Approximation by p -Faber-Laurent Rational Functions in the Weighted Lebesgue Spaces

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Let L ⊂ C be a regular Jordan curve. In this work, the approximation properties of the p-Faber-Laurent rational series expansions in the ω weighted Lebesgue spaces L p(L, ω) are studied. Under some restrictive conditions upon the weight functions the
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  Czechoslovak Mathematical Journal Daniyal M. IsralovApproximation by  p -Faber-Laurent rational functions in the weighted Lebesguespaces Czechoslovak Mathematical Journal , Vol. 54 (2004), No. 3, 751--765Persistent URL:  http://dml.cz/dmlcz/127926 Terms of use: © Institute of Mathematics AS CR, 2004Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese  Terms of use .This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project  DML-CZ: The Czech Digital Mathematics Library   http://project.dml.cz  Czechoslovak Mathematical Journal, 54 (129) (2004), 751–765 APPROXIMATION BY  p -FABER-LAURENT RATIONAL FUNCTIONSIN THE WEIGHTED LEBESGUE SPACES  ✂✁✄☎✝✆✞✁✠✟☛✡☞✌✍✏✎✁✑☎✒✟✓✔ , Balikesir ( Received December 18, 2001) Abstract.  Let  L  ⊂  C   be a regular Jordan curve. In this work, the approximationproperties of the  p -Faber-Laurent rational series expansions in the  ω  weighted Lebesguespaces  L  p ( L,ω ) are studied. Under some restrictive conditions upon the weight functionsthe degree of this approximation by a  k th integral modulus of continuity in  L  p ( L,ω ) spacesis estimated. Keywords  : Faber polynomial, Faber series, weighted Lebesgue space, weighted Smirnovspace,  k -th modulus of continuity MSC 2000  : 41A10, 41A25, 41A58, 41A30, 30E10 1. Introduction Let  L  be a rectifiable Jordan curve in the complex plane  C  ,  G  := int L  and  G − :=ext L . Without loss of generality we assume that  0  ∈  G . Let also  U   :=  { w :  | w |  <  1 } , T   :=  ∂U  ,  U  − :=  { w | :  | w |  >  1 } , and let  ϕ  and  ϕ 1  be the conformal mappings of   G − and  G  onto  U  − respectively, normalized by ϕ ( ∞ ) =  ∞ ,  lim z →∞ ϕ ( z ) /z >  0 and ϕ 1 (0) =  ∞ ,  lim z → 0 zϕ 1 ( z )  >  0 . The inverse mappings of   ϕ  and  ϕ 1  will be denoted by  ψ  and  ψ 1 , respectively.Later on we assume that  p  ∈  (1 , ∞ ) , and denote by  L  p ( L )  and  E   p ( G )  the set of all measurable complex valued functions such that  | f  |  p is Lebesgue integrable withrespect to arclength, and the Smirnov class of analytic functions in  G , respectively.751  Each function  f   ∈  E   p ( G )  has a nontangential limit almost everywhere (a.e.) on  L ,and if we use the same notation for the nontangential limit of   f  , then  f   ∈  L  p ( L ) .For  p >  1 ,  L  p ( L )  and  E   p ( G )  are Banach spaces with respect to the norm  f   E  p ( G )  =   f   L p ( L )  :=   L | f  ( z ) |  p | d z |  1 p . For the further properties, see [5, pp. 168–185] and [8, pp. 438–453].The order of polynomial approximation in  E   p ( G ) ,  p    1  has been studied byseveral authors. In [17], Walsh and Russel gave results when  L  is an analytic curve.For domains with sufficiently smooth boundary, namely when  L  is a smooth Jordancurve and θ ( s ) , the angle between the tangent and the positive real axis expressed as afunction of arclength  s , has modulus of continuity  Ω( θ,s )  satisfying the Dini-smoothcondition(1)    δ 0 Ω( θ,s ) s  d s <  ∞ , δ >  0 , this problem, for  p >  1 , was studied by S.Y. Alper [1].These results were later extended to domains with regular boundary which wedefine in Section 2, for  p >  1  by V.M. Kokilashvili [13], and for  p    1  by J.E. An-dersson [2]. Similar problems were also investigated in [10]. Let us emphasize that inthese works, the Faber operator, Faber polynomials and  p -Faber polynomials werecommonly used and the degree of polynomial approximation in  E   p ( G )  has been stud-ied by applying various methods of summation to the Faber series of functions in E   p ( G ) . More extensive knowledge about them can be found in [7, pp. 40–57] and[16, pp. 52–236].In [11], for domains with a regularboundary we have constructed the approximantsdirectly as the  n th partial sums of   p -Faber polynomial series of   f   ∈  E   p ( G ) , and laterapplying the same method in [3], we have investigated the approximation propertiesof the  n th partial sums of   p -Faber-Laurent rational series expansions in the Lebesguespaces  L  p ( L ) . The approximation properties of the  p -Faber series expansions in the ω -weighted Smirnov class  E   p ( G,ω )  of analytic functions in  G  whose boundary is aregular Jordan curve are studied in [12].In this work, when  L  is a regular Jordan curve, the approximation properties of the  p -Faber-Laurent rational series expansions in the  ω -weighted Lebesgue spaces L  p ( L, ω )  are studied. Under some restrictive conditions upon weight functions thedegree of this approximation is estimated by a  k th ( k    1 ) integral modulus of continuity in  L  p ( L,ω )  spaces. The results to be obtained in this work are also newin the nonweighted case  ω  = 1 .752  We shall denote by  c  constants (in general, different in different relations) depend-ing only on numbers that are not important for the questions of our interest. 2. New results For the formulation of new results in detail it is necessary to introduce somedefinitions and auxiliary results. Definition 1.  L  is called regular if there exists a number  c >  0  such that forevery  r >  0 ,  sup {| L  ∩  D ( z,r ) | :  z  ∈  L }    cr , where  D ( z,r )  is an open disk withradius  r  and centered at  z  and  | L ∩ D ( z,r ) |  is the length of the set  L ∩ D ( z,r ) .We denote by  S   the set of all regular Jordan curves in the complex plane. Definition 2.  Let  ω  be a weight function on  L .  ω  is said to satisfy the Muck-enhoupt  A  p -condition on  L  if  sup z ∈ L sup r> 0  1 r   L ∩ D ( z,r ) ω ( ζ  ) | d ζ  |  1 r   L ∩ D ( z,r ) [ ω ( ζ  )] − 1 /p − 1 | d ζ  |   p − 1 <  ∞ . Let us denote by  A  p ( L )  the set of all weight functions satisfying the Muckenhoupt A  p -condition on  L .For a weight function  ω  given on  L  we also define the following function spaces. Definition 3.  The set  L  p ( L,ω ) :=  { f   ∈  L 1 ( L ):  | f  |  p ω  ∈  L 1 ( L ) }  is called the ω -weighted  L  p -space. Definition 4.  The set  E   p ( G,ω ) :=  { f   ∈  E  1 ( G ):  f   ∈  L  p ( L,ω ) }  is called the ω -weighted Smirnov space of order  p  of analytic functions in  G .Let g  ∈  L  p ( T,ω )  and ω  ∈  A  p ( T  ) . Since L  p ( T,ω )  is noninvariantwith respect to theusual shift, we consider the following mean value function as a shift for  g  ∈  L  p ( T,ω ) : σ h g ( w ) := 12 h    h − h g ( w e it )d t,  0  < h <  π , w  ∈  T. As follows from the continuity of the Hardy-Littlewood maximal operator in weighted L  p ( T,ω )  spaces, the operator  σ h  is bounded in  L  p ( T,ω )  if   ω  ∈  A  p ( T  )  and thefollowing inequality holds:  σ h g  L p ( T,ω )   c (  p )  g  L p ( T,ω ) ,  1  < p <  ∞ . The last relation is equivalent [15] to the property lim h → 0  σ h g  − g  L p ( T,ω )  = 0 . Starting from the last two relations we can give the following definition.753  Definition 5.  If   g  ∈  L  p ( T,ω )  and  ω  ∈  A  p ( T  ) , then the function  Ω  p,ω,k ( g, · ):[0 , ∞ ]  →  [0 , ∞ )  defined by Ω  p,ω,k ( g,δ  ) := sup 0 <h i  δi =1 , 2 ,...,k  k  i =1 ( E   − σ h i ) g  L p ( T,ω ) ,  1  < p <  ∞ , is called the  k th integral modulus of continuity in the  L  p ( T,ω )  space for  g . Here E   is the identity operator.Note that the idea of defining such a modulus of continuity srcinates from [18].In [9] this idea was used for investigations of the approximation problems in L  p ([0 , 2 π ] ,ω )  spaces. Recently, in [12], to obtain direct theorems of the approxi-mation theory in the weighted Smirnov spaces  E   p ( G,ω ) , we have used the same ideafor the case  k  = 1 .It can be shown easily that  Ω  p,ω,k ( g, · )  is a continuous, nonnegative and nonde-creasing function satisfying the conditions(2)  lim δ → 0 Ω  p,ω,k ( g,δ  ) = 0 ,  Ω  p,ω,k ( g 1  + g 2 , · )  Ω  p,ω,k ( g 1 , · ) + Ω  p,ω ( g 2 , · ) . For an arbitrary function  f   ∈  L  p ( L,ω )  and a weight function given on  L  we alsoset f  0 ( w ) :=  f  [ ψ ( w )]( ψ ′ ( w )) 1 /p , f  1 ( w ) :=  f  [ ψ 1 ( w )]( ψ ′ 1 ( w )) 1 /p w 2 /p , (3) ω 0 ( w ) :=  ω [ ψ ( w )] , ω 1 ( w ) :=  ω [ ψ 1 ( w )] . The condition  f   ∈  L  p ( L,ω ) , implies that  f  0  ∈  L  p ( T,ω 0 )  and  f  1  ∈  L  p ( T,ω 1 ) .Then if   ω  ∈  A  p ( L )  and  ω 0 ,ω 1  ∈  A  p ( T  )  we can define the weighted integral moduliof continuity  Ω  p,ω,k ( f  0 ,δ  )  and  Ω  p,ω,k ( f  1 ,δ  ) , using the procedure given above.Main result in our work is the following theorem. Theorem 1.  Let  L  ∈  S   and   f   ∈  L  p ( L,ω ) ,  1  < p <  ∞ . If   ω  ∈  A  p ( L )  and  ω 0 ,ω 1  ∈  A  p ( T  ) , then for every natural number   n  there are a constant  c >  0  and a rational function R n ( z,f  ) := n  k = − n a ( n ) k  z k such that  f   − R n ( · ,f  )  L p ( L,ω )   c  Ω  p,ω 0 ,k  f  0 ,  1 n  + Ω  p,ω 1 ,k  f  1 ,  1 n  , 754
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