Numerische Mathematik

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1 Numer. Math. 2002) 91: Digital Object Identifier DOI) /s Numerische Mathematik L 2 -Approximations of power and logarithmic functions with applications to numerical conformal mapping V.V. Maymeskul, E.B. Saff, N.S. Stylianopoulos 1 Department of Mathematics, University of South Florida, 4202 East Fowler Ave., Tampa, FL 33620, USA; vmaymesk@math.usf.edu 2 Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, 4202 East Fowler Ave., Tampa, FL 33620, USA; esaff@math.usf.edu 3 Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, Nicosia, Cyprus; nikos@ucy.ac.cy Received September 1, 1999 / Published online August 17, 2001 c Springer-Verlag 2001 Dedicated to Nick Papamichael in honor of his sixtieth birthday and in recognition of his substantial contributions to the subject of this paper. Summary. For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best L 2 G) polynomial approximation to functions of the form z τ) β, β > 1, and z τ) m log l z τ), m> 1, l /=0, where τ L. Furthermore, Andrievskii s lemma that provides an upper bound for the L G) norm of a polynomial p n in terms of the L 2 G) norm of p n is extended to the case when a finite linear combination independent of n) of functions of the above form is added to p n. For the case when the boundary of G is piecewise analytic without cusps, the results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained. Mathematics Subject Classification 1991): 30E10, 65E05, 30C30 The research of this author was supported, in part, by the U.S. National Science Foundation under grant DMS and by a University of Cyprus collaborative research grant The research of this author was supported, in part, by a grant from the University of Cyprus Correspondence to: E.B. Saff

2 504 V.V. Maymeskul et al. 1Introduction and notations If G a bounded Jordan domain whose boundary L is piecewise analytic without cusps and f 0 is a normalized conformal mapping of G onto a disk, then Bieberbach polynomials π n can be used to approximate f 0 see Sect. 4). In [9], [10], and [11], D. Gaier obtained estimates of the form ) log n 1.1) f 0 π n L G) = O n s n ) for the uniform norm of the error in approximation, where s is explicitly determined from the interior angles α j π, 0 <α j < 2, at the corners τ j of L see [11, Theorem 2]). To obtain 1.1), two essential ingredients are the Lehman formulas [13] for the asymptotic expansion of f 0 near the corners τ j, and Andrievskii s lemma [2] which provides an estimate for the L G) norm of a polynomial in terms of the L 2 G) norm of its derivative. Since the Lehman formulas involve power functions of the form 1.2) f β,τj z) =z τ j ) β and logarithmic functions of the form 1.3) g m,l,τj z) =z τ j ) m log l z τ j ), it is natural to expect an improvement in the convergence rate 1.1) if the ordinary Bieberbach polynomials are replaced by augmented Bieberbach polynomials that include suitable singular functions of the above power and logarithmic type. This was first observed by Levin, Papamichael and Sideridis [14] and subsequently used by Papamichael, Kokkinos, Hough and Warby for improving the convergence rates of certain orthonormalization methods associated with the mapping of interior, exterior and doublyconnected domains; see e.g. [16], [15], [19] and [18]. One goal of the present paper is to obtain sharp estimates for the improvement gained in using such augmented Bieberbach polynomials. For this purpose, upper and lower bounds are derived for the error in the best L 2 G) n-th degree polynomial approximation to functions of the form 1.2) and 1.3). These estimates cf. Corollary 2.2) are, in fact, obtained in the more general setting when the boundary L is a quasiconformal curve. In Lemma 2.3 and Corollary 2.5 we present extensions of Andrievskii s lemma to the case when one or several singular functions of the form 1.2), 1.3) are adjoined to ordinary polynomials to form augmented polynomials. In Sect. 3 we apply the above results to obtain upper and lower estimates for the error in approximating f 0 by augmented Bieberbach polynomials

3 L 2 -Approximations of power and logarithmic functions with applications 505 see Theorems 3.1 and 3.2). The lower estimates provide new sharpness results, even in the case of classical Bieberbach polynomials cf. 3.19)). Finally, in Sect. 4, we present numerical computations that illustrate our theoretical results. For convenient reference we provide here a listing of the main notations used throughout the paper. 2 Approximation of power and logarithmic functions In what follows we denote by C, c, C 1,... constants whose values either are absolute or depend on parameters not essential for arguments; at least, they are independent of n. For quantities A>0, B>0, which depend on some parameters, we use the notation A B inequality with respect to the order) if A CB; the expression A B means that A B and B A simultaneously. Unless otherwise specified, we assume throughout that G is a bounded Jordan domain with quasiconformal boundary L, and z 0 G. Denote by yζ) a quasiconformal reflection with respect to L, i.e., an orientationchanging quasiconformal mapping of the extended plane C onto itself that carries G into its complement Ω := C \ G and conversely, leaves the points of the curve L fixed, and satisfies y z 0 )=, y ) =z 0 [1, Chapter IV, D]). Let Φz) denote the Riemann function that conformally and univalently maps Ω onto the complement of the unit disk ID normalized by the conditions Φ ) =, Φ ) > 0. This function can be extended to a homeomorphism between closed domains, and we keep the previous notation for the extension. We extend Φz) to a quasiconformal map of the plane onto itself by setting for z G { 1/Φ [yz)], z /= z0, Φz) = 0, z = z 0. Denote by L r, r>0, the r-th level curve of the function Φz), i.e., L r := {ζ : Φζ) = r}. Also, let G r := Int L r and Ω r := C \ G r. For z C and u>0we define [ d u z) := max z Ψ Φz)+ue iϕ ]. 0 ϕ<2π Remark 2.1 Recall that any quasiconformal map F of C onto itself satisfies the so-called D-property, i.e., 2.1) min F ζ) F z) max F ζ) F z), z C. ζ: ζ z =r ζ: ζ z =r

4 506 V.V. Maymeskul et al. This implies that for any z L 1 u, 0 <u<1, 2.2) dist z,l) d u z) and, for any fixed 0 <ε<1, z G 1+ε/n \G 1 ε/n, the quantity d 1/n z) has the same order as n ) as the traditional quantity in the approximation theory of functions of complex variable - the distance from z to L 1+1/n see [5]). In particular, it follows from Warschawski s results in [21] that if L consists, near τ L, of two analytic arcs meeting at an interior angle απ, 0 <α<2, atτ, then 2.3) d 1/n τ) n α 2. For z C, r>0, denote by Dz,r) :={ζ : ζ z <r} the open disk centered at z with the radius r. Let ω := Φτ), γ τ := Ψ {w : arg w = arg ω, w 1}). For noninteger β > 1 and arbitrary m> 1, l /=0denote by f β,τ z) and g m,l,τ z) branches of z τ) β and z τ) m log l z τ), respectively, which are analytic in C \ γ τ.givenm, l we also define { l = l l 1, if m 0, l 1 are integer; 2.4) m, l) = l, otherwise. In this paper we obtain two-sided estimates for the error in the best L 2 -approximation E n,2 f,g) := min f p p:deg p n L 2 G) of f β,τ z) and g m,l,τ z) by polynomials of degree at most n, n =1, 2,..., and apply these estimates to the problem of approximation of the Riemann function f 0 z) that conformally and univalently maps the domain G onto the disk D 0,r 0 ) with f 0 z 0 )=0, f 0 z 0)=1r 0 is the conformal radius of G with respect to z 0 ). In our arguments we need the following two auxiliary results. The first lemma describes the properties of d 1/n z) cf. [5, Lemma 2]). Lemma 2.1 There exists a constant c = cg) > 0 such that for arbitrary points z G and ζ Ω 1 1/n : 1) if ζ z d 1/n z), then 2.5) d 1/n ζ) d 1/n z); 2) if ζ z d 1/n z), then 1/c d 1/n z) ζ z d c 1/n ζ) d 1/n z) 2.6) ζ z ζ z ; 3) for any 0 <v<u<1, 2.7) u/v) c d u z)/d v z) u/v) 1/c.

