Lacunary non-continuable boundary-regular holomorphic functions with universal properties

iZWESTIQ nan aRMENII . mATEMATIKA , 41 , } 1, 2006, 27{40 LACUNARY NON{CONTINUABLE BOUNDARY{REGULAR HOLOMORPHIC FUNCTIONS WITH UNIVERSAL PROPERTIES L. Bernal{Gonzalez, M. C. Calderon{Moreno 1 and W. Luh 2 Facultad De Matematicas, Avenida Reina Mercedes, 41080 Sevilla, Spain Fachb ereich Mathematik Universitat Trier, D-54286 Trier, Germany E{mails : lb [email protected], [email protected], [email protected] Abstract. A holomorphic function ' in a Jordan domain G in the complex plane is constructed with all its derivatives extending continuously up to the b oundary @G that happ ens to b e a natural b oundary of ' . In addition, the action of a certain class of op erators on ' presents some universal prop erties related to the overconvergence phenomenon. x 1. INTRODUCTION AND NOTATION In this pap er, we are concerned with the problem of the existence of holomorphic functions dened on a Jordan domain G of the complex plane that enjoy simultaneously several prop erties, namely : { The b oundary of G is the natural b oundary of those functions. { They are b oundary-regular, that is, their derivatives of all orders extend continuously up to the b oundary of G . {Thepower series expansion of eachsuch a function around a prexed p ointof G presents gaps outside a prescrib ed sequence S of integers with upp er density d ( S )=1. { The action of a certain class of op erators {including, for instance, the identity and the dierentiation op erators of all orders{ on the partial sums of their Taylor 1 The rst two authors have b een partially supp orted by Plan Andaluz de Investigacion de la Junta de Andaluca FQM-127 and by Ministerio de Ciencia y Tecnologa Grant BFM2003-03893-C02-01. 2 Corresp onding author. 28 L. Bernal{Gonzalez, M. C. Calderon{Moreno, W. Luh expansions satisfy some kind of \external universality" which is, in fact, a strong version of overconvergence. The aim of this note is the construction of a function with all ab ove prop erties. The precise statement together with its pro of will b e p ostp oned till Section 3. In the remainder of this section, the p ertinent terminology will b e xed, and some historical or bibliographical notes will b e p ointed out. A numb er of preparatory results will b e stated in Section 2, where we also intro duce a new class of op erators, which are rather \natural" to our goal. As usual, by C , D , Q , N , N 0 we denote the complex plane, the op en unit disk, the set of rational numb ers, the set of p ositiveintegers and N f 0 g , resp ectively. A subsequence f n j g j  1 in N or N 0 will always mean a strictly increasing sequence n 1 <n 2 <  .If M  C then M 0 M @M will stand for the interior, the closure and the b oundary, resp ectively,of M in C .If G is a domain (i.e. a nonempty, connected op en subset) of C , then H ( G ) represents the set of holomorphic functions on G . Let b e given a function f 2 H ( G ), then wesaythat f is holomorphic exactly on G (or G is the domain of holomorphyof f ,or @G is the natural b oundary of f )if f is analytically noncontinuable across any p ointof @G or, more precisely, for every a 2 G , the radius of convergence of the Taylor series of f with center at a equals the Euclidean distance between a and @G .By H e ( G )we abbreviate the class of all functions which are exactly holomorphic on G . Mittag-Leer discovered in 1884 that H e ( G ) 6 =  for all domains G , see 15, Chapter 10]. It is clear that if f 2 H e ( G )then f has no holomorphic extension to any domain containing G strictly. Let G  C b e a domain. Then A 1 ( G ) denotes the class of holomorphic functions in G with very regular b ehavior at the b oundary,that is, A 1 ( G )=  f 2 H ( G ): f ( ` ) has a continuous extension to G for all ` 2 N 0  : Notice that while H ( G )isaFrechet space (i.e. a completely metrizable lo cally convex space) when endowed with the top ology of uniform convergence on