APPROXIMATION BY GENERALIZED FABER SERIES IN BERGMAN SPACES ON INFINITE DOMAINS WITH A QUASICONFORMAL BOUNDARY

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volue , 11 APPROXIMATION BY GENERALIZED FABER SERIES IN BERGMAN SPACES ON INFINITE DOMAINS WITH A QUASICONFORMAL BOUNDARY Daniyal M. Israfilov and Yunus E. Yildirir Received April 004 Abstract. Using an integral representation on infinite doains with a quasiconforal boundary the generalied Faber series for the functions in the Bergan space A G are defined and their approxiative properties are investigated. 1. Introduction and New Results Let G be a siple connected doain in the coplex plane C and let ω be a weight function given on G. For functions f analytic in G we set A G, ω := f : f ωdσ <, G where dσ denotes the Lebesgue easure in the coplex plane C. If ω = 1, we denote A G := A G, 1. The space A G is called the Bergan space on G. We refer to the spaces A G, ω as weighted Bergan spaces. It becoes a nored spaces if we define f A G,ω := 1/ f ωdσ. G Hereafter, we consider only the special weight ω := 1/ 4 in this work. Now let L be a finite quasiconforal curve in the coplex plane C. We recall that L is called a quasiconforal curve if there exists a quasiconforal hoeoorphis of the coplex plane onto itself that aps a circle onto L. We denote by and G the bounded and unbounded copleents of C \L, respectively. It is clear that if f A G, then it has ero in at least second order. As in the bounded case [7, p. 5], A G is a Hilbert space with the inner product f, g := fgdσ, G 1991 Matheatics Subject Classification 30E10, 41A10, 41A5, 41A58. Key words and phrases: Faber polynoials, Faber series, Error of approxiation, Quasiconforal curves, Bergan spaces.

2 1 DANIYAL M. ISRAFILOV and YUNUS E. YILDIRIR which can be easily verified. Moreover, the set of polynoials of 1/ are dense in A G with respect to the nor f A G := f, f 1/. Indeed, let f A G. If we substitute = 1/ζ and define 1 f = f =: f ζ, ζ then G aps to a finite doain G ζ, and f A G ζ, because f ζ dσ ζ = G ζ G dσ f 4 c G f dσ <, with soe constant c > 0. Since f has ero in at least second order, the point ζ = 0 is the ero of f at least second order and f ζ ζ dσ ζ = f dσ <. G ζ Hence f ζ/ζ A G ζ. If P n ς is a polynoial of ς, then we have P nζ f ζ ζ dσ ζ = Pn ζζ f ζ 1 ζ 4 dσ ζ G ζ = G G ζ G P n 1 1 f dσ. This iplies that the set of polynoials of 1/ are dense in A G, since the set of polynoials P n ζ are dense in A G ζ with respect to the nor f A G ζ := f, f 1/, see, for exaple: [7, Ch. 1]. Also, for n = 1,,... there exists a polynoial Pn1/ of 1/, of degree n, such that E n f, G = f Pn A G see for exaple, [6, p. 59, Theore 1.1.], where } E n f, G := Inf { f P A G : P is a polynoial of 1/, of degree n denotes the inial error of approxiation of f by polynoials of 1/ of degree at ost n. The polynoial P n1/ is called the best approxiant polynoial of 1/ to f A G. Let D be the open unit disc and w = ϕ be the conforal apping of onto := C \D, noralied by the conditions and let ψ be the inverse of ϕ. expansion ϕ0 = and li 0 ϕ > 0, In the neighborhood of the origin we have the ϕ = α + α 0 + α 1 + ldots + α k k +....

3 GENERALIZED FABER SERIES 13 Raising this function to the power we obtain [ϕ] = F 1/ + Q for, 1.1 where F 1/ denotes the polynoial of negative powers of and the ter Q contains non-negative powers of and is analytic in the doain. The polynoial F 1/ of negative powers of is called the generalied Faber polynoial for the doain G. If G, then integrating in the positive direction along L, we have F 1 = 1 πi L [ϕζ] ζ dζ = 1 πi w =1 w ψ w ψw dw. This forula iplies that the functions F 1/, = 1,,... are the Laurent coefficients in the expansion of the function ψ w ψw G, w in the neighborhood of the point w =, i.e. the following expansion holds ψ w ψw = 1 1 F w +1 G, w, which converges absolutely and uniforly on copact subsets of G. Differentiation of this equality with respect to gives ψ w 1 ψw = F 1 1 w +1 or ψ w ψw = F 1 1 w for every, w G, where the series converges absolutely and uniforly on copact subsets of G. More inforation for Faber and generalied Faber polynoials can be found in [1, p. 55] and [7, p. 4]. In this work, for the first tie, we obtain Section, Lea.1 an integral representation on the infinite doain G with a quasiconforal boundary for a function f A G. By eans of this integral representation in Section we define a generalied Faber series of a function f A G to be of the for 1 a ff, with the generalied Faber coefficients a f, = 1,,.... Our ain results are presented in the following theores, which are proved in Section 3. Theore 1.1. Let f A G. If a ff is a generalied Faber series of f, then this series converges uniforly to f on the copact subsets of G.

4 14 DANIYAL M. ISRAFILOV and YUNUS E. YILDIRIR Corollary 1.. Let P n 1/ be a polynoial of degree n of 1/ and P n 1/ A G. If a P n are its generalied Faber coefficients, then a P n = 0 for all n + and 1 P n = A uniqueness theore for the series a ff a P n F 1, 1. which converges to f A G with respect to the nor A G is the following. Theore 1.3. Let {a } be a coplex nuber sequence. If the series 1 a F converges to a function f A G in the nor A G, then the a, = 1,,..., are the generalied Faber coefficients of f. The following theore estiates the error of the approxiation of f A G by the partial sus of the series 1.3 in the weighted nor A G,ω for the special weight ω := 1/ 4, regarding to the inial error E n f, G. Theore 1.4. If f A G, ω := 1/ 4 and S n f, 1 1 = a F is the nth partial su of its generalied Faber series 1 a F, then f S n f, A G c nen,ω 1 k f, G, for all natural nubers n and with a constant c independent of n. Siilar results for the bounded doains with a quasiconforal boundary were stated and proved in [8] and [5], respectively. These probles in the weghted cases were studied in [9] and [10]. We shall use c, c 1,..., to denote constants depending only on paraeters that are not iportant for the questions of interest.. Definitions and Soe Auxiliary Results In [4], V.I. Belyi gave the following integral representation for the functions f analytic and bounded in the doain f = 1 f yζ π ζ y ζ ζdσ ζ,. G

5 GENERALIZED FABER SERIES 15 Here y is a K-quasiconforal reflection across the boundary L, i.e., a sensereversing K-quasiconforal involution of the extended coplex plane keeping every point of L fixed, such that y = G, yg =, y0 = and y = 0. Such a apping of the plane does exist [11, p. 99]. As follows fro Ahlfors theore [1, p. 80] the reflection y can always be chosen canonical in the sense that it is differentiable on C alost everywhere, except possibly at the points of the curve L, and for any sufficiently sall fixed δ > 0 it satisfies the relations y ς + y ς c 1, if ζ {ζ δ < ς < 1/δ, ς / L} y ς + y ς c ς, if ς 1/δ or ς δ, with soe constants c 1 and c, independent of ζ. Considering only the canonical quasiconforal reflections, I.M. Batchaev [3] generalied the integral representation above to functions f A. The accurate proof of the Batchaev s result is given in [, p. 110, Th. 4.4]. A siilar integral representation can also be obtained for functions f A G. The following result holds. Lea.1. Let f A G. If y is a canonical quasiconforal reflection with respect to L, then f = 1 f yζ π ζ [yζ] y ζ ζdσ ζ, G..1 Proof. Let y a canonical quasiconforal reflection and f A G. substitute ζ = 1/u for ζ G and define fζ = f 1/u =: f u, If we then G aps to a finite doain G u and f A G u. If y t is a canonical quasiconforal reflection with respect to G u, then fro the Batchaev s result we have f t = 1 f y u π u t yuudσ u, t G u, CG u where CG u := C G u. Substituting u = 1/ζ in this integral representation we get f = f1/t = f t = 1 f y 1/ς π 1/ς 1/ y u 1/ς Jdσ ζ = 1 f[1/y 1/ζ] π ζ y 1/ς dσ ζ ζ, G. If we define 1 yζ := y 1/ς, then yζ becoes a canonical quasiconforal reflection with respect to L. Consequently, for f A G we get f = 1 π f yζ ζ [yζ] y ζ ζdσ ζ, G.