5 L 2 -Approximations of power and logarithmic functions with applications 507 The next lemma is a modification of the Tamrazov s result [12, Theorem 1]. Lemma 2.2 Suppose that for a polynomial p n of degree at most n, n = 1, 2,..., some positive constants M, ρ, and a point τ L the inequality ) p n z) M 1+ z τ C 1 ρ, C 1 > 0, is satisfied for all points z L. Then, for any fixed positive constant C 2 and all z D τ,ρ) Int L 1+C2 /n, there holds p n z) C 3 M, C 3 = C 3 C 1,C 2 ) 1. We remark that this result holds more generally for any bounded continuum with connected complement cf. [4, Theorem 6.1]). Let ωδ) be continuous, monotonic and positive on 0, ). Suppose there exist constants c 1,C 1 > 0, c 2 [0, 1) and C 2 0 such that for all δ>0 and t 1 2.8) c 1 t c 2 ωδ) ωtδ) C 1 t C 2 ωδ). Note that 2.8) is clearly satisfied with C 1 = c 1 =1)ifforx>0 small enough the functions ωx)/x C 2 and ωx)x c 2 are decreasing and increasing, respectively. In particular, if we specify the monotonicity of ωδ), one of the inequalities in 2.8) becomes obvious say, for increasing functions - the left one). Note also that the left-hand side of 2.8) implies 2.9) 1 0 ωx)dx 1 c ω1) x c dx <. 2 Theorem 2.1 Let gz) be an analytic and single-valued branch in C \ γ τ of some multi-valued analytic function having branch points at τ L and. Denote by g ± ζ), ζ γ τ, the boundary values of gz) on γ τ. Suppose the functions g ± ζ), ζ γ τ, ζ τ small enough, satisfy the inequality 2.10) Then 2.11) g + ζ) g ζ) ω ζ τ ). E n,2 g, G) d 1/n τ) ω d 1/n τ) ). Proof. Fix R>1 such that 2.10) holds true on the subarc γ := γ τ G R of γ τ. It was shown in [3] that γ is a quasi-smooth arc i.e., for any subarc, its length and the distance between endpoints have the same order) and, for any ζ γ, z G 2.12) dist ζ,l) ζ τ, dist z,γ) z τ

6 508 V.V. Maymeskul et al. and, hence, 2.13) ζ z ζ τ + z τ. Because of 2.9), it is easy to verify that 2.10) implies the validity of the Cauchy formula for the function gz) in the domain G R \ γ, i.e., under suitable orientation of all the arcs, gz) = 1 g + ζ) g ζ) gζ) dζ + 2πi ζ z ζ z dζ. γ L R The last integral represents a function that is analytic up to the R-th level line of the domain G and, therefore, can be approximated by polynomials geometrically fast with the rate R1 n for any fixed R 1 1,R)). So, we can restrict ourselves to the approximation of the Cauchy-type integral along γ. Set fz) := γ g + ζ) g ζ) dζ. ζ z In order to estimate E n,2 f,g) we use Dzjadyk s polynomial kernels to approximate the Cauchy kernel 1/ζ z). By [5, Lemma 3], for every fixed m>0and R>1and for all n =1, 2,...there exists a polynomial kernel K n,m ζ,z) of degree in z) at most n with the following property: for any z G and ζ G R \ G 1 ζ z K n,mζ,z) 1 ) d 1/n z) m 2.14). ζ z ζ z + d 1/n z) We define approximating polynomials p n by the formula p n z) := g + g ) ζ)k n,m ζ,z)dζ. γ Set r := z τ and d n := d 1/n τ). First, suppose that r d n, and denote γ := γ D τ,d n ). Then, by using 2.14), 2.13), 2.5) and taking into account the quasi-smoothness of γ,weget fz) p n z) 2.15) γ d n 0 ω ζ τ ) ζ τ + z τ dζ + d 1/n z) ) m γ\ γ ω ζ τ ) dζ ζ τ m+1 ωt) t + r dt +d n) m d n ωt) t m+1 dt = I 1 +d n ) m I 2.

7 L 2 -Approximations of power and logarithmic functions with applications 509 For the first integral we have r d n I 1 = + ωt) t + r dt 1 r 0 ωr) r 1 c 2 r r 0 dt t c 2 r 0 d n ωt) ωt)dt + t + r dt d n + max {ωr),ωd dt n)} t + r 2.16) ωr)+ωr)+ω d n )) log d n r. To estimate I 2 we choose m>c 2, where the constant C 2 is taken from 2.8). Then and 2.17) ωt) t m+1 = ω t/d n) d n ) t m+1 ω d n) d n ) C 2 tc 2 m 1 I 2 ω d n) d n ) C 2 d n t C2 m 1 dt Substituting 2.16) and 2.17) into 2.15) we get r r ω d n ) d n ) m. 2.18) fz) p n z) ωr)+ω d n )) log ed n r. Suppose now that r>d n, and set γ := γ Dτ,r). Using 2.6), in a manner similar to the previous case we obtain 2.19) fz) p n z) d 1/n z) ) m r m+1) + γ\ γ γ ω ζ τ ) dζ ζ τ m+1 ω ζ τ ) dζ d 1/n z) ) r m r m+1) ωt) ωt)dt + t 0 r ) d1/n z) m ) cm dn ωr) ωr). r r m+1 dt

8 510 V.V. Maymeskul et al. Then from 2.18) and 2.19) we conclude f p n 2 L 2 G) ωr)+ωd n )) 2 log 2 ed n r dxdy Dτ,d n) + C\Dτ,d n) ω 2 r) ) 2mc dn dxdy r 2.20) d n 0 rω 2 r) log 2 ed n r dr + ω d n) +ω 2 d n ) d n 0 d n r log 2 ed n r dr +d n) 2mc 0 rωr) log 2 ed n r dr = J 1 + ω d n ) J 2 + ω 2 d n ) J 3 +d n ) 2mc J 4. d n ω 2 r) dr r2mc 1 According to 2.8), for all r 0,d n ), there holds ) c2 dn ωr) ω d n ). r So, using integration by parts, we easily get Similarly, J 1 ω 2 d n )d n ) 2c 2 d n 0 r 1 2c 2 log 2 ed n r dr d nω d n )) 2. J 2 d n ) 2 ω d n ) and J 3 d n ) 2. Furthermore, to estimate J 4 we apply the right-hand side of 2.8) to deduce that J 4 ω2 d n ) dr d n ) 2C 2 r 2mc C 2) 1 d n) 21 mc) ω 2 d n ), provided m>c 2 +1)/c. Combining these estimates we finally get d n f p n 2 L 2 G) d nω d n )) 2, and 2.11) follows.