compacta. In the case that G is b oundes then the class A 1 ( G ) also b ecomes a Frechet space under the metric top ology dened by f n ! f in A 1 ( G ) if and only if f ( ` ) n ! f ( ` ) uniformly in G for every ` 2 N 0 . Foradomain G  C (a compact set L  C , resp ectively) we denote by M ( G )( M ( L ), resp ectively) the collection of all compact sets K  G c ( K  L c , resp ectively) with connected complementin C .If K  C is compact, then by A ( K )we mean the family of all functions which are continuous on K and holomorphic in its interior K 0 .The class A ( K ) b ecomes a Banach space under the maximun norm. Lacunary non{continuable b oundary{regular holomorphic functions ... 29 Note that A 1 ( G ) \ H e ( G )maywell b e empty or not. For instance, the function ' with ' ( z ):= P 1 n =0 exp( ; 2 n= 2 ) z 2 n b elongs to A 1 ( D ) \ H e ( D ) (see 30, Chapter 16]), but A ( G ) \ H e ( G )=  if G := D n 0  1). Another interesting example of a function ' 2 A 1 ( D ) \ H e ( D ) is given by ' ( z ):=2 z + 1 X n =0 z 2 n 2 n 2 : It turns out that ' is one-to-one on D and hence mapping D conformally onto a Jordan domain G whose b oundary @G is a C 1 -curvewhichisnowhere analytic. Let us recall that J. Siciak proved in 31] a strong statement ab out noncontinuability in a N -dimensional setting (his pro of leans on typical metho ds of several complex variables) whose one-dimensional instance asserts that if G  C is a b ounded domain such that G = G 0 and G c is connected then H e ( G ) \ A 1 ( G ) 6 =  . Supp ose that S = f s j g j  1 b e a subsequence of N 0 and let  S ( n )bethenumber of m 2 S with m  n . Then the upp er and lower densityof S are dened as d ( S ) := lim sup n !1  S ( n ) n  d ( S ) := lim inf n !1  S ( n ) n : If d ( S )= d ( S )=: d ( S ) then S is said to have the density d ( S ). If now G  C is a domain and z 0 2 G then by H Sz 0 ( G )we mean the class of holomorphic functions in G whose p ower series expansion around z 0 presents gaps outside S or, equivalently, H Sz 0 ( G )=  f 2 H ( G ): f ( n ) ( z 0 )=0 for all n 62 S  : Therefore if f 2 H Sz 0 ( G )wehave in a neighb orho o d of z 0 that f ( z )= 1 X  =0 a  ( z ; z 0 )  with a  =0 for all  62 S: For the sake of simplicity,we set H S 0 ( G )= H S ( G ). Moreover, P S will stand for the family of lacunary p olynomials P ( z )= 1 X  =0  2 S c  z  with gaps outside S . If f 2 H ( G ), z 0 2 G and n 2 N 0 ,thenwe denote by S ( f z 0 n ) the partial sum of order n of the Taylor expansion f ( z )= P 1  =0 f (  ) ( z 0 )  ! ( z ; z 0 )  of f around z 0 , that is, S ( f z 0 n )( z ):= P n  =0 f (  ) ( z 0 )  ! ( z ; z 0 )  . A century ago Porter discovered that certain Taylor series with radius of convergence 1 enjoy the prop erty that some subsequences of their sequences of partial sums (with 30 L. Bernal{Gonzalez, M. C. Calderon{Moreno, W. Luh z 0 =0)converge at some p oints outside the closed unit disk D . This phenomenon is called overconvergence. Starting from 1970, this idea has b een largely develop ed and strengthened along various ways, as for instance : the partial sums havebeen replaced by the action of certain innite matrices {with constant or non-constant entries{  the overconvergence has b een reinforced to the universal prop erty of uniform approximation to any function f 2 A ( K ) for certain compact sets K {with K \ G =  or even K \ G =  ,where G is a domain{  the Taylor series have b een generalized to Laurent series or Fab er series  and some prop erties havebeen shown to b e generic in the space X  H ( G ) where they are studied (that is, the subset of functions of X satisfying each of such prop erties is residual in X ). These improvements are contained in a numb er of pap ers byChui-Parnes, Melas, Nestoridis, Costakis, Katsoprinakis, Papadop erakis, Vlachou, Gehlen, M uller and the authors, among others (see 18, 7, 19, 27, 20, 13, 16, 24, 25, 32, 33, 2, 3, 28, 9, 4] and the references contained in them). Finally holomorphic functions