Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl ) Contents lists available at ScienceDirect Journal of Mathematical Analysis Applications Approximation by rational functions in Smirnov Orlic classes Sadulla Z. Jafarov Department of Mathematics, Faculty of Art Sciences, Pamukkale University, 20017, Denili, Turkey article info abstract Article history: Received 14 September 2010 Available online 12 February 2011 Submitted by S. Ruscheweyh Let G be a doubly-connected domain bounded by Dini-smooth curves. In this work, the approximation properties of the Faber Laurent rational series expansions in Smirnov Orlic classes E M G) are studied Elsevier Inc. All rights reserved. Keywords: Faber Laurent rational functions Conformal mapping Dini-smooth curve Smirnov Orlic class Modulus of smoothness 1. Introduction, some auxilary results main results Let G be a doubly-connected domain in the complex plane C, bounded by the rectifiable Jordan curves the closed curve is in the closed curve ). Without loss of generality we assume 0 int. Let G 0 1 := int, G 1 := ext, G 0 2 := int, G 2 := ext. We denote by ω = φ) the conformal mapping of G 1 onto domain D 1 := {ω C: ω > 1} normalied by the conditions φ ) =, φ) lim = 1 let ψ be the inverse mapping of φ. We denote by ω = φ 1 ) the conformal mapping of G 0 2 onto domain D 2 := {ω C: ω > 1} normalied by the conditions φ 1 0) =, lim.φ1 ) ) = 1, 0 let ψ 1 be the inverse mapping of φ 1. Let us take C ρ0 := { : φ) = ρ 0 > 1 }, L r0 := { : φ 1 ) = r 0 > 1 }. φ has the Laurent expansion in some neighbourhood of the point = has the form φ) = γ γ 0 γ 1 γ 2 2 address: sjafarov@pau.edu.tr X/$ see front matter 2011 Elsevier Inc. All rights reserved. doi: /j.jmaa

2 S.Z. Jafarov / J. Math. Anal. Appl ) hence we have [ ] n n 1 φ) = γ n n The polynomial γ n,k k k<0 n 1 Φ n ) := γ n n γ n,k k γ n,k k. is called the Faber polynomial of order n for the domain G 0 1. The function φ 1 has an expansion in some neighbourhood of the point origin: φ 1 ) = 1 β 0 β 1 β k k. Raising this function to the power n, weobtain [ φ1 ) ] ) n 1 = Fn Q n ), G 0 2, where F n 1 ) denotes the polynomial of negative powers of the term Q n) contains non-negative powers of ; hence this is an analytic function in the domain G 0 2. For Φ n ) F n 1 ) the following integral representations hold [21]: 1. If int C ρ0,then Φ n ) = 1 Cρ 0 [φζ)] n ζ dζ = 1 ω =ρ 0 ψ ω)ω n dω. 1.1) ψω) 2. If ext C ρ0,then Φ n ) = [ φ) ] n 1 Cρ 0 [φζ)] n ζ dζ. 1.2) 3. If int C r0,then ) 1 F n = [ φ 1 ) ] n 1 C r0 [φ 1 ζ )] n dζ. 1.3) ζ 4. If ext C r0 then ) 1 F n = 1 C r0 [φ 1 ζ )] n dζ = 1 ζ ω =r 0 ψ 1 ω)ωn dω. 1.4) ψ 1 ω) If a function f ) is analytic in a doubly-connected domain bounded by the curves C ρ0 r0, then the following series expansion holds [17,27]: ) 1 f ) = a k Φ k ) b k F k 1.5) where a k = 1 C r1 f )φ ) 1 d = [φ)] k1 ω =ρ 1 f [ψω)] ω k1 dω 1 < ρ 1 < ρ 0 ) b k = 1 C r1 f )φ 1 ) 1 d = [φ 1 )] k1 ω =r 1 f [ψ 1 ω)] ω k1 dω 1 < r 1 < r 0 ). 1.6)

3 872 S.Z. Jafarov / J. Math. Anal. Appl ) For G use of Cauchy theorem, gives f ) = 1 f ζ ) ζ dζ 1 f ξ) ξ dξ. If int ext,then 1 f ζ ) ζ dζ 1 f ξ) dξ = ) ξ Let us consider I 1 ) = 1 f ζ ) ζ dζ, I 2) = 1 f ξ) ξ dξ. The function I 1 ) determines the functions I 1 ) I 1 ) while the function I 2) determines the functions I 2 ) I 2 ). The functions I 1 ) I 1 ) are analytic in int ext, respectively. The functions I 2 ) I 2 ) are analytic in int ext, respectively. Let us further assume that B is a simply-connected domain with a rectifiable Jordan boundary B := ext, further let T := { ω C: ω =1 }, D := ext T. Also, φ st for the conformal mapping of B onto D normalied by φ ) = φ ) lim > 0 let ψ be the inverse of φ. Let also χ be a continuous function on 2π. Its modulus of continuity is defined by ωt, χ) := sup χt 1 ) χt 2 ), t 0. t 1,t 2 [0,2π], t 1 t 2 <t The curve is called Dini-smooth curve if it has the parametriation : χt), 0 t 2π, such that χ t) is Dini-continuous, i.e. π 0 ωt,χ ) dt < t χ t) 0 [24, p. 48]. A convex continuous function M :[0, ) [0, ) is called an N-function if the conditions M0) = 0, Mx)>0 for x > 0, Mx) Mx) lim = 0, lim = x 0 x x x are satisfied. The complementary N-function to M is defined by ) Ny) := max xy Mx), for y 0. x 0

4 S.Z. Jafarov / J. Math. Anal. Appl ) Let us denote by L M ) the linear space of Lebesgue measurable functions f : C satisfying the condition [ M α ] f ) d < for some α > 0. The space L M ) becomes a Banach space with Orlic norm f LM ) := sup f ).g) d, ρg,n) 1 where g L N ), N is the complementary N-function to M ρg; N) := [ ] N g) d. The Banach space L M ) is called an Orlic space [26, pp ]. It is known, cf. [26, p. 50], that every function in L M ) is integrable, i.e. L M ) L 1 ). An N-function M satisfies the 2 -condition if M2x) lim sup x Mx) <. The Orlic space L M ) is reflexive if only if the N-function M its complementary function N both satisfy 2 - condition [26, p. 113]. Detailed information about Orlic spaces can be found in the books [18,26]. Let r be the image of the circle {ω C: ω =r, 0 < r < 1} under some conformal mapping of D onto B let M be an N-function. If an analytic function f in B satisfies [ ] M f ) d <, r uniformly in r, then it belongs to Smirnov Orlic class E M B). We remark that if Mx) := Mx, p) := x p,1< p <, then the Smirnov Orlic class E M B) coincides with the usual Smirnov class E p B). The space E M B) becomes a Banach space with the Orlic norm. Every function in the class E M B) has [19] the non-tangential boundary values almost everywhere a.e.) on the boundary function belongs to L M ), hence for f E M B) the norm E M B) can be defined as f E M B) := f LM ). Let B be a finite domain in the complex plane by a rectifiable Jordan curve f L 1 ). Then the functions f f defined by f ) = 1 f ζ ) ζ dζ, B f ) = 1 f ζ ) ζ dζ, B are analytic in B B respectively, f ) = 0. Thus the limit 1 f ζ ) S f )) := lim ε ζ dζ {ζ : ζ >ε} exists is finite for almost all. The quantity S f )) is called the Cauchy singular integral of f at. According to the Privalov theorem [5, p. 431], if one of the functions f or f has the non-tangential limits a.e. on, then S f )) exists a.e. on also the other one has the non-tangential limits a.e. on.conversely,ifs f )) exists a.e. on, then the functions f ) f ) have non-tangential limits a.e. on. In both cases, the formulae

5 874 S.Z. Jafarov / J. Math. Anal. Appl ) f ) = S f )) 1 2 f ), f ) = S f )) 1 2 f ) hence f = f f holds a.e. on. From the results in [20], it follows that if is a Dini-smooth curve L M ) is a reflexive Orlic space on,the singular operator S is bounded on L M ). For r > 0ther-th modulus of smoothness of a function f L M T ) is defined as ω r f,δ) M := sup r h f LM, δ>0, r > 0, T ) h δ where r h f x) := r ) 1) k r f x r ) k)h. k If are Dini-smooth, then from the results in [30], it follows that 0 < c 1 < φ ) < c 2 <, 0 < c 3 < φ 1 ) < c 4 <, 0 < c 5 < φ ω) < c 6 <, 0 < c 7 < φ 1 ω) < c 8 < 1.8) where the constants c 1, c 2, c 3, c 4 c 5, c 6, c 7, c 8 are independent of Ḡ ω 1, respectively. We will say that the doubly connected domain G is bounded by the Dini-smooth curve if the domains G 0 1 G0 2 are bounded by the closed Dini-smooth curves. Let i i = 1, 2) be a Dini-smooth curve let f 0 := f ψ for f L M ) let f 1 := f ψ 1 for f L M ).Thenby 1.8) we obtain f 0 L M T ) f 1 L M T ). Using the non-tangential boundary values of f 0 f 1 on T we define ω r, f,δ) M := ω r f 0,δ) M, δ>0, ω r, f,δ) M := ω r f 1,δ) M, δ>0, for r > 0. Since f 0, f 1 L M T ), wehave f 0, f 1 E MD) f 0, f 1 E MD ) such that f 0 ) =, f 1 ) = 0 f 0 ω) = f 0 ω) f 0 ω), } f 1 ω) = f 1 ω) f 1 ω) 1.9) a.e. on T. Lemma 1. See [4].) Let both an N-function M its complementary function satisfy the 2 condition. Then there exists a constant c 9 > 0, suchthatforeveryn N, gω) d k ω k LM c 9 ω r g, 1/n) M, α > 0 T ) where d k k = 0, 1, 2,...)are the k-th Taylor coefficients of g E M D) at the origin. We set R n f, ) := ) 1 a k Φ k ) b k F k. The rational function R n f, ) is called the Faber Laurent rational function of degree n of f. Since series of Faber polynomials are a generaliation of Taylor series to the case of a simply connected domain, it is natural to consider the construction of a similar generaliation of Laurent series to the case of a doubly-connected domain. In this work direct theorem of approximation theory in the Smirnov Orlic classes, defined in the doubly-connected domains with the Dini-smooth boundary are proved. Similar problems were studied in [17,28,29,31].

6 S.Z. Jafarov / J. Math. Anal. Appl ) We remark that problem of approximation theory in Orlic classes defined on the simply connected domain with boundary has been investigated by Akgün Israfilov [4] in the case that is a closed Dini-smooth curve. Similar problems for the different spaces defined on the simply connected domain of the complex plane were investigated by several authors see for example, [1 4,6 17,22,23]). Note that the approximation of functions by polynomials rational functions in Orlic spaces defined on the intervals of the real line have been investigated by Ramaanov [25]. Now, in the doubly-connected domain we define Smirnov Orlic class. Let r be the image of the circumference ω =r r 2 < r < r 1 ) regard to a conformal mapping of the ring 0 < r 2 < ω < r 1 < 1 onto doubly-connected domain G let M be an N-function. The class of analytic functions f ) defined on the domain G will be called Smirnov Orlic class E M G) if [ ] M f ) d <, r uniformly in r. Our main results are as follows. Theorem 1. Let G be a finite doubly-connected domain with the Dini-smooth boundary =,lete M G) be a reflexive Smirnov Orlic class on G. If r > 0 f E M G) then for any n = 1, 2, 3,...there is a constant c 10 > 0 such that f Rn., f ) LM ) c 10{ ωr, f, 1/n) M ω r, f, 1/n) M }, where R n., f ) is the n-th partial sum of the Faber Laurent series of f. 2. Proof of the new results Proof of Theorem 1. We take the curves, T := {ω C: ω =1} as the curves of integration in the formulas 1.1) 1.4) 1.6), respectively. This is possible due to the conditions of Theorem 1.) Let f E M G). Then f 0, f 1 L M T ). According to 1.9) f ζ ) = f 0 φζ) ) f 0 φζ) ), f ξ) = f 1 Let ext.thenfrom1.2) 2.1) we have a k Φ k ) = = [ ] k 1 a k φ) [ ] k 1 a k φ) φ1 ξ) ) f 1 a k[φζ)] k dζ ζ a k[φζ)] k f 0 [φζ)] dζ 1 ζ φ1 ξ) ). 2.1) f ζ ) ζ dζ f [ ] 0 φ). 2.2) For ext, consideration of 1.4) 2.1) gives ) 1 b k F k = 1 b k[φ 1 ξ)] k dξ ξ = 1 = 1 b k[φ 1 ξ)] k dξ ξ f 1 [φ 1ξ)] b k[φ 1 ξ)] k ξ dξ 1 f ξ) dξ. 2.3) ξ For ext, using 2.2), 2.3) 1.7) we have [ a k Φk ) ] ) k 1 [ ] k a k F k = a k φ) f [ ] 1 f 0 φ) 0 [φζ)] a k[φζ)] k dζ ζ 1 f 1 [φ 1ξ)] b k[φ 1 ξ)] k dξ. ξ

7 876 S.Z. Jafarov / J. Math. Anal. Appl ) Taking the limit as along all non-tangential paths outside,weobtain f ) a k Φ k ) ) 1 b k F k = f 0 [ φ )] [ a k φ )] k 1 2 S 1 [ f 0 φ) f 0 ] a k φ k 1 [ φ )] [ a k φ )] ) k f 1 [φ 1ξ)] b k[φ 1 ξ)] k a.e. on. Now using 2.4), Minkowski s inequality the boundedness of S 1 we have f Rn f, ) LM ) c 11 f 0 ω) a k ω k LM T ) ξ dξ 2.4) c 12 f 1 w) b k ω k LM T ). 2.5) That is, the Faber Laurent coefficients a k b k of the function f are the Taylor coefficients of the functions f 0 f 1, respectively. Then by Lemma 1 2.5) we have f Rn f, ) LM ) c 13{ ωr, f, 1/n) M ω r, f, 1/n) M }. Let int.thenby1.4) 2.1) we have ) 1 b k F k = = [ b k φ1 ) ] k 1 [ b k φ1 ) ] k 1 1 For int, from 1.1) 2.1) we obtain b k[φ 1 ξ)] k dξ ξ b k[φ 1 ξ)] k f 1 [φ 1ξ)]) dξ ξ f ξ) ξ dξ f [ 1 φ1 ) ]. 2.6) a k Φ k ) = 1 = 1 a k[φζ)] k dζ ζ a k[φζ)] k f 0 [φζ)]) dζ 1 ζ f ζ ) dζ. 2.7) ζ The use of 2.6) 2.7) for int gives ) 1 a k Φ k ) b k F k = 1 1 a k[φζ)] k f 0 [φζ)]) dζ ζ b k[φ 1 ξ)] k f 1 [φ 1ξ)]) dξ f [ 1 φ1 ) ] ξ [ b k φ1 ) ] k.

8 S.Z. Jafarov / J. Math. Anal. Appl ) Taking the limit as along all non-tangential paths inside,wereach f ) a k Φ k ) ) 1 b k F k = f [ 1 φ1 )] [ 1 [ b k φ1 )] k f [ 1 φ1 )]] 2 [ S 2 b k φ k 1 f 1 φ ) ] 1 1 a k[φζ)] k f 0 [φζ)]) ζ dζ 2.8) a.e. on. Consideration of 2.8), the Minkowski s inequality the boundedness of S 2 give rise to f Rn f, ) LM ) c 14 f 1 ω) b k ω k LM c 15 f 0 w) a k ω k LM. 2.9) T ) T ) Use of Lemma 1 2.9) leads to f Rn f, ) LM ) c 16{ ωr, f, 1/n) M ω r, f, 1/n) M }. Then the proof of Theorem 1 is completed. Acknowledgment The author is greatly indebted to Daniyal M. Israfilov for discussion of this paper. References [1] S.Ja. Al per, Approximation in the Mean of Analytic Functions of Class E p, Gos. Idat. Fi.-Mat. Lit., Moscow, 1960, pp in Russian). [2] J.E. Andersson, On the degree of polynomial approximation in E p D), J. Approx. Theory ) [3] M.I. Andrasko, On the approximation in the mean of analytic functions in regions with smooth boundaries, in: Problems in Mathematical Physics Function Theory, vol. 1, Idat. Akad. Nauk Ukrain. SSR, Kiev, 1963, p. 3 in Russian). [4] R. Akgün, D.M. Israfilov, Approximation moduli of fractional orders in Smirnov Orlic classes, Glas. Mat ) 2008) [5] G.M. Goluin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Monogr., vol. 26, Amer. Math. Soc., Providence, RI, [6] I.I. Ibragimov, D.I. Mamedkhanov, Constructive characteriation of a certain class of functions, Sovrem. Math. Dokl ) [7] D.M. Israfilov, Approximation properties of generalied Faber series in an integral metric, Iv. Akad. Nauk A. SSR Ser. Fi.-Tekh. Mat. Nauk ) in Russian). [8] D.M. Israfilov, Approximation by p-faber polynomials in the weighted Smirnov class E p G, w) the Bieberbach polynomials, Constr. Approx. 17 3) 2001) [9] D.M. Israfilov, Approximation by p-faber Laurent rational functions in the weighted Lebesque spaces, Cechoslovak Math. J. 54 3) 2004) [10] D.M. Israfilov, A. Guven, Approximation in weighted Smirnov classes, East J. Approx. 11 1) 2005) [11] D.M. Israfilov, R. Akgün, Approximation in weighted Smirnov Orlic classes, J. Math. Kyoto Univ. 46 4) 2006) [12] D.M. Israfilov, B. Oktay, R. Akgün, Approximation in Smirnov Orlic classes, Glas. Mat ) 2005) [13] S.Z. Jafarov, D.M. Israfilov, Approximation on curves by means of rational functions, in: Materials of VIII Republican Conference of Young Scientists on Mathematics Mechanics, Elm, Baku, 1988, pp in Russian). [14] S.Z. Jafarov, D.M. Israfilov, Approximation of continuous functions given on curves, in: S.M. Nikol skii Ed.), Theory of Functions Approximations. Part 1, Proceedings of the Second Saratov Winter School, 1990, Saratov. Gos. Univ., Saratov, Russia, 1990, pp in Russian). [15] S.Z. Jafarov, S.M. Nikol skii type inequality estimation between the best approximation of a functions in norms of different spaces, Math. Balkanica N.S.) ) 2007) [16] S.Z. Jafarov, Approximation by polynomials rational functions in Orlic spaces, J. Comput. Anal. Appl ) [17] G.S. Kocharyan, On a generaliation of the Laurent Fourier series, Iv. Akad. Nauk Arm. SSR Ser. Fi.-Mat. Nauk 11 1) 1958) [18] M.A. Krasnoselskii, Ya.B. Rutickii, Convex Functions Orlic Spaces, P. Norrdhoff Ltd., Groningen, [19] V. Kokilashvili, On analytic functions of Smirnov Orlic classes, Studia Math ) [20] A.Yu. Karlovich, Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlic spaces, Math. Nachr ) [21] A.I. Markushevich, Analytic Function Theory: vols. I, II, Nauka, Moscow, [22] Dh.I. Mamedkhanov, Approximation in complex lane singular operators with a Cauchy kernel, Dissertation Doct. Phys.-Math. Nauk, University of Tbilisi, 1984 in Russian). [23] Mokhamad Ali, The problem of approximation theory on complex plane, Author s summary of cidate dissertation, Baku, 1990 in Russian). [24] Ch. Pommmerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin, [25] A.-R. K Ramaanov, On approximation by polynomials rational functions in Orlic spaces, Anal. Math ) [26] M.M. Rao, Z.D. Ren, Theory of Orlic Spaces, Marcel Dekker, New York, [27] P.K. Suetin, Series of Faber Polynomials, vol. 1, Gordon Breach, Reading, [28] H. Tiet, Laurent-Trennung weifach unendliche Faber-Systeme, Math. Ann ) 1955) [29] H. Tiet, Faber series the Laurent decomposition, Michigan Math. J. 4 2) 1957) [30] S.E. Warschawski, Über das ranverhalten der Ableitung der Abildunggsfunktion bei Konfermer Abbildung, Math. Z ) [31] Xie Ji-feng, Zhang Ming, A complex approximation on doubly-connected domain, J. Fudan Univ. Nat. Sci. 44 2) 2005)