9 L 2 -Approximations of power and logarithmic functions with applications 511 Since for all ζ γ τ, ζ τ small enough, f + β,τ ζ) f β,τ ζ) f + f β,τ ζ) + β,τ ζ) ζ τ β, g + m,l,τ ζ) g m,l,τ ζ) ζ τ m log + ζ τ) ) l log ζ τ) ) l,m,l IN; g + g m,l,τ ζ) + m,l,τ ζ), otherwise ζ τ m log 1 l ζ τ, applying Theorem 2.1 we get Corollary 2.1 For all n 1 large enough there holds 2.21) 2.22) E n,2 f β,τ,g) d 1/n τ) ) β+1, E n,2 g m,l,τ,g) d 1/n τ) ) m+1 log l 1 d 1/n τ). Theorem 2.2 For all n 1 large enough there holds 2.23) 2.24) E n,2 f β,τ,g) d 1/n τ) ) β+1, E n,2 g m,l,τ,g) d 1/n τ) ) m+1 log l 1 d 1/n τ). Proof. Suppose that E n,2 := E n,2 f β,τ,g) d 1/n τ) ) β+1. Denote by p n z), deg p n n, the polynomial of best L 2 -approximation to the function f β,τ z). Then, for any fixed k IN and any point z L 1 1/n, applying Lemma 1 of [8, p. 4] and taking into account 2.2), we get f β,τ p n ) k) z) k! E n,2 k +1 dist z,l)) k+1 E n,2 d1/n z) ) k ) Let τ n := Ψ [1 1/n) Φτ)] L 1 1/n. It follows from 2.1) that 2.26) and 2.5) implies 2.27) d 1/n τ n ) τ τ n d 1/n τ n ), d 1/n τ n ) d 1/n τ).

10 512 V.V. Maymeskul et al. Denote D n := D τ n, 2d 1/n τ n ) ). Setting k := [β] +1, for z L 1 1/n we conclude: 1. if z D n, it follows from 2.5), 2.25), 2.26) and 2.27) that f β,τ p n ) k) z) E n,2 d1/n τ n ) ) k+1 d 1/n τ) ) β k d 1/n τ) ) {β} 1, where {β} := β [β]. Since for z D n L 1 1/n, n large enough, 2.28) z τ d 1/n z) d 1/n τ n ) d 1/n τ), we have p k) n z) f k) β,τ z) fβ,τ + p n ) k) z) d 1/n τ) ) {β} ) ; 2. if z D n then 2.5), 2.6) and 2.25) imply ) f β,τ p n ) k) E n,2 d1/n z) d1/n τ n ) ) τ n ) k k d 1/n z) d 1/n τ) ) ) {β} 1 z τn k d 1/n z) d 1/n τ) ) {β} 1 z τ n k/c d 1/n τ n ) and, therefore, p k) n z) z τ {β} 1 + d 1/n τ) ) {β} 1 z τ n k/c d 1/n τ n ) z τ n {β} 1 + d 1/n τ) ) {β} 1 z τ n k/c d 1/n τ n ) d 1/n τ) ) {β} 1 1+ z τ n k/c) 2.30). d 1/n τ n ) Combining the estimates 2.29) and 2.30) we get for all z L 1 1/n p k) n z) d 1/n τ) ) {β} 1 1+ z τ n k/c). d 1/n τ n ) Note that we can choose the constant C 2 such that D n lies inside the 1 + C 2 /n)-th level line for domain G 1 1/n see the proof of [5, Lemma 1]). Applying Lemma 2.2 to the polynomial p k) n, considered in G 1 1/n,weget p k) z) C 4 d1/n τ) ) {β} ), z D τn,d 1/n τ n ) ), n

11 L 2 -Approximations of power and logarithmic functions with applications 513 with some constant C 4 = C 4 β,g). On the other hand, for z G f k) 2.32) β,τ z) ={β})k z τ {β} 1, where {β}) k := β β 1) {β}. For ε>0 small enough take 2.33) z ε D τ,εd 1/n τ) ) Ψ [Φτ),Φτ n )]). As Φz) is a quasiconformal mapping, the arc Ψ [Φτ),Φτ n )]) satisfies an inequality similar to the first one in 2.12), and we have 2.34) dist z ε,l) c 1 εd 1/n τ). Since 1 < {β} 1 < 0, 2.31) and 2.32) allow us to choose ε>0 take, for instance, ε ={β}) k /2C 4 )) 1/1 {β}) ) such that f β,τ p n ) k) z ε ) 1 2 {β}) k εd1/n τ) ) {β} ). At the same time, by Lemma 1 of [8, p. 4] applied in the disk Dz ε, dist z ε,l)) f β,τ p n ) k) z ε ) E n,2 = f β,τ p n L2 G) k! dist z ε,l)) k+1. k +1 This inequality together with 2.34) and 2.35) implies E n,2 {β}) k ck+1 1 k! εd1/n τ) ) {β}+k = c2 β,g) d 1/n τ) ) β+1, k +1 which completes the proof of 2.23). In 2.24), the case when l = l can be treated similarly. So, we assume that l = l 1, i.e., m 0 and l 1 are both integers. Suppose that E n,2 := E n,2 g m,l,τ,g) d 1/n τ) ) m+1 log l 1 d 1/n τ), and let p n z), deg p n n, be the polynomial of best L 2 -approximation to the function g m,l,τ z). Then, similarly to 2.25), for any point z L 1 1/n we have 2.36) g m,l,τ p n ) m+1) z) In the previous notations, we get for z L 1 1/n : E n,2 d1/n z) ) m+2.

12 514 V.V. Maymeskul et al. 1. if z D n, it follows from 2.5), 2.26), 2.27) and 2.36) that g m,l,τ p n ) m+1 E n,2 z) d1/n τ n ) ) m+2 logl 1/d1/n τ) ). d 1/n τ) Note that g m) m,l,τ z) is, in fact, a polynomial on logz τ) of degree l, i.e., 2.37) g m) m,l,τ z) =q llogz τ)). Therefore, using 2.28) we get g m+1) m,l,τ z) log l 1/ z τ ) z τ logl 1/d1/n τ) ) d 1/n τ) and 2.38) p n m+1) z) g m+1) m,l,τ z) gm,l,τ + p n ) m+1) z) logl 1/d1/n τ) ) ; d 1/n τ) 2. if z D n, then 2.5), 2.6), and 2.36) imply g m,l,τ p n ) m+1 z) ) E n,2 d1/n d1/n τ n ) ) τ n ) m+2) m+2) d 1/n z) logl 1/d1/n τ) ) d1/n τ) d 1/n τ n ) d 1/n τ n ) Thus, p m+1) n 2.39) logl 1/d1/n τ) ) d 1/n τ) z) q l logz τ)) z τ logl C/d1/n τ n ) ) d 1/n τ n ) logl 1/d1/n τ) ) d 1/n τ) z τ n d 1/n τ n ) ) m+1) ) z τn m+2) d 1/n z) m+2)/c + logl 1/d1/n τ) ) z τ n d 1/n τ) d 1/n τ n ) + logl 1/d1/n τ) ) z τ n d 1/n τ) d 1/n τ n ) ) 1+ z τ n C 1 d 1/n τ n ),. m+2)/c m+2)/c

13 L 2 -Approximations of power and logarithmic functions with applications 515 where C 1 =m +2)/c. Estimates 2.38) and 2.39) imply p m+1) n z) logl 1/d1/n τ) ) 1+ z τ n d 1/n τ) d 1/n τ n ) C 1 ), z L 1 1/n, and, consequently, p m+1) n z) logl 1/d1/n τ) ) 2.40), z D τ n,d d 1/n τ) 1/n τ n ) ). On the other hand, because of 2.37), for z τ small enough there holds g m+1) m,l,τ z) log l 1/ z τ ) 2.41). z τ This inequality and 2.40) allow us to choose ε>0such that at the point z ε, defined by 2.33), we have g m,l,τ p n ) m+1 z ε ) logl 1/d1/n τ) ) 2.42). εd 1/n τ) Now applying Lemma 1 of [8, p. 4] we get E n,2 = g m,l,τ p n L2 G) g m,l,τ p n ) m+1 z ε ) m + 1)! dist z ε,l)) m+2, m +2 which together with 2.34) and 2.42) implies E n,2 εd 1/n τ) ) m+1 log l 1 d 1/n τ). From Corollary 2.1 and Theorem 2.2 we immediately deduce Corollary 2.2 If G is a quasidisk, τ L, then 2.43) E n,2 f β,τ,g) d 1/n τ) ) β+1, E n,2 g m,l,τ,g) d 1/n τ) ) m+1 log l 1 d 1/n τ) as n. In particular, if near τ) L consists of two analytic arcs forming an interior angle απ, then see 2.3)) where l is defined by 2.4). E n,2 f β,τ,g) n α 2)β+1), E n,2 g m,l,τ,g) n α 2)m+1) log l n,