satisfying b oth prop erties of universalityoverconvergence and lacunarityhave b een found by Gharibyan, M uller and the third author in 14]. x 2. PRELIMINARIES AND A NEW CLASS OF OPERATORS This section is devoted to state several auxiliary results to b e used later, and to consider certain classes of op erators which are adequate for the statementofour main result. Let b e given a xed  2 R and consider the logarithmic  -spirals L  := f z = e (1+ i ) t t 2 R gf 0 g : Then a set M  C is called  -starlike with resp ect to z 0 =0 if M  ( L  \ D ):= f z = w  2 M w 2 L  \ D g = M and M is called  -starlike with resp ect to z 0 2 M if M z 0 := f z =  ; z 0  2 M g is  -starlike with resp ect to the origin. If  = 0 then M is starlike in the traditional sense. The content of the following lemma can b e found in 13] and 20]. Lemma 2.1. Let S b e a subsequence of N 0 with d ( S )=1 and supp ose that K is a compact set with connected complementand 0 2 K 0 .Assumethat f is holomorphic on K with f ( z )= 1 X  =0 f  z  where f  =0 for all = 2 S Lacunary non{continuable b oundary{regular holomorphic functions ... 31 near the origin. Supp ose in addition that one of the two conditions is satised : (a) d ( S )=1 , (b) d ( S )=1 and the comp onentof K whichcontains the origin is  -starlike with resp ect to the origin. Then for every "> 0 there exists a lacunary p olynomial P 2P S such that max z 2 K j f ( z ) ; P ( z ) j <": Recall that a p ower series P 1  =0 a  ( z ; z 0 )  is said to have Ostrowski gaps ( p k q k ) ( k 2 N )if p k , q k are p ositiveintegers suchthat p 1 <q 1  p 2 <q 2   lim k !1 q k p k = 1  and lim  !1  2 I j a  j 1 = =0  where I =  k 2 N ( p k q k ) : Lemma 2.2. Assume that G is a domain. Let z 0 2 G and let f 2 H ( G ) such that the Taylor expansion of f around z 0 has Ostrowski gaps ( p k q k )( k 2 N ) .Then sup  2 L sup z 2 K j S ( f z 0 p k )( z ) ; S ( f   p k )( z ) j! 0 ( k !1 )(1) for every pair K , L of compact sets with K  C , L  G . Pro of : In 19, Theorem 1] it is shown that the expression in (1) without \sup  2 L " tends to zero for each compact set K . But its pro of reveals in fact that such convergence to zero holds uniformly with resp ect to  whenever  b elongs to a compact subset of G . Next we are going to consider two kinds of op erators (i.e. continuous linear selfmappings) on the space E := H ( C )ofentire functions. This rst kind is that of op erators T : E!E having dense range. For instance, if T ( E ) f p olynomials g then T has dense range. Trivially, T has dense range if it is surjective. The second kind of op erators is less usual, and it is xed in the following denition. Denition 2.1. Let L  C b e a compact set and T b e an op erator on E .Then wesay that T is compactly L -externally controlled if the following prop erty is satised : Given "> 0 and a compact set K M ( L ), there are > 0and M 2M ( L ) suchthat  h 2E and sup z 2 K j h ( z ) j <  implies sup z 2 K j ( Th )( z ) j <": 32 L. Bernal{Gonzalez, M. C. Calderon{Moreno, W. Luh Examples 2.3. 1. Let ( z )= P 1 n =0 a n z n be an entire function. Then is said to b e of exp onential typ e provided that there are p ositive constants A , B such that j ( z ) j A exp ( B j z j ) for all z 2 C . Consider its asso ciated formal linear (in general, innite order) dierential op erator ( D )= P 1 n =0 a n D n dened as ( D ) f = P 1 n =0 a n f ( n ) ( f 2E ). Then ( D )isinfactawell-dened op erator on E . This is easy to see just by taking into account the Cauchy estimates as well as the fact that is of exp onential typ e if and only if the sequence f ( n ! j a n j ) 1 =n g n  1 is b ounded. By the Malgrange-Ehrenpreis theorem (see 11] or 23]) wehave that ( D ) is surjective (so it has dense range) as so on as 6 0. Assume now that is of sub exp onential typ e, that is, for given "> 0 there is a p ositive constant A suchthat j ( z ) j A exp( " j z j ) for all z 2 C  equivalently, lim n !1 ( n ! j a n j ) 1 =n = 0 (see for instance 6]  see also 1] for a go o d exp osition ab out the corresp onding op erators ( D )). Then ( D ):= T is compactly L -externally controlled for every compact set L  C . Indeed, if "> 0 and K 2M ( L ) are xed, we can cho ose a Jordan domain J such that K  J 0 , L \ J =  and  := @J is rectiable. Recall that ( n ! j a n j ) 1 =n ! 