6 16 DANIYAL M. ISRAFILOV and YUNUS E. YILDIRIR Fro now on, the reflection y will be a canonical K-quasiconforal reflection with respect to L. Let f A G. Substituting ζ = ψw in.1, we get f = 1 π f yψw ψ wy ζ ψw [y ψw ] ψ w ψw dσ w, G.. Thus, if we define the coefficients a f, = 1,,..., by a f := 1 f yψw ψ w π w +1 [y ψw ] y ζ ψw dσw,.3 then, by 1. and., we can associate a foral series a ff 1/ with the function f A G, i.e., f a ff 1/. We call this foral series a generalied Faber series of f A G, and the coefficients a f, = 1,,..., generalied Faber coefficients of f. Lea.. Let {F 1/}, = 1,,..., be the generalied Faber polynoials of 1/ for G. Then n F, A G nπ. Proof. Since yζ is a canonical K-quasiconforal apping of the extended co- plex plane onto itself, we have y ζ / y ζ k and y ζ yζ > 0. Also, it is known that y ζ = yζ and y ζ. yζ = y ζ Therefore, yζ / y ζ k and yζ > 0. Hence f yζ yζ ζ dσ ζ = f yζ 1 y ζ / 1 yζ y ζ yζ dσ ζ 1 1 k yζ f yζ yζ dσ ζ. Since y ζ y ζ is the Jacobian of yζ, substituting ζ for yζ on the right side of the last inequality we get f yζ yζ ζ dσ ζ f A G 1 k.

7 GENERALIZED FABER SERIES Proofs of the New Results Proof of Theore 1.1. Let M be a copact subset of G and y a canonical K-quasiconforal reflection with respect to L. Since by Lea.1 f = 1 f yζ π ζ [yζ] y ζ ζdσ ζ = 1 π f yψw ψ wy ζ ψw [y ψw ] ψ w ψw dσ w, for M, by eans of.3, Hölder s inequality and Lea 4 we obtain n f a ff 1/ c 3 f A G π ψ n w 1 k ψw + F 1/ w +1 dσ w for every M, where the constant c 3 depends only on L. Let 1 < r < R <. In view of 1. ψ w n F 1/ + ψw w +1 dσ w r< w <R = = π r< w <R 4π = = = 1 F 1/ w +1 dσ w 1 r 1 R F 1/ + 1 and by letting r 1 + and R we get ψ w n F 1/ + ψw w +1 dσ w 4π F 1/ = Therefore, by 3.1, 3. and Lea 3 we conclude that a ff 1/ converges uniforly to f on M. Proof of Corollary 1.. Let G. By Theore 1.1 we have P n 1/ = a P n F 1/. 1/, 3.1 F 1/

8 18 DANIYAL M. ISRAFILOV and YUNUS E. YILDIRIR On the other hand, P n 1/ can be written in the for P n 1/ = A k F k 1/, k=1 with the specific coefficients A k, k = 1,,..., n + 1. Let y be a canonical K- quasiconforal reflection relative to L. Since y is identical on L, by Green s forulae we get a P n = 1 [ ] P n 1/y ψw ψ w π w +1 [y ψw y ] ζ ψw dσw By 1.1 = = = k=1 A k π k=1 A k π k=1 A k πi w =1 F k [ ] 1/y ψw ψ w w +1 [y ψw y ] ζ ψwdσ w w Fk [1/yψw] w +1 F k [1/ψw] w +1 dw. F [1/ψw] = w Q ψw, where Q ψw is analytic in, and then 1 πi w =1 F k [1/ψw] w +1 dw = QAT OP D dσ w { 1, if k =, 0, if k, 3.3 which iplies that a P n = A, for = 1,..., n + 1, and a P n = 0 for all n +. Hence P n 1/ = a P n F 1/. Proof of Theore 1.3. Let y be a canonical K-quasiconforal reflection relative to L and be the nth partial su of S n f, 1/ := a F 1/. a F 1/ Using 3.3 it can be shown that [ ] 1 S n 1/y ψw ψ w li n π w +1 [y ψw y ] ζ ψw dσw = a, = 1,,

