =1, 2,..., l, note that l H (z) ( r m,n l 1/z) 2 2 = H(z)+1/z r m,n l (z) 2 dσ z j =1 A s j (4.24) + H (z)+1/z r m,n l (z) 2 dσ z, A\ l j =1 sj where, as before, dσ z denotes 2-dimensional Lebesgue measure. Then, for the integrals in (4.24), the triangle inequality gives that j=1 page 506 of Numer. Math. (1997) 76:

19 Approximation of conformal mappings of annular regions 507 ( ) 1/2 ( ) 1/2 A s j H(z)+1/z r m,n l (z) 2 dσ z H(z)+1/z 2 dσ z A s j ( ) 1/2 + r m,n l (z) 2 dσ z, A s j j =1, 2,..., l, where from (4.15) (1.3), H(z)+1/z 2 dσ z C z z j 2 ν 2 j dσz = O(δm,n), 2 A s j A s j from (4.23), r m,n l (z) 2 dσ z = O(δm,n). 2 A s j Also, from (4.21) (4.22), 2 H (z)+1/z r m,n l (z) 2 dσ z = h(z) r m,n l (z)ω(z) A\ l j =1 sj A\ l ω(z) dσ z j =1 sj C h rm,n 2 dσ z ω(z) 2, A\ l j =1 sj where it is easy to see that dσ z ω(z) 2 C A\ l j =1 sj 1 δ 2 m,n dr r = O( log δ m,n ). Hence, by combining all the above estimates we find that H ( r m,n l 1/z) 2 2 = O( log δ m,n ) h r m,n 2 + O(δ 2 m,n), i.e. (4.25) H ( r m,n l 1/z) 2 = O( log δ m,n 1/2 h rm,n +δ m,n ). (ii) In order to investigate the rate of decrease of h rm,n we need to consider the smoothness properties of h on A, i.e. we need to determine the behavior of h in the neighborhood of every corner z j, j =1, 2,..., l. For this we proceed as follows: By differentiating (1.3) applying induction we find, from (4.15), that for z A, near z j, j =1, 2,..., l, (H(z)+1/z) (k) =O( z z j 1 ν j k 1 ), k =0, 1, 2,.... Hence, by using (4.19), we can verify that for z A, near z j, j =1, 2,..., l, (4.26) h (k) (z) =O( z z j 1 ν j k ), k =0, 1, 2,... ; see the proof of Theorem 3 in [7]). Next, with the help of Cauchy s formula we express h as page 507 of Numer. Math. (1997) 76:

20 508 N. Papamichael et al. (4.27) h(z) =h e (z)+h i (z), z A, where h e is analytic inside continuous on Γ e := Ω h i is analytic outside continuous on Γ i := G. Moreover, since h e is analytic on Γ i, the function h i has the same boundary properties on Γ i as the function h. Similarly, the behavior of h on Γ e is inherited by h e. Thus, in particular, from (4.26) we have that h e (k) (z) = O( z z j 1 ν k (4.28) j ), k =0, 1, 2,..., for z Γ e near a corner point z j Γ e. Let Ψ be the conformal mapping of D = {w : w > 1} onto the exterior of Γ e normalized by Ψ( ) = lim w Ψ(w)/w > 0. Then, by following the proof of Theorem 3 of [7], we find that (h e Ψ) (qj 1) Lip β j in a neighborhood of w j = Ψ 1 (z j ), z j Γ e, where 2 = q j + β j, q j N, 0 <β j 1. ν j Further, since λ e =2 max zj Γ e ν j, there exists a polynomial P n of degree at most n in z such that (4.29) h e P n Ω = O(n γe ), n =1, 2,..., where γ e = λ e /(2 λ e ); see Equation (2.3) of [7]. Similarly, there exists a polynomial Q m of degree at most m in 1/z such that (4.30) h i Q m (1/z) G = O(m γi ), m =1, 2,..., where γ i = λ i /(2 λ i ), λ i =2 max zj Γ i ν j. Thus, since h rm,n h e P n Ω + h i Q m (1/z) G, (4.29) (4.30) give that h rm,n = O(max(m γi, n γe )), m, n =1, 2,.... Finally, we recall that the quantity δ m,n in (4.25) can be made sufficiently small so that δ m,n max(m γi, n γe ). Therefore, ( ) {log max(m, n)} 1/2 (4.31) H ( r m,n l 1/z) 2 = O, m, n =2, 3,.... min(m γi, n γe ) (iii) Let r m,n l (z) 1/z = n l k= m ã k z k. Then, from Lemma 4.2 we have that H ( r m,n l 1/z ã 1 /z) 2 K H ( r m,n l 1/z) 2, where K depends only on the geometry of A. Therefore, (4.18) follows from (4.31) the extremal property (1.9) of Rm,n. page 508 of Numer. Math. (1997) 76:

21 Approximation of conformal mappings of annular regions 509 (II) Relating the norms The result of this subsection (i.e. of Lemma 4.6) holds under much weaker assumptions about the geometry of A than those of Theorem 2.3. In fact, for our purposes here we need only assume that the two boundary components Γ i Γ e of A are arbitrary Jordan curves. We note that Lemma 4.6 is the doubly-connected analogue of Andrievskii s lemma [1] for simply-connected domains with quasiconformal boundaries. We also note that a simpler proof some extensions of Andrievskii s lemma were given by Gaier [5], that Gaier s approach served as the basis for generalizing the lemma to arbitrary Jordan domains in [13] that our method of proving Lemma 4.6 is also based on Gaier s approach. Lemma 4.6. Let E := {w : 1 < w < M} assume that the mapping function τ := f 1 : E A satisfies a Hölder condition (4.32) τ(w 1 ) τ(w 2 ) C w 1 w 2 ν,w 1,w 2 E, for some ν (0, 1]. Also, let r m,n be a rational function of the form (4.33) n r m,n (z) = a k z k, m, n=2, 3,..., k= m such that there exist a point ζ A a constant b > 0 for which r m,n (ζ) b. Then, there exists a constant K, depending only on ζ, b the geometry of A, such that (4.34) r m,n K {log max(m, n)} 1/2 r m,n 2. Proof. In the proof we shall use the letter K to denote constants (not necessarily the same), depending only on ζ, b the geometry of A. We assume, without loss of generality, that r m,n 2 1 note that the function h(w) :=r m,n (τ(w)) is analytic in E. Therefore, h has a Laurent expansion (4.35) h(w) = c k w k,w E, k= hence (4.36) h (w) 2 dσ w = π k c k 2 (M 2k 1). E k= On the other h, (4.37) h (w) 2 dσ w = r m,n(z) 2 dσ z 1. E A Thus, for w = ρ, 1 <ρ<m, the use of Schwarz s inequality gives that page 509 of Numer. Math. (1997) 76:

22 510 N. Papamichael et al. (4.38) h(w) k= c k ρ k c π k= k/=0 ρ 2k k(m 2k 1) Regarding the sum on the right of (4.38), this can be estimated by means of (4.39) k=1 1 k= ρ 2k k(m 2k 1) ρ 2k k(m 2k 1) = M 2 M 2 1 k=1 M 2 M 2 1 log M M ρ M 2 M 2 1 1/2 ρ 2k km 2k M 2 M 2 1 log 1 1 ρ/m k=1 M 2 (4.40) M 2 1 log M ρ 1. Also, for any fixed ρ 0,1<ρ 0 <M, we have that (4.41) c 0 1 2π w =ρ 0 1 kρ 2k M 2 M 2 1 log 1 1 1/ρ h(w) dw max r m,n (t). w f (t) =ρ 0 Further, it is easy to see that the point ζ can be connected to any other point z {t: f(t) =ρ 0 }by a path γ z such that γ z < K 1 dist(γ z, A) > K 2, where K 1 K 2 are positive constants independent of z. Thus, with the help of we find that (4.42) r m,n(t) 1 r 1 πdist(t, A) m,n 2, t A, πdist(t, A) r m,n (z) r m,n (ζ) + z for z {t: f(t) =ρ 0 }. Hence, from (4.38) (4.42), (4.43) h(w) b+ K 1 πk2 + ζ ( M log π(m2 1) r m,n(t)dt b + K 1, πk2 M 2 ) 1/2, (M ρ)(ρ 1) for w = ρ, 1 <ρ<m. Consider now an annular region A ρi,ρ e := {z : ρ i < f (z) <ρ e }, where 1 <ρ i <ρ e <M, denote by L ρe L ρi its outer inner boundary components. Also, let Ω ρe G ρi denote respectively the domains exterior to L ρe interior to L ρi, let Φ ρe φ ρi denote the conformal mappings Φ ρe : Ω ρe D = {w : w > 1}, with Φ ρe ( ) = lim z Φ ρe (z)/z > 0, φ ρi : G ρi D = {w : w < 1}, with φ ρi (0) = 0 φ ρ i (0) > 0. Next, denote by A m,n the annular region bounded by the level curves L ρe,n := {z :. page 510 of Numer. Math. (1997) 76:

23 Approximation of conformal mappings of annular regions 511 Φ ρe (z) =1+1/n} L ρi,m := {z : φ ρi (z) =1 1/m}, note that ρ i ρ e can be chosen so that the region A lies entirely within A m,n. The latter can be shown as follows: From [15, p. 181] we have that dist(l ρe,n, L ρe ) diaml ρ e 4n 2 dist(l ρi,m, L ρi ) dist(0, L ρ i ) 4m 2. On the other h, τ(m e iθ ) τ(ρ e e iθ ) C(M ρ e ) ν τ(e iθ ) τ(ρ i e iθ ) C(ρ i 1) ν, where θ [0, 2π). Therefore, by choosing ρ i ρ e so that diaml ρe 4n 2 > C (M ρ e ) ν dist(0, L ρi ) 4m 2 > C (ρ i 1) ν, we can ensure that A A m,n. Finally, from Lemma 4.1 (4.43) we find that r m,n max r m,n (z) z A m,n ( (1+1/n K { log max(m, n)} 1/2 ) n, ( ) ) m max 1 1/m K{log max(m, n)} 1/2. We remark that the result of Lemma 4.6 is sharp in the sense that the factor { log max(m, n)} 1/2 cannot be improved even for E = {w :1< w <M}. This can be seen by considering the functions r 0,n (z) = n k=1 z k km k r m,0 (z) = 1 k= m z k k. page 511 of Numer. Math. (1997) 76:

24 512 N. Papamichael et al. (III) Estimating the errors f f m.n M M m,n Proof of Theorem 2.3. We now use the letter C to denote constants, not necessarily the same, that depend only on the geometry of A. We first integrate along some rectifiable path in A, from z 0 to z, obtain the single-valued analytic functions z (4.44) g(z) g(z 0 )= H(ζ)dζ, z A, z 0 z (4.45) g m,n (z) = Rm,n(ζ)dζ, z A, m, n =2, 3,.... z 0 We then seek to estimate the error g g(z 0 ) g m,n, by following the technique of [1] [6]. For this we proceed as follows: For any m, n N satisfying (2.8) we choose k(m) k(n) so that 2 k(m) m < 2 k(m)+1 2 k(n) n < 2 k(n)+1. Then, it follows from Lemma 4.5 that g 2 k(m)+1,2 { log max(m, n)}1/2 (4.46) m,n k(n)+1 g 2 C. min(m γi, n γe ) Next, we note that the function g 2 k(m)+1,2 k(n)+1 g m,n satisfies all the conditions of Lemma 4.6. (This follows from Lemma 4.4 the fact that Rm,n converges to H locally uniformly in A.) Therefore, (4.47) Similarly, g 2 k(m)+1,2 k(n)+1 g m,n C log max(m, n) min(m γi, n γe ). (4.48) g 2 j +1,2 k+1 g 2 j,2 k C max(j +1,k+1) min(2 j, j, k =1, 2,.... γi, 2 kγe ) Since, for any z A, g(z) g(z 0 ) g m,n (z) = {g 2 k(m)+1,2 k(n)+1(z) g m,n(z)} + {g 2 j +1,2 +1(z) k(n) k(m)+j g 2 j,2k(n) k(m)+j (z)}, j =k(m)+1 it follows, from (4.47) (4.48), that log max(m, n) g g(z 0 ) g m,n C min(m γi, n γe ) max(j +1,k(n) k(m)+j +1) + C min(2 j γi, 2 (k(n) k(m)+j. )γe ) Therefore, in view of (2.8), j =k(m)+1 page 512 of Numer. Math. (1997) 76:

25 Approximation of conformal mappings of annular regions 513 (4.49) log max(m, n) g g(z 0 ) g m,n K min(m γi, n γe ), where K depends on the geometry of A the numbers c 1 c 2 in (2.8). Finally, by recalling f (z) =ze g(z) = z z 0 e g(z) g(z0), z A taking into account (1.10), we obtain f f m,n f 1 f m,n /f M 1 exp{g m,n (g g(z 0 ))} C g m,n (g g(z 0 )). Thus, (2.9) follows from (4.49). Proof of Theorem 2.4. This follows at once from Lemmas References 1. Andrievskii, V.V. (1983): Convergence of Bieberbach polynomials in domains with quasiconformal boundary. Ukrainian Math. J. 35, Curtiss, J.H. (1966): Solutions of the Dirichlet problem in the plane by approximation with Faber polynomials. SIAM J. Numer. Anal. 3, Gaier, D. (1964): Konstructive Methoden der Konformen Abbildung. Springer, Berlin, Göttingen, Heidelberg 4. Gaier, D. (1987): Lectures on Complex Approximation. Birkhäuser, Boston 5. Gaier, D. (1987): On a polynomial lemma of Andrievskii. Arch. Math. 49, Gaier, D. (1988): On the convergence of the Bieberbach polynomials in regions with corners. Constr. Approx. 4, Gaier, D. (1992): On the convergence of the Bieberbach polynomials in regions with piecewise analytic boundary. Arch. Math. 58, Kokkinos, C.A., Papamichael, N., Sideridis, A.B. (1990): An orthonormalization method for the approximate conformal mapping of multiply-connected domains. IMA J. Numer. Anal. 9, Papamichael, N. (1993): Two numerical methods for the conformal mapping of simply- multiply-connected domains. In: Lipitakis, E.A. (ed.) Advances in computer mathematics its applications, pp World Scientific, New Jersey 10. Papamichael, N., Kokkinos, C.A. (1984): The use of singular functions for the approximate conformal mapping of doubly-connected domains. SIAM J. Sci. Stat. Comput. 5, Papamichael, N. Warby, M.K. (1984): Pole-type singularities the numerical conformal mapping of doubly-connected domains. J. Comp. Appl. Math. 10, Papamichael, N. Warby, M.K. (1986): Stability convergence properties of Bergman kernel methods for numerical conformal mapping. Numer. Math. 48, Pritsker, I.E. (1992): Order comparison of norms of polynomials in regions of the complex plane. Ukrainian Math. J. 43, Tamrazov, P.M. (1973): The solid inverse problem of polynomial approximation of functions on a regular compactum. Math. USSR Izv. 7, Tamrazov, P.M. (1975): Smoothness Polynomial Approximations. Naukova Dumka, Kiev 16. Walsh, J.L. (1969): Interpolation Approximation by Rational Functions in the Complex Domain. Colloquium Publications, vol. 20, Amer. Math. Soc. Providence, RI 17. Warby, M.K. (1992): BKMPACK User Guide, Dept of Maths Stats, Brunel University Uxbridge, UK 18. Warschawski, S.E. (1955): On a theorem of L. Lichtenstein. Pacific J. Math. 5, This article was processed by the author using the LaT E X style file pljour1m from Springer-Verlag. page 513 of Numer. Math. (1997) 76: