 Advances in Pure Mathematics, 2011, 1, 305-314 doi:10.4236/apm.2011.16056 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Uniform Convergence of Extremal Polynomials When Domains Have Corners and Special Cusps on the Boundary Cem Koşar1, Mehmet Kucukaslan2, Fahreddin G. Abdullayev2 1Department of Mat hem at i cs, Facult y of S cience and L iterat ur e , Bitlis Eren University, Bitlis, Turkey 2Department of Mat hem at i cs, Facult y of S cience and L iterat ur e , Mersin University, Mersin, Turkey E-mail: ckosar77@gmail.com, {mkucukaslan, fabdul}@mersin.edu.tr Received May 26, 2011; revised September 13, 2011; accepted Septemb er 22, 2011 Abstract We study the approximation properties of the extremal polynomials in Ap-norm and C-norm. We prove esti-mates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners and special cusps. Keywords: Uniform Approximation, Extremal Polynomials, Conformal Mapping, p-Bieberbach Polynomials 1. Introduction and Main Results 1.1. Statement of Problem and Some Definitions Let be a finite domain bounded by Jordan curve and 0GG:=LG be an arbitrary fixed point. Let be conformal mapping of onto the disk wz=G000, :=Dr0=0,: 0.p1pAG the set of functions fz=0,f analytic in and normalized by such that G0f0=111:=d< ,pp'zAGpGffz where dz is a two-dimensional Lebesque measure on .GWe denote by n of all algebraic polynomials satisfying nn,degnPz P0=0,nP0=1.nPLet us consider following extremal problem: 1:nnnAGpPPmin (1.1) Using a method similar to the one given in (, p. 137), it is seen that there exists an extremal polynomial nPz furnishing to the problem (1.1), these polyno- mials nPz are determined uniquely in case of (, p. 142). This extremal problem was first considered by Kucukaslan M. and Abdullayev F.G. and they were called the p-Bieberbach polynomial of degree for the pair >1pn0,G in , and denoted by ,.npBzThe main goal of this paper is to investigate the approximation rate of convergence to the function ,npBz in uniform norm for some domains when has some certain singularity, i.e. ,,():= max,np npCG zGcBzBzn (1.2) where :=,> 0,pG the constant is independent of c.nIn case of the solution of (1.1) coincides with the well known Bieberbach polynomial =2,p-nthz,2nn (see, for example, [3,4] ). The appro- ximation properties in the uniform norm of Bz BnBz on G first was observed by Keldysh in 1939  for the domains with sufficiently smooth boundary. A consi- derable progress in this area has been achieved by Mergelyan , Suetin , Simonenko , Andrievskii [7,8] Gaier [9,10], Abdullayev [11-13], Israfilov [14,15] and the others. In this paper, we are going to consider the case the problem in (1.2). First, we will investigate the approximation rate of to the function >1p,npBz in C. KOŞAR ET AL. 306 1-pAnorm and using well known Simonenko and Andrievskii method (see, for example, [6,7]), the appro- ximation rate of to the function ,npBz in uniform norm will be obtained. Now we need some definitions: Definition 1.1. (, p. 97)The Jordan arc or a curve is called a quasiconformal () arc or curve if there is a quasiconformal mapping L-K1K-Kf of a region containing such that D LfL is line segment or circle. Let FL denote the set of all sense-preserving plane homeomorphisms f of regions such that DLfL is a line segment or circle and let =inf :,KLKffFL where Kf. is the maximal dilatation of such a map- ping f Then is L-Kquasiconformal if and only if If is a <.KL L-Kquasiconformal, then .KLK D gives the global definition of a -Kqua- siconformal arc or curve consequently. This definition is common in the literature. At the same time, we can consider the domain as the neighborhood of the curve In this case, Definition 1.1 will be called local definition of quasiconformal arc or curve. Through this work we consider the local definition. The local defi- nition has an advantage in determining the coefficients of quasiconformality for some simple arcs or curves. D=,zzsL.LLet us denote 0,smesL natural repre- sentation of := .LGDefinition 1.2. () We say that GC if G has a continuous tangent  :=szs,GC for every points . zsCorollary 1.1. () If  then =1K for all >0. Definition 1.3. () We say that  if is expressed as a union of a finite number of ;, 0  0<<2,,GC :=L-CGarcs, connecting at the points 0, 1 such that is locally smooth at 0 and in the local coordinate system ,,,mzzz L,zxy with origin at ,jz the following conditions are satisfied: 1j,m,1) For every jz the domain has 1jp ,mGπ,j 0< 0 for some constants 12<< 0. ,5np CGBcn for each Corollary 1.7. Let and assume that =2p;GC for some 0< <2 and 1101ww :<1,ww normalized by =, >0 (00ˆˆ=0, >0 ). The level curve (exterior or interior) can be defined for as >0t 1ˆ:=:=,if<1;=,if> 1,.tLzzttztt LL Let us denote t and := ,ttGintL :=textL,:=inf :dzLz L. Let be a L-Kquasiconformal curve and .D Then the region can be chosen to be the region 0D0\,RrG,,Gˆ for a certain number 01< depending on 2Rf and 10R0=r. In this case, it is known that the function  11.= .ff2 is a -K quasi- conformal reflection across as shown in (, p. 28) by analogously in (, p. 75), that is, is a L.2-K quasiconformal mapping leaving points on fixed and Lsatisfying the conditions \RGG G0\rG, 0\rRGGG G\ for some 01< <,RR0<<1.rrBy using the facts in (, p. 97) and (, p. 76, , p. 26) we can find a -CKquasiconformal reflection . across such fies the following L that it satisCopyright © 2011 SciRes. APM C. KOŞAR ET AL. 308 111221,,<<,zzzzzLz11, << ,1,<, ,>,zzzzzzzzz   (2.1) and Jacobian 22=zzJof . satisfied by means of thef a quDLemma 2.1. () Let be a 1.J Therefore, exn theorem otensioasiconformal mapping, without loss of generality we may assume that  =, .zzz L-Kquasiconformal curve; 123 1110;,:, ,RLzzG zzzcdzL zˆ=jjwz 231210,: ,, ==1,2,3.jjorzzGzzzc dzLwzj  Then, ents 1) The statem12 13zz zz and 12 13w ww are are w equivalent. So12 13zz zz and 1213ww ww 2) If 12 13zz,zz then 221313 131212 12KKwwzz wwwwzz ww and, consequently, for any 3300RzLzL 2121Kww z 2212,Kz wwwhere and are fixed cotants. al cu01<< 2Rma 2.2. ([2100=R t :=LnsLem 1]) LeG be a quasiconformrve. Then, For every zL 0there exists an arc 0,z in G joining  to z with following p: 1) roperties,dLz for every If0,z 2) 12, is joining the subarc of ,z01 to 2, then 12 1 ,es 2m very pair 2 0,,.z  ( a -Kqua-  for e) Let be sie aritrary Jan domain and le arc except Lemma 2.3. 1:=LG mesconformal curve. Then  mes, fvery rectifiable arc .G Let G bbordor e0:=,, , zL  an rectifiab0L which satisfies the following conditions: 1) mes0zfor one of its endpoints of its z12, 1 2, for all 12,. ng function 2) There exists a monotone increasigt suhat ch t0,dLg z for all . Lemma 2.4. () Let ;GC for some 0< <2,  0 and ofunc- tion n the arc  a measurable f n such that be give120zf for all .en for all z. Th211222122ln1<<2,d22<2 1.pAGppfzpp . Approximation in the 3-1pAGNorm Suppose that ;GC f0<<2or some  and 0 =zis giveof simput n. For the sake , we assumelicity, but witho=1,z loss of generality that =2,m 121; 1,1 G and let the local coordinate axes be parallel to OX and OY in the coordinate system; ,z 2:=:,0 .LzzLImz Then 0z is taken a arbitrary point on 2L (or on 1L We recall that the domain 1:=Lzsubject t:o, 0LImzs anhosen directio the cn). ;GC has exteri r πo 0< <2 non zero angle and 1xd of the po type interior zero angle in the neighborhooints 1=1z 2ectively. We can say that the function ˆ=wz domain and =1,z respfor the;CG satisfies the escribed inconditions dint 1z Lemma 2 in the neighborhood of po1=. So, we can e Lemma 2  asily get from 1ˆ1,z (3.1) for all 221ˆˆ,1; 1dzL zz1:=:1 >.zM zGz On the other hand, using properties of the function zˆ=w in the neigbourhood of the point 1=1z (see, [7,23]) we obtain 1ˆˆ1ln 1zz  (3.2)Because each is a ,=1,2jLj1-iquasi- .j muconformal arc, st be rethe quasiconformal flection across .jLsider the Let us con curve 1112112:=: =1;3cczxiyyx= 1212122:==: =1;3cczxiyy x 123:==:=1 ;zxiyycx 224:==:=1;zxiyycx Copyright © 2011 SciRes. APM C. KOŞAR ET AL.309 for some where a constnts and frommma 2.2 that 43<<<,cc Definition 1.3. It is easy to check from Lea1c2c 121 2 ,,jmes ifoLet fficiently large ntural number. For arbitrary   r all 12,, ,=1,2.ijij  =NNR su0>nN and a0< <1, let us choose 1cn0=Rr such =1, 2 such that and points 00<0  12 1112212 1122212121ln,1<2,=1, 12, 2<<2,=1,11d12,=2, 1, ppjppnpjnpjnjnn    12lniRjAGpzn 2, 1,:= .iij jmes R wtoThen, from (2.1), (3.1) and (3.2), for Lemma 2.1 and Corollary 1.0, we get  C. KOŞAR ET AL.311 1111,11,=1,, =11.1,, =2,=iijijij jiijjdzL jnzdz Ljjn  2 (3.13) Then, the following inequalities are obtained from (3.12) and (3.13)   12 111222ln << 2,1dnpz  111211,112,2 <2.1piRAGpnpn2p (3.14) and  222121iRpz211221ln,1 <<2,1d2,2 <2.1pAGpnpnpn  (3.15) Combining (3.14) and (3.15) the proof is completed. □ Now, for sufficiently small 010<< ,1233VV where we are going to use following notations: 0121222:=RUVVVVj111:=jRV 01,:>0, =1,2UDzImzj  20:=1,:<0, =1,2jjRVUDzImzj  130:=:>0\1,1,,RVUzImz DD 230:=:< 0\1,1,.RVUzImzDD0 Lemma 3.2. (, p. 10) Let ;GC for some 0< <2, 0.ll >0 Then for a 111311 ,iiimes V ,imes Vnn  and 221, =1,2iimesVin where 1:= min,.22 Lemma 3.3. Let ;GC for some 0< <2, 0. Then for all >0 min 21=Rm Ui,1 , 1, 2.ies n Proof. We have 3iNow, let us estimate  212=1= iiiR iiUmesVmes VV 1 iis V by choosing me=1, >0K and from Lemma 2.1 we have 1121311,.iiiimesVdzLn  1121111 ,iiiimesVdzLn On the other hand if we use but except the conformal mappingthe method of Gaier in   in method, then we obtain min 2,121.imes Vn After then using the second and third inequality in the we obtain the desireroof. □ Lemma 3.4. Let first oned p;GC for some 0< <2, 0. Then for all >0 12 11d, URppiin  =1,2where 12<21p=min ,2. and  Proof. Let ,zG zthe point of nearest to:=LG .z Then we have 11,zzzz zz and from (, Lemma 3) Copyright © 2011 SciRes. APM C. KOŞAR ET AL. 312  100,1.oowwwwdwLwrwrwrwrw After this estimation, we get 12121 121001()1 d,UURiRi'UiRpppwrw n2dpwddpp wrrUiR       wh  ere :zzL:= minr and iR1<2 .12p□ By using Lemma 3.1, 3.2, 3.3, 3.4 and (3.11) we get the following results. We need this notation:  12 1112222111222131ln,max,,=1, 2,3.11, <,, 0 <.2,:=11, <,, <.2312, <,, <<2.3pipnppnpnpnppnpn    2iLemma 3.5. Let and assume that 1< 2p;GC for some 0<<2, 0. Then, for and arbitrary smany 3nall >0 1,,.np nAGpBp (3.16) Lemma 3.6. Let 22< <21pand assume that ;GC 11,12 11,0<<1,.11<2.pnp pAGppnBn  (3.17) Proof. The proof of Lemma 3.5-3.6 is similar. So, we give them together. From (3.11) and Lemma 3.1-3.4 we obtain for some 0<<2, 0. Then, for arbitrary smany 3n andall >0 12 1112212 11221122ln n2121ln1 <<2,= 1,1122<2 ,=1,111 <<2,=2, 1pnAGppppnpjnQnpjnpjnn 221< 2,2112<2 ,=2,,2 <2,11pppppj pn12 1,1 < pn  (3.18)   Copyright © 2011 SciRes. APM C. KOŞAR ET AL.313 where 1:= min,22, :=min2;1,:=min1, 2  Case 1. Let In this case from (3.18) we get 1< <2.p212 1112 2211ln ,.pppnAGpnQnnnp    Case 2. 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Kucukaslan, “On the Conver-gence of the Fourier Series of Orthonormal Polynmials in the Domain with Piecewise Smooth Boundary,” Pro-ceeding of IMM2001, pp. 3-13. Copyright © 2011 SciRes. APM