9 GENERALIZED FABER SERIES 19 If and n are natural nubers, then by using Hölder s inequality and Lea 4 we get a f a 1 fyψw S n [1/yψw] ψ w π w +1 [yψw] y ζ ψwdσ w [ ] 1 S n 1/y ψw ψ w + π w +1 [y ψw y ] ζ ψw dσw a 1 π f yψw S n [ 1/y ψw ] ψ w yζ ψw y ψw 4 [ ] 1 S n 1/y ψw ψ w + π w +1 [y ψw y ] ζ ψw dσw a dσ w w + dσ w 1/ 1/ c 4 π f S n y ζ yζ ζ dσ ζ 1/ [ ] 1 S n 1/y ψw ψ w + π w +1 [y ψw y ] ζ ψw dσw a c 4 f S n A G π1 k [ ] 1 S n 1/y ψw ψ w + π w +1 [y ψw y ] ζ ψw dσw a. 3.5 Since li f S n n A G = 0, 3.4 and 3.5 show that a f = a, = 1,,.... Proof of Theore 1.4. Let y be a canonical K-quasiconforal reflection with respect to L, and P n1/ the best approxiant polynoial to f A G in the nor A G. For G, by eans of Hölder s inequality, Lea 4 and

10 0 DANIYAL M. ISRAFILOV and YUNUS E. YILDIRIR Corollary 1. we obtain f S n f, 1/ f P n 1/ + P n1/ S n f, 1/ f Pn 1/ + a P n a f F 1/ f P n 1/ + 1 f y Pn y ψw ψ wy ζ ψw π [y ψw ] f P n 1/ + 1 π F 1/ w +1 f y P n y ψw ψ w yζ ψw F 1/ w +1 dσ w y ψw 4 1/ dσ w dσ w 1/ f P n 1/ + c 5 π f y Pn yζ yζ ζ dσ ζ 1/ F π 1/ f P n 1/ + 1/ c 5 π1 k f P n A G F 1/ 1/ = f Pn c 5 1/ + π1 k E F 1/ nf, G for all natural nubers n. This shows that f S n f, 1/ f P n 1/ + c 5 π1 k E nf, G 1/ F 1/.

11 GENERALIZED FABER SERIES 1 Multiplying both sides by 1/ 4 and take into account that 1/ 4 c 6 for G and with a constant c 6, we get f S n f, 1/ 1 4 c 7 f P n 1/ + c 8 π1 k E nf, G F, 1/. Now, by integrating both sides over G and by virtue of Lea. we get f S n f, A G,ω c 7 Enf, c 8 F, G + π1 k E nf, G i.e., c 7 + c 8n k Enf, G c 9 n 1 k E nf, G, f S n f, A G,ω c nen 1 k f, G for all natural nubers n. References A G 1. L. Ahlfors, Lectures on Quasiconforal Mapping, Wadsworth, Belont, CA, 1987, p V.V. Andrievskii, V.I. Belyi and V.K. Dyadyk, Conforal Invariants in Constructive Theory of Functions of Coplex Variable, Adv. Series in Math. Sciences and Engineer., WEP Co., Atlanta, Georgia, 1995, p I.M. Batchaev, Integral Representations in a Region with Quasiconforal Boundary, and Soe Applications, Ph. D. Thesis, Baku State Univ. Baku, V.I. Belyi, Conforal Mappings and Approxiation of Analytic Functions in Doains with Quasiconforal Boundary, USSR-Sb , A. Çavus, Approxiation by generalied Faber series in Bergan spaces on finite regions with a quasi-conforal boundary, J. Approx. Theory, , R.A. DeVore, G.G. Lorent, Constructive Approxiation, Springer Verlag, New York/Berlin, 1993, p D. Gaier, Lectures on Coplex Approxiation, Birkhäuser, Boston/Basel/Stuttgart, 1987, p D.M. Israfilov, Approxiative properties of generalied Faber series, in Proc. All-Union Scholl on Approxiation Theory, Baku, May, D.M. Israfilov, Approxiative properties of generalied Faber series in weighted Bergan spaces on finite doains with quasi-conforal boundary, East Journal on Approxiatios, , 1 13.

12 DANIYAL M. ISRAFILOV and YUNUS E. YILDIRIR 10. D.M. Israfilov, Faber Series in Weighted Bergan Spaces, Coplex Variables, , O. Lehto and K.I. Virtanen, Quasiconforal Mappings in the Plane, Springer Verlag, New York/Berlin, 1973, p P.K. Suetin, Series of Faber Polynoials, Gordon and Breach Science Publishers, Asterda, 1998, p Daniyal M. Israfilov Departent of Matheatics Faculty of Art and Sciences Balikesir University Balikesir TURKEY Yunus E. Yildirir Departent of Matheatics Faculty of Art and Sciences Balikesir University Balikesir TURKEY