14 516 V.V. Maymeskul et al. We can prove even more. Theorem 2.3 Let τ L, {a n,j } s j=1, n =1, 2,..., be arbitrary complex numbers, and, for each j, let h j denote either f βj,τ, β j > 1 noninteger, or g mj,l j,τ, m j > 1. Suppose also that h j /= h k for j /= k. Then for the function s 2.44) hz) =hn, z) := a n,j h j the inequalities 2.45) are satisfied. E n,2 h, G) j=1 s a n,j E n,2 h j,g) n =1, 2,..., j=1 Proof. The upper estimate in 2.45) is trivial. Proving the lower one, we restrict ourselves, for simplicity, to the case s =2, i.e., we have to show that for 2.46) E n,2 a n,1 h 1 + a n,2 h 2,G) a n,1 E n,2 h 1,G)+ a n,2 E n,2 h 2,G). If either a n,1 =0or a n,2 =0, then 2.46) is trivially satisfied. So assume that neither of a n,1, a n,2 is zero. Taking into account 2.43) we can further assume that E n,2 h 2,G)=o E n,2 h 1,G)) as n. Without loss of generality, we take a n,2 =1.If a n,1 < 1 E n,2 h 2,G) 2 E n,2 h 1,G) or a n,1 > 2 E n,2 h 2,G) E n,2 h 1,G), then 2.46) holds with c =1/4. For instance, in the first case, the triangle inequality yields E n,2 a n,1 h 1 + h 2,G) E n,2 h 2,G) a n,1 E n,2 h 1,G) Suppose now that 2.47) 1 2 E n,2 h 2,G) 1 4 a n,1 E n,2 h 1,G)+E n,2 h 2,G)). 1 E n,2 h 2,G) 2 E n,2 h 1,G) a n,1 2 E n,2 h 2,G) E n,2 h 1,G).

15 L 2 -Approximations of power and logarithmic functions with applications 517 In this case, using 2.43), we can apply the arguments, given in the proof of Theorem 2.2, to the function h = a n,1 h 1 + h 2 to get an estimate similar to either 2.31), if h 2 = f β,τ, or 2.40), if h 2 = g m,l,τ. Further, instead of 2.32) or 2.41)) we have h k) z) = a n,1 h k) h 1 z)+hk) 2 z) k) = 2 z) a h k) 1 z) n,1 h k) +1 2 z), where k is chosen see proof of Theorem 2.2) for h 2. If we show that, for every r>0small enough, one can find a point ζ G satisfying ζ τ = r, the first inequality in 2.12), and such that h k) z) c h 1,h 2,G) h k) 2.48) 2 z), then we can again follow the proof of Theorem 2.2 to obtain a lower estimate for E n,2 h, G) in terms of d 1/n τ), which is similar to that for E n,2 h 2,G). Then, in view of 2.43), we get E n,2 h, G) E n,2 h 2,G) 1 4 a n,1 E n,2 h 1,G)+E n,2 h 2,G)) because of the assumption 2.47). To prove 2.48), one can proceed as follows. Let ω := f 0 τ), := f0 1 {w : ω w < 5/4, arg ω w) <π/6}). Since G is a quasidisk, so is with the coefficient of quasiconformality depending only on that of G). Moreover, for any point ζ the first inequality in 2.12) holds. The quasiconformality of also implies that, for r<r 0 small enough, for the circular arc l r = { z : z τ = r, θ 1 r) arg z τ) θ 2 r) }, separating in points τ and z 0,wehave θ 2 r) θ 1 r) δ = δl). Further arguments depend on the explicit forms of h 1 and h 2, and are left to the reader. Note that, for each particular pair of functions h 1, h 2, the above arguments can be substantially simplified. For example, if h 1 and h 2 are both power functions, i.e., h j = f βj,τ, j =1, 2, β 2 >β 1, the case we are mostly interested in), then using the left inequality in 2.47), 2.43), and applying 2.32) to h 1 and h 2 with k =[β 2 ]+1,weget a h k) 1 z) n,1 h k) 2 z) c β d 1,β 2 ) 1/n τ) 2, β2 β1 z τ

16 518 V.V. Maymeskul et al. if z τ = εd 1/n τ) with any ε<ε 0 β 1,β 2,G). Hence, for the point z ε defined by 2.33), we have h k) z ε ) h k) 2 z ε ) for any ε<ε 0, which is sufficient to then follow the proof of Theorem 2.2. Corollary 2.3 Let h be defined by 2.44). Then, for any fixed R>0, E n,2 h, G Dτ,R)) E n,2 h, G). Proof. In view of 2.45), it is sufficient to get the desired inequality in the case when h is either f β,τ or g m,l,τ. Let r>0be chosen in such a way that D Φτ),r) Φ Dτ,R)). Denote G = Ψ D Φτ),r) ID). Then G is a quasidisk and G G Dτ,R). Obviously, E n,2 h, G Dτ,R)) E n,2 h, G ) 2.49). The domains G and G have the same local structure near the point τ. So, it is not difficult to verify that 2.50) d 1/n τ) d 1/n τ) as n. Thus, applying lower estimate in 2.43) to the function h in G, using 2.50) and upper estimate in 2.43) for G, we easily get E n,2 h, G ) 2.51) E n,2 h, G), which together with 2.49) gives the required. Corollary 2.4 Suppose fz) is analytic in G and, for some R>0, onthe set G Dτ,R), τ L, can be represented in the form fz) =hz)+gz), where g is analytic on Dτ,R), g L Dτ,R)) 1, and h is of the form 2.44). Then E n,2 f,g) ce n,2 h, G), where c is a constant independent of n and g.

17 L 2 -Approximations of power and logarithmic functions with applications 519 Proof. We follow the previous proof and construct, for R/2 instead of R, the domain G, for which 2.49) and 2.50) are satisfied. Observe now that E n,2 g, G ) E n,2 g, Dτ,R/2)) 2.52) R g L Dτ,R)) 2 n 2 n R. It follows from 2.7) that, for any τ G, 2.53) d 1/n τ) n 1/c. Hence, 2.45) and 2.43) imply 2.54) E n,2 h, G) n C 1. Similar estimates hold for G. Using the triangle inequality, 2.52), and 2.51) we get E n,2 f,g) E n,2 h, G ) E n,2 g, G ) E n,2 h, G ) E n,2 h, G ). This completes the proof. The following result is an analog of Andrievskii s lemma [2]). Lemma 2.3 Let z 0 G and hz) be analytic on G, continuous on G and such that h L 2 G). Suppose that for some positive integer constant C 0 and every k IN 2.55) E kc0,2 h,g ) 1 2 E k,2 h,g ). If for some p IP n+1 n 2) and constants c n C, M>0, 2.56) p + c n h L 2 G) M and p z 0 )+c n h z 0 )=0, then p + c n h L G) CM log n, where C is independent of c n, M, and n. Proof. If c n =0, the assertion reduces to Andrievskii s lemma, so we assume that c n /=0. Let Q k, deg Q k k, k =1, 2,..., be the best L2 -approximants to h. Then from 2.56) we have 2.57) E n,2 h,g ) = h Q n L 2 G) h + p c n L 2 G) M c n,

18 520 V.V. Maymeskul et al. and so p + c n Q n L 2 G) M + c n E n,2 h,g ) 2M. Selecting Q k so that p z 0 )+c n Q k z 0 )=0, i.e., Q k z 0 )=h z 0 ) for all k, Andrievskii s lemma yields 2.58) p + c n Q n L G) C 1M log n. 2.59) Next we claim that h Q n L G) C 2 log n M c n. Indeed, following a well-known scheme see e.g. [2]) we choose k satisfying C0 k n<ck+1 0. Since 2.60) Q C j+1 0 Q C j 2E 0 L 2 C j h G) 0,2,G ) and, thanks to 2.55), the series j E C j 0,2 h,g) converges, we get for z G ) ) h Q n )z) = Q C k+1 Q n z)+ Q 0 C j+1 Q 0 C j z). 0 j=k+1 From 2.60) we conclude via Andrievskii s lemma) that Q C j+1 Q 0 C j C 0 L 1 log C j+1 0 E G) C j h 0,2,G ). In the same way, L Q C k+1 Q n C 0 1 log C k+1 0 E n,2 h,g ). G) Hence h Q n L G) C 1 log C0 k+1 E n,2 h,g ) + j=k+1 C 3 log C0 k+1 E n,2 h,g ) + E C k+1 log C j+1 j=k k+1 j 0,2 log C j+1 0 E C j h 0,2,G ) h,g )

19 L 2 -Approximations of power and logarithmic functions with applications 521 C 4 log nen,2 h,g ) j k +1 2k+1 j j=k+1 C 4 log nen,2 h,g ) ) 1+2 m2 m m=1 C 5 log nen,2 h,g ) M C 5 log n c n, which proves the claim. Finally, from 2.58) and 2.59) we obtain p + c n h L G) p + c nq n L G) + c n h Q n L G) C 1 M log n + C 2 c n log n M c n = CM log n. This completes the proof. We will need the following application of Lemma 2.3. Corollary 2.5 Let τ j L, j = 1,r, and, for each j and k = 1,k j, let h j,k denote either f βj,k,τ j, β j,k > 0 noninteger, or g mj,k,l j,k,τ j, m j,k > 0. Suppose that for some constants c n,j,k, k = 1,k j, j = 1,r, and a polynomial p IP n+1 n 2) the inequality k r j p + c n,j,k h j,k M holds. If then p + p z 0 )+ j=1 k=1 L 2 G) k r j c n,j,k h j,k z 0 )=0, j=1 k=1 k r j c n,j,k h j,k j=1 k=1 L G) where C is a constant independent of n and CM log n, { } {c n,j,k } k r j k=1. j=1 Proof. Let c n := max k,j c n,j,k. For all j, k denote c n,j,k := c n,j,k /c n, and set k r j h := c n,j,k h j,k. j=1 k=1

20 522 V.V. Maymeskul et al. It is sufficient to show that the function h satisfies 2.55). Since β j,k f βj,k h 1,τ j if h j,k = f βj,k,τ j, j,k = m j,k g mj,k 1,l j,k,τ j + l j,k g mj,k 1,l j,k 1,τ j if h j,k = g mj,k,l j,k,τ j, we have r r h = b n,j,k hj,k =: hj, j=1 k j=1 where h j,k is one of the functions f βj,k 1,τ j, g mj,k 1,l j,k,τ j, and g mj,k 1,l j,k 1,τ j. Hence, h j is of the form 2.44). Note that the set of all b n,j,k s is bounded by a constant independent of n. Clearly, 2.61) E n,2 h,g ) r ) b n,j,k E n,2 hj,k,g. k=1 k On the other hand, by 2.12), there is an R, depending only on the set {τ j } r j=1 and the coefficient of quasiconformality of L, such that for j = 1,r the function h j is analytic on D τ k,r), for k/= j. Therefore, by Corollary 2.4 and Theorem 2.3, we get E n,2 h,g ) { )} c 1 max E n,2 hj,g 1 j r r ) c 2 E n,2 hj,g 2.62) c 3 j=1 r ) b n,j,k E n,2 hj,k,g. k=1 If h j,k = f βj,k 1,τ j, then by Corollary 2.2 and 2.7) we have ) E nc0,2 hj,k,g C 1 d1/nc0 ) τ j ) ) β j,k Therefore, taking C j,k = from 2.62), we get k C 2 C c 0 d 1/n τ j ) ) β j,k C 3 C cβ j,k 0 E n,2 hj,k,g ). [ 2C 3 /c 3 ) 1/cβ j,k) ] +1, where c 3 is the constant ) E nc0,2 hj,k c 3 2 E n,2 hj,k,g).

21 L 2 -Approximations of power and logarithmic functions with applications 523 A similar inequality holds if h j,k is any of the other possible functions. With C 0 := max j,k C j,k, 2.61) and 2.62) yield E nc0,2 h,g ) r ) b n,j,k E nc0,2 hj,k,g k=1 1 2 c 3 k r ) b n,j,k E nc0,2 hj,k,g 1 2 E n,2 h,g ), k=1 k which establishes 2.55). The result now follows from Lemma Approximation of the Riemann mapping Now we apply the results obtained to the approximation of Riemann mapping function f 0 z). Suppose that the boundary L of a domain G is a piecewise analytic curve without cusps, i.e., L is composed of a finite number of analytic arcs meeting at corners τ j and forming there interior angles α j π, 0 <α j < 2, j = 1,M. Obviously, G is a quasidisk. With z 0 G, let w = f 0 z) denote the conformal mapping of G onto the disk D0,r 0 ), normalized so that f 0 z 0 )=0and f 0 z 0)=1, where r 0 := r 0 G, z 0 ) is the conformal radius of G with respect to z 0. The behavior of f 0 at an analytic corner has been considered in [13] and applied to the problem of approximation of the Riemann mapping function in [11]. We shall assume throughout this section that no logarithmic terms occur in the asymptotic expansions of f 0 near the corners τ j, j = 1,M, where M is the total number of corners of the boundary L. This would be the case if, for every j, j = 1,M, either the corner τ j is formed by two straight-line segments or two circular arcs, or if α j is irrational; see [13, Theorem 2], [7, p. 170] and [19, pp ]. Suitable modifications of the analysis for the cases when logarithmic terms appear in the asymptotic expansions of f 0 near some corner τ j are left to the reader. Let m denote the number of corners for which α j is not of the form 1/N, N IN, and assume m 1. For convenience, such corners τ j will be indexed by j = 1,m. That is, if j>m, then the mapping function f 0 has an analytic continuation in some neighborhood of the corner τ j.) { For each k = 1,mdenote by γ k) j } j=1 the increasing arrangement of the possible powers p + q/α k p IN 0, q IN)ofz τ k ) that appear in the asymptotic expansion of f 0 z) near τ k. In particular, if τ k is formed by two straight-line segments, then γ k) j = j/α k, j =1, 2,...

22 524 V.V. Maymeskul et al. Also, if α k is irrational, or the corner τ k is formed by two circular arcs, then γ k) 1 =1/α k ; γ k) 2 =1/α k + min 1/α k, 1) ; 1/α k +2, 0 <α k < 1/2, γ k) 3 = 2/α k, 1/2 <α k < 1, 1/α k +1, 1 <α k < 2 ;. The mentioned asymptotic expansion near τ k can thus be written in the form 3.1) f 0 z) = a k) j z τ k ) γk) j = a k) j f k) γ z),,τ k where 3.2) j=0 j=0 γ k) 0 := 0, f 0,τk z) 1, and a k) 1 /= 0. In 3.1) we remark that γ k) 1 =1/α k > 1/2 and that in the case when α k is rational, it is possible that γ k) j IN for indices j 2, so that f k) γ j,τ k is analytic at τ k. For each k = 1,mchoose a number p k IN 0 and denote { } ν k := min j>p k γ k) j / IN,a k) j /= 0. In what follows, we assume that at least one of ν k s is finite; otherwise, results become trivial. Consider the function fz) :=f 0 z) m ν k k=1 j=0 a k) j f k) γ j j,τ k z)+hz), where, for m>1, Hz) is the polynomial interpolating each of the functions m k =1 k/= l ν k j=0 a k) j f k) γ j,τ k z) and its derivatives up to and including the order l = 1,m.Form =1, we take Hz) 0. Clearly, deg H m l=1 [ ] γ l) ν l +1 + m 1 [ ] γ l) ν l +1 at the point τ l,

23 L 2 -Approximations of power and logarithmic functions with applications 525 and the function f z) =f 0z) m ν k k=1 j=1 a k) j has, near each τ k, the asymptotic expansion f γ k) j,τ k z)+h z) f z) =γ k) ν k +1 ak) ν k +1 z τ k) γk) ν k Proceeding as in [11], we denote gz) :=f z) m z τ k ) k=1 and conclude that h := g Ψ Λ s) with s := min 1 k m { 2 α k ) γ k) ν k +1 This means that for s = p + γ, p IN 0, 0 <γ 1, h p) Lip γ,ifγ<1, and h p) belongs to the Zygmund class, if γ =1,on ID. Since G is a Faber domain, it immediately follows that 3.3) E n, g, G) := min g p L p IP G) n s, n =1, 2,.... n Then see [10, Theorem 2]) there is a polynomial sequence {Q n } n>m, Q n IP n, such that f Q L n 2G) n s log n. }. That is, 3.4) f 0 m ν k k=1 j=1 a k) j f γ k) j + H Q n,τ k L 2 G) n s log n. Note now that f β,τ z) =βf β 1,τ z). So, using Corollary 2.2 and 2.3) we construct polynomials P k,n IP n, k = 1,m, such that f P γ ν k) k,n n 2 α k)γ ν 3.5) k) k. k,τ k Then for the polynomials ˆP n z) = ˆP n p 1,...,p m ; z) m ν k 1 := a k) j f z)+ γ k),τ k k=1 j=p k +1 j L 2 G) m k=1 a k) ν k P k,n z) H z)+q n z)

24 526 V.V. Maymeskul et al. we get from 3.4) and 3.5) m f 0 3.6) where 3.7) p k k=1 j=1 a k) j n s log n + f γ k) j m k=1 s = s p 1,...,p m ) := and an empty sum has value zero. ˆP n,τ k L 2 G) n 2 α k)γ k) ν k n s, { } min 2 α k ) γ ν k) 1 k m k, Remark 3.1 Taking, for instance, all the p k =0, k = 1,m, and setting ˇP n z) := ˆP n 0,...,0; z) we get from 3.6): f 0 ˇP n L 2 G) n s, where s := s 0,...,0) = { min 1 k m 2 α k ) γ k) 1 } = min 1 k m { 2 αk 3.8) Because of the minimal property of Bieberbach polynomials π n z), it follows that 3.9) f 0 π n L 2 G) = O n s) α k }. and, consequently, 3.10) f 0 π n L G) = O log n n s ) as n. This rate of convergence is an improvement of [10, Theorem 1] and, particularly, [11, Theorem 2], where the corresponding results contain the factor log n instead of its square root. Such improvement was also shown in [6] for domains with piecewise quasianalytic boundary. In view of the asymptotic expansion of f 0 near the corners τ k, k = 1,m, and the estimate 3.6), it is reasonable to extend the power system {z n }, n IN 0, by adjoining the functions f z), j = 1,p γ k) k, k = 1,m cf. j,τ k [14]). For this purpose, put l r 0 := 0, r l := p k, k=1 l = 1,m,

25 L 2 -Approximations of power and logarithmic functions with applications 527 and consider the system {η j } 1 defined by η j z) :=f z), j = r γ l) l 1 +1,r l, l = 1,m, j r,τ l l 1 η rm+1z) :=1, η rm+2z) :=2z,... η rm+nz) :=nz n 1,.... If some α k is rational, it is possible that some η j, 1 j r m,isa polynomial, in which case we avoid redundancy in the basis by omitting such η j. For convenience in exposition, we assume that this situation does not arise. Set 3.11) µ j z) := z z 0 η j ζ)dζ, z G. Next we orthonormalize the system {η k } 1 by means of the Gram-Schmidt process to get {ηk } 1. The functions η k have the representation η k z) = where b k,k > 0, k =1, 2,... Let { IP A n 1 := p A : p A z) = = r m+n k=1 k b k,j η j z), j=1 t k η k z), t k C { t 1 f z)+...+ t γ 1) 1,τ rm f z)+t 1 γ pm m) rm+1 +2t rm+2z,τm nt rm+nz n 1 }. Also, for z 0 G, let Kz,z 0 ) denote the Bergman kernel function of G, which has the reproducing property 3.12) gz 0 )= gz)kz,z 0 )dxdy, for any g L 2 G). Then see [8, p. 34]) 3.13) G f 0z) = K z,z 0) K z 0,z 0 ), 1 f 0 z) = K z 0,z 0 ) z z 0 } K ζ,z 0 ) dζ.

26 528 V.V. Maymeskul et al. We form the partial Fourier sum for K z,z 0 ): K n z,z 0 )= = r m+n j=1 r m+n j=1 K,z0 ),ηj ) η j z) η j z 0)η j z) = r m+n j=1 h n,j η j z) = h n,1 f z)+...+ h γ 1) 1,τ n,rm f z)+h 1 γ pm m) n,rm+1,τm nh n,rm+nz n 1. Obviously, augmented polynomials K n z,z 0 ) are the best approximants to K z,z 0 ) in L 2 G) out of the space IP A n 1, i.e., K,z 0 ) K n,z 0 ) K,z 0 ) p A L 2 G) L for any 2 G) p A IP A n 1. Following 3.13) we approximate f 0 z) and f 0z) respectively by 3.14) and π n z) := = = 3.15) π nz) := K r n z,z 0 ) K n z 0,z 0 ) = 1 m+n h n,j η j z) K n z 0,z 0 ) 1 K n z 0,z 0 ) z z 0 Kn ζ,z 0 ) dζ r 1 m+n h n,j µ j z) K n z 0,z 0 ) j=1 1 K n z 0,z 0 ) + n j=1 h n,rm+j m r l l=1 j=r l 1 +1 z j z j 0). j=1 ) h n,j f l) γ z) f l) j r,τ l γ z l 1 j r,τ 0 ) l l 1 Clearly, π n z 0 )=0, π n z 0 )=1, n =1, 2,... It is natural to call the functions π n the augmented Bieberbach polynomials over the system {µ k } 1. Hence, from 3.6) and the minimum property of the Fourier sum we get f 0 π n 3.16) L 2 G) n s.

27 L 2 -Approximations of power and logarithmic functions with applications 529 Now we follow the method of Andrievskii [2]. Applying Corollary 2.5 we derive an estimate for the uniform norm of π 2 k π 2 k 1, k =1, 2,..., from an estimate of the L 2 -norm of π 2 k π 2 k 1) and get Thus, we have established f 0 π n L G) n s log n. Theorem 3.1 Suppose L = G is a piecewise analytic curve with interior angles α j π, 0 <α j < 2, at the corners τ j, j = 1,M, where we assume that no logarithmic terms occur in the asymptotic expansions of f 0 near τ j. With m 1) described as at the beginning of this section, we have: For any fixed numbers p j IN 0, j = 1,m, the augmented Bieberbach polynomials π n, defined by 3.15), approximate f 0 z) with the estimate ) log n 3.17) f 0 π n L G) = O as n, n s where s = s p 1,...,p m ) is defined by 3.7). Let En, A f 0,G) denote the error in best uniform approximation to f 0 z) out of the space ĨPA n spanned by the functions {µ j } rm+n 1 cf. 3.11)). Theorem 3.2 Let G and s be as in Theorem 3.1, and let κ be the index for which the minimal value in 3.7) is attained. Then 3.18) f 0 π n L G) EA n, f 0,G) n s. In particular since a κ) 1 /= 0and γ κ) 1 =1/α κ / IN), for the classical Bieberbach polynomials, f 0 π n L 2 G) 1 n s and 1 log n 3.19) n s f 0 π n L G) n s, where s is given by 3.8). Proof. Since G is a Faber domain, the upper estimate in 3.18) follows from the fact that for the function hz) :=f 0 z) we have h Ψ Λ s ) on ID. m p k k=1 j=0 a k) j f k) γ j,τ k z)

28 530 V.V. Maymeskul et al. Let q A n ĨPA n, n ν κ, be an arbitrary augmented polynomial. Then, for r small enough, using the expansion 3.1) of f 0 z) in G r := G D τ κ,r) we get 3.20) f 0 z) q A n z) = p κ j=0 + c n,j f κ) γ z)+a κ) j,τ κ ν κ f κ) γ,τκz) νκ j=ν κ+1 c j f κ) γ z) hz) q j,τ n z), κ where the function h is analytic in D τ κ, 2r), q n IP n is an algebraic polynomial of degree at most n. By arguments similar to those used for the estimate 3.3) see also [11, Sec. 1.3]), the second sum in 3.20) can be approximated with the rate n ακ 2)γκ) νκ+1. Also, h can be approximated on G r geometrically fast. Therefore, it is sufficient to show that for any constants {C n,j } pκ j=0 the function gz) := p κ j=0 C n,j f κ) γ z)+f κ) j,τ κ γ,τκz) νκ cannot be uniformly approximated on G r by algebraic polynomials essentially faster than f κ) γ for which the rate, according to the Corollary 2.2, νκ,τκ is n s. Estimating the uniform norm from below by the L 2 -norm and applying Theorem 2.3, we arrive at the desired lower estimate in 3.18). The assertions of 3.19) follow from 3.9) by applying a similar argument to get a lower bound for f 0 π n L 2 G). We remark that we can also apply Gaier s method [11, p. 39]) to get the pointwise estimate 3.21) ) 1 ηnz 0 ) = O, n z 0 G), n s for the augmented orthonormal polynomials η n. For the nonaugmented case, i.e. when η n = P n 1 is the ordinary Bergman orthonormal polynomial, Gaier [11] raised the question of finding more precise estimates for P n z 0 ). In our case, 3.21) gives 3.22) ) 1 P n z 0 ) = O n s, n )

29 L 2 -Approximations of power and logarithmic functions with applications 531 with s given by 3.8). On the other hand, from 3.19) and Lemma 4.4 of [17], P k z 0 ) 2 = K,z 0 ) K n,z 0 ) 2 L 2 G) k=n+1 f 0 π n 2 L 2 G) n 2s, from which it follows that for each ε>0, there exists a subsequence Λ ε IN such that ) P n z 0 ) n, n Λ ε. s ε Numerical results in Sect. 4 indicate that for certain regions G, the precise rate of decrease is indeed P n z 0 ) 1. n s Numerical experiments 4.1The test regions Let G α denote the circular sector of radius 2 with interior angle απ: G α := {z : z < 2, απ/2 < arg z<απ/2}, 0 <α<2; let f 0 be the conformal map G α D 0,r 0 ), normalized by the conditions f 0 1) = 0 and f 01)=1; recall that r 0 is the conformal radius of G α with respect to z 0 =1); and, as in Sect. 3, let Kz,1) denote the Bergman kernel function of G α with respect to z 0 = 1. Also, let π n z) denote the Bieberbach polynomial obtained, as indicated in Sect. 3, by integrating the best approximant to Kz,1) in L 2 G α ) out of the space of polynomials of degree n 1, and similarly, let π n z) denote the augmented Bieberbach polynomial obtained by integrating the best approximant to Kz,1) in L 2 G α ) out of the space { IP A n 1 = t 1 z 1/α 1 + t 2 + t 3 z + + t n+1 z n 1, } 4.1) t j C,j= 1,n+1 ; see 3.15). Finally, for noninteger β> 1 and m IN, let 4.2) f β z) =z β and g m z) =z m log z.

30 532 V.V. Maymeskul et al. In this section we present numerical results illustrating the orders of approximation predicted by the theory, regarding the following six errors: 4.3) 4.4) and 4.5) E n,2 f β,g α ) := min p IP n f β p L 2 G α), E n,2 g m,g α ) := min p IP n g m p L 2 G α) E n,2 f 0,G α ):= f 0 π n L 2 G α), E n, f 0,G α ):= f 0 π n L G α), Ẽ n,2 f 0,G α ):= f 0 π n L 2 G 4.6), α) Ẽ n, f 0,G α ):= f 0 π n L G. α) In addition, we present numerical results illustrating the rate of decrease of the Bergman orthonormal polynomials P n z) of G α,asn. We do these by considering: a) an Orthonormalization Method ONM) for constructing both P n z) and the polynomials that realize the minimum in 4.3) and 4.4), see e.g. [8, Chapter I]; b) the application of the Bergman Kernel Method BKM) for computing approximations to the conformal map f 0, with respect to the errors in 4.5) and 4.6), see e.g. [8, Chapter I] and [14]. For each value of the parameter α, the mapping w = f 0 z) can be computed by means of the transformation ) 2α4 1/α 4.7) 1) t d f 0 z) = 4 1/α +1 td 1, where ) 2 ) 2 iz 1/α +2 1/α i +2 1/α 4.8) t = iz 1/α 2 1/α and d = i 2 1/α, Thus, the value of the conformal radius r 0 is given by r 0 = f 0 2) = 2α41/α 1) 4 1/α. +1 Since the kernel function Kz,1) is related to f 0 z) by 4.9) Kz,1) = 1 πr0 2 f 0z); see e.g. [8, p. 34], it follows, in particular, that 4.10) K1, 1) = 1 πr 2 0 ) 2 = 1 4 1/α +1 π 2α4 1/α. 1)

31 L 2 -Approximations of power and logarithmic functions with applications 533 We consider now the asymptotic behavior of the map f 0 near the three corners τ 1 =0, τ 2 =2e iαπ/2 and τ 3 =2e iαπ/2 of G α. The two arms forming the corner at the origin are both straight lines, and therefore from the Schwarz-Christoffel transformation, as z 0, 4.11) f 0 z) =f 0 0) + a j z j/α, a 1 /=0; j=1 see e.g. [19, pp ]. We note that the coefficients a j in 4.11) can be computed explicitly in terms of α from 4.7) 4.8). In particular, 4.12) a 1 =2r /α ) and a 2 = 2r /α ) 2 4 2/α. Each of the two other corners τ j, j =2, 3, has interior angle π/2 and is formed by a straight line and a circular arc; hence, as remarked in Sect. 3, f 0 is regular at τ j, j =2, 3. Furthermore, f 0 has a branch point singularity at 0 whenever 1/α / IN. In other words, the only singularity of f 0 on the boundary L of G α occurs at the origin, in cases when 1/α is not an integer and, in such cases, the dominant term of the asymptotic expansion of f 0 at 0 is z 1/α. This observation, and the fact that in the BKM one constructs best approximations with respect to the kernel function 4.9) explains the particular choice of the space 4.1); see also [14], [16]. 4.2 Computational details η k z)η j z) dxdy, k = 1,l, j = 1,l. Let {η j } j=1 be a complete set of functions in L2 G α ). In both ONM and BKM the set {η j } l j=1 is orthonormalized by means of the Gram-Schmidt { } l process to produce the orthonormal set the computation of the inner products ηj. This, in particular, requires j=1 4.13) η k,η j )= G α For the application of the ONM we use the computational convenient monomial set η j z) =z j 1, j = 1,n+1. { } n+1 Once the orthonormal system ηj has been constructed, the values of j=1 the Bergman polynomials are obtained from P j 1 z) =η j z), j = 1,n+1.

32 534 V.V. Maymeskul et al. Also, the two errors 4.3) and 4.4) can be computed from the Fourier coefficients of the functions f β and g m, since the minimum property of finite Fourier sums implies n+1 En,2 2 f β,g α )= f β 2 L 2 G α) f β,ηj ) ), and 4.15) j=1 n+1 En,2g 2 m,g α )= g m 2 L 2 G α) g m,ηj ) 2. For our purposes here it is important to note that each one of the inner products involved in 4.13), 4.14), and 4.15) is given explicitly in terms of sines and cosines of the opening angle απ. In the BKM, the approximation to f 0 is obtained from 4.9) after first approximating the kernel Kz,1) by a finite Fourier sum. The reason for doing so is that, due to the reproducing property 3.12), the Fourier coefficients of Kz,1) can be computed without requiring the explicit knowledge of Kz,1). Therefore, in order to construct the approximations π n z) and π n z) we orthonormalize, respectively, the monomial set MB), 4.16) and the augmented set AB) 4.17) η j z) =z j 1, j=1 j = 1,n, η 1 z) =z 1/α 1, η j z) =z j 2, j = 2,n+1. As with the ONM, the inner products in 4.13) are given explicitly in terms of sines and cosines of the opening angle απ. The details of the BKM are as follows: Let {ηj }n j=1 and { η j }n+1 j=1 denote respectively the two orthonormal systems obtained from 4.16) and 4.17). Then, because of 3.12), n 4.18) K n z,1) = ηj 1)η j z), and 4.19) j=1 n+1 K n z,1) = η j 1) η j z), j=1 are, respectively, the n-th BKM/MB and the n-th BKM/AB approximation to Kz,1), and from 3.15) we set 4.20) π n z) = 1 z K n ζ,1)dζ, K n 1, 1) 1

33 L 2 -Approximations of power and logarithmic functions with applications ) π n z) = 1 z K n 1, 1) 1 K n ζ,1)dζ. Note that π n z) and π n z) are normalized so that π n 1) = π n 1)=0and π n1) = π n1)=1.) Regarding the four errors 4.5) 4.6) we observe the following: The order of approximation in each of the two errors f 0 π n L 2 G α) and f 0 π n L 2 G α), can be computed with the orthonormal functions, using 4.10), 4.18) and 4.19). This follows by noting that: i) f 0 π n L 2 G K, 1) K α) n, 1) L 2 G α) and f 0 π n L 2 G α) K, 1) Kn, 1) ; L 2 G α) see [17, Lemma 4.4]. ii) The minimum property of finite Fourier sums and Parseval s identity imply that 4.22) and 4.23) K, 1) K n, 1) 2 L 2 G α) = K1, 1) K n1, 1), K, 1) K n, 1) 2 = K1, 1) K n 1, 1); L 2 G α) see e.g. [8, p. 25]. Estimates for the two other errors f 0 π n L G α) and f 0 π n L G α) can be obtained from 4.7) 4.8) and 4.18) 4.21), by using a number of test points on the boundary L. 4.3 Numerical results The results of Corollary 2.2 indicate that for any α, 0 <α<2, 4.24) E n,2 f β,g α ) 1 n λβ+1), E n,2 g m,g α ) 1 n λm+1), with λ =2 α, for the polynomial approximations to the special functions 4.2). Furthermore, since for any 0 <α<2, the coefficients a 1 and a 2 in

34 536 V.V. Maymeskul et al. the asymptotic expansion 4.11) of f 0 near τ 1 =0are different from 0 cf. 4.12)), we have from Theorems 3.1, 3.2 and 3.16) that, for 1/α / IN, E n,2 f 0,G α ) 1 n λ/α, 1 log n 4.25) n λ/α E n, f 0,G α ) n λ/α, Ẽ n,2 f 0,G α ) 1 n 2λ/α, 1 log n 4.26) n 2λ/α Ẽn, f 0,G α ) n 2λ/α. Finally, regarding the Bergman polynomials {P n } n=0, for any α, 0 <α<2, with 1/α / IN, and any ζ G α, we have from 3.22) and 3.23): for all n IN, 4.27) P n ζ) 1 n λ/α ; for any ε>0, there exist infinitely many n such that, 4.28) 1 P n ζ) n. λ/α ε For the remainder of this section we present numerical results that illustrate the laws in the above errors and rates. All the results were obtained with Maple V, using the systems facility for 128-digit floating point arithmetic, on an IBM RS/6000. We have chosen to perform numerical work using this high accuracy, in order to postpone the breakdown of the Gram-Schmidt process. This was essential for our purposes here, because we needed to use a large number of basis functions, typically up to 100, to observe a distinct behavior for the orders of approximation in 4.25) and 4.26). We note in passing that numerical experiments with G α, using the double precision Fortran conformal mapping package BKMPACK of Warby [20], failed to produce precise conclusions for the orders in 4.25) 4.26); see also [18, Example 5.3].) We recall that the Gram-Schmidt process is required by the application of both ONM and BKM for the construction of the orthonormal system. See [18] for a comprehensive study regarding the stability properties of Bergman Kernel Methods and a characterization of the level of instability in the Gram-Schmidt process, in terms of the geometry of the domain under consideration. In particular, [18, Theorem 3.1] implies that the level of instability for the domain G α, increases with decreasing α. Also, see [9, Sect. 6] for a report on numerical experiments regarding the errors f 0 π n L 2 G) and f 0 π n L G), when G is the image of {t : t 1 < 1} under the mapping z = t α, 0 <α<2. In presenting the numerical results we use the following notations: σ: This denotes the order of approximation the exponent of 1/n)inthe errors 4.3) 4.6), or the rate of decrease of P n ζ), as they are predicted by the theory of Sects. 2 and 3; see 4.24) 4.28).

35 L 2 -Approximations of power and logarithmic functions with applications 537 σ n : This denotes the estimate of σ corresponding to the use of n basis functions and is determined as follows: With E n denoting any of the errors E n,2 f β,g α ), E n,2 g m,g α ), E n,2 f 0,G α ), Ẽn,2 f 0,G α ),orthe value of P n ζ), we assume that 4.29) E n C 1 n σ, and seek to estimate σ by means of the formula ) En ) σ n = log / log E n n n 20 If E n denotes either of the uniform errors E n, f 0,G α ) or Ẽn, f 0,G α ), then we assume that 4.31) E n C log n 1 n σ, ). and seek to estimate σ by means of the formula ) En 20 σ n = log 1 logn 20) E n 2 log log n ) n 4.32) log. n 20 )) / σn: With E n denoting either of the uniform errors E n, f 0,G α ) or Ẽ n, f 0,G α ), we also test the law 4.33) E n C 1 n σ ; thereby estimating σ by means of 4.34) σ n = log En 20 E n ) ) n / log. n 20 L 2 -approximations to special functions. The numerical results for the values 0.5 and 1.5 of the parameter α, and for n = 2020)100, are given in Tables More precisely, Table 4.1 and Table 4.2 contain the results for the orders in the L 2 -approximations to z β, corresponding to the values β = 0.5, β =0.5and β =1.5. Table 4.3 and Table 4.4 contain the results for the orders in the L 2 -approximations to z m log z, corresponding to the values m =1, m =2and m =3. The presented results indicate clearly a close agreement between the theoretical and the computed order of approximation, thus providing experimental confirmation of the results in Sect. 2.