0( n !1 ). Therefore given e " := dist ( K ) 2 there is a constant A 2 (0  + 1 )such that n ! j a n j A e " n ( n 2 N 0 ). Let us dene M := J and  := "  dist( K  ) A  length(  ) : Then M 2M ( L ) and > 0. Now, if wemake  oriented counterclo ckwise, weget from the Cauchyintegral formula for derivatives that for every z 2 K and every h 2E one has j ( Th )( z ) j =      1 X n =0 a n h ( n ) ( z )      = =      1 X n =0 a n  n ! 2 i I  h ( t ) ( t ; z ) n +1 dt       1 X n =0 A e " n 2   sup t 2  j h ( t ) j length(  ) (dist( K  ) n +1   A  length(  ) sup z 2  j h ( t ) j 2  dist( K  )  1 X n =0  1 2  n = A  length(  ) dist( K  )  sup w 2 M j h ( w ) j : Hence sup z 2 K j ( Th )( z ) j <" whenever sup z 2 M j h ( z ) j < , as required. 2. The second part of the ab ove example covers the cases T = D n ( n 2 N 0 ), where D 0 := I = the identity op erator. Indeed, just take ( z ):= z n .However, if is of exp onential typ e then ( D ) is not always controlled in the sense of Denition 2.1. For instance, if we take ( z ):= e z then ( D ) is the translation op erator that takes a function h 2E to the function z 7! h ( z + 1), which is not controlled for some compact set L  C . In fact, more is true : If ' 2E is not the identitythen the Lacunary non{continuable b oundary{regular holomorphic functions ... 33 comp osition op erator C ' : E!E dened as C ' h = h  ' is not compactly L -externally controlled for some compact set L . Indeed, x  2 C suchthat  := ' (  ) 6 =  and cho ose L := f  g , " 0 := 1, K = f  g . Observethat K 2M ( L ). Nowx > 0 and M 2M ( L ). By the Runge approximation theorem (see 12]) {applied to the compact set M f  g {we can nd a p olynomial h suchthat j h ( z ) ; 0 j < ( z 2 M ) and j h (  ) ; 2 j < 1. Hence sup z 2 M j h ( z ) j < but sup z 2 K j ( C ' h )( z ) j = j h (  ) j >" 0 ,as required. It is clear that C ' is L -controlled for all compact sets if ' is the identity.If ' is not the identity but it is a nonexpansive similarity {that is, ' ( z )  a ( z ; b )+ b , where j a j 1 and b is the (unique) nite xed p ointof ' {then C ' is compactly L -externally controlled, where L is any closed ball with center at b . As for the density of the range, we claim that if ' 2E then C ' has dense range if and only if ' is a similarity ' ( z )  az + b ( a b 2 C with a 6 = 0). Indeed, the part \if" is evident b ecause C ' would b e surjective. Finally, supp ose that C ' has dense range and that, by the wayofcontradiction, ' is not one-to-one. Then there are p oints a b 2 C with a 6 = b such that ' ( a )= ' ( b ). By density, there is sequence f f n g n  1 E for which f n  ' ! g ( n !1 )in E , where g ( z )  z . In particular, lim n !1 f n ( ' ( a )) = a and lim n !1 f n ( ' ( b )) = b ,which is absurd b ecause ' ( a )= ' ( b ). Therefore ' is an injectiveentire function, so it is a similarity, whichproves the claim. 3. Let  2E and consider the multiplication op erator M  : f 2E 7! f 2E . It is easy to see that M  is always compactly L -externally controlled for all compact sets L  C and that, in addition, M  has dense range if and only if  has no zeros. 4. Given a compact set L  C , the family A of compactly L -externally controlled op erators is a vector algebra in the space of all op erators on E , that is, if   are complex numb ers and T 1 T 2 are in A , then the op erators T 1 + T 2 and T 1  T 2 are in A to o. Indeed, this is evidentfor T 1 + T 2 . As for the comp osition T 1  T 2 ,x a number "> 0 together with a compact set K 2M ( L ). Then there are  1 > 0 and M 1 2M ( L ) such that k T 1 f k K <" whenever f 2E and k f k M 1 < 1 . By using now that T 2 is controlled, there are > 0 and M 2M ( L )suchthat h 2E and k h k S < ] implies k T 2 h k M 1 < 1 .Then if k h k M < we obtain k T 1 T 2 h k K <" ,and we are done. x 3. CONSTRUCTION OF A UNIVERSAL FUNCTION We are now ready to construct the promised universal function with resp ect to overconvergence having moreover additional prop erties of lacunarity, b oundaryregular b ehavior and non-continuability. Theorem 3.1. Supp ose that G is a Jordan domain, that z 0 2 G and that S is a subsequence of N 0 satisfying at least one of the following conditions : 34 L. Bernal{Gonzalez, M. C. Calderon{Moreno, W. Luh (a) d ( S )=1 , (b) d ( S )=1 and G is  -starlike with resp ect to z 0 2 G . Then there exist a function ' 2 A 1 ( G ) \ H e ( G ) \ H Sz 0 ( G ) and a subsequence f p k g k  1  N 0 for which the following prop erties hold : (A) For each compact set L  G wehave S ( '   p k ) ! ' ( k !1 ) in A 1 ( G ) uniformly for all  2 L . (B) For each compact set K 2M ( G ) , each compactly G -externally controlled op erator T on E with dense range, and each f 2 A ( K ) , there exists a subsequence f k j g j 2 N  N 0 such that lim j !1 sup  2 L sup z 2 K j ( TS ( '   p k j ))( z ) ; f ( z ) j =0 for every compact set L  G . Pro of : 1. Without loss of generality,we can assume that z 0 =0.Let f K ?  g   1 be an exhausting sequence for M ( G ), that is, K ?  2M ( G ) for each  and, given K 2M ( G ), there is  2 N dep ending on K suchthat K  K  (see for instance Lemma 2.9]5]). Let f " ?  g   1 be an enumeration of all p olynomials with co e#cients in Q + i Q . Supp ose that f ( K n  " n ) g n  1 is an arrangementofall K ?  and " ?  in whichany combination ( K ?   " ?  ) o ccurs innitely many often. Wecho ose a sequence of Jordan domains G n with rectiable b oundary satisfying G  G n +1  G n +1  G n ( n 2 N )  G n \ K n =  ( n 2 N ) and 1 \ n =1 G n = G: In the case that G is  -starlike with resp ect to z 0 = 0 then we assume in addition that all G n are  -starlike also (see for instance Duren 10], Theorem 2.19). 2. We construct sequences f p n g n  1 , f q n g n  1  N 0 and a sequence f P n g n  1 of p olynomialsby induction. First, wedene  n := dist ( G@G n )   n := length ( @G n )  " n :=  n n n ! n 2  n ( n 2 N ) : Without loss of generalitywemay assume  n < 1( n 2 N ). By Lemma 2.1 there exists a p olynomial P 1 ( z )= p 1 X  =0 a  z  with a  =0 for  62 S Lacunary non{continuable b oundary{regular holomorphic functions ... 35 which satises max z 2 G 1 j P 1 ( z ) j <" 1 and max z 2 K 1 j P 1 ( z ) ; " 1 ( z ) j < 1 : We assume that P 1 :::P n have already b een determined and that P n has the form P n ( z )= p n X  = q n ; 1 a  z  with a  =0 for  62 S: Wehave set q 0 := 0. Cho ose q n 2 N with q n >np n . Observing that S n := f t 2 S : t  q n g also satises d ( S n ) = 1 if (a) holds and d ( S n ) = 1 if (b) holds wecan ndby Lemma 2.1 again a p olynomial P n +1 ( z )= p n +1 X  = q n a  z  with a  =0 for  62 S (1) which satises max z 2 G n +1 j P n +1 ( z ) j <" n +1 (2) and max z 2 K n +1      P n +1 ( z ) ; ( " n +1 ( z ) ; n X  =1 P  ( z ) )      < 1 n +1 : (3) By induction weget f p n g n  1 , f q n g n  1 and f P n g n  1 . 3. For xed l 2 N 0 and n>l we obtain from the Cauchyintegral formula for derivatives (we can assume that @G n is oriented counterclo ckwise) that max z 2 G j P ( l ) n ( z ) j = max z 2 G     l ! 2 i I @G n P n (  ) (  ; z ) l +1 d      l ! 2    n  " n (  n ) l +1 <n !  n  " n  n n = 1 n 2 : Therefore the series P 1 n =1 P ( l ) n ( z ) converges for each l 2 N 0 uniformly on G , and it follows that the function ' , which is dened by ' ( z ):= 1 X n =1 P n ( z )  is holomorphic on G and that each derivative ' ( l ) has a continuous extension to G . In other words, ' 2 A 1 ( G ). 4. We consider the p ower series of ' around the origin. By the sp ecial form (1) of the p olynomials P n and by the prop erty q n >np n ( n 2 N ), the p owers in P n and P m do not overlap if n 6 = m and therefore the p ower series of ' is given by ' ( z )= 1 X  =0 a  z  with a  =0 for  62 S: (4)