Universal functions with prescribed zeros and interpolation properties
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Michigan Math. J. 58 (2009) Universal Functions with Prescribed Zeros and Interpolation Properties Luis Bernal-González, Antonio Bonilla, & Markus Nieß Dedicated to Professor José Méndez on the occasion of his sixtieth birthday 1. Introduction Roughly speaking, universality means “existence of a dense orbit”. Thus, in some sense, universal functions are “uncontrolled”. In this paper, we study the existence of functions that are universal with respect to differential operators and that are, at the same time, “controlled” by prescribed interpolation properties, including prescribed zeros and multiplicities. Precise definitions are given in what follows. We denote by N,Z,C, and N0the set of positive integers, the set of all integers, the complex plane, and the set N∪{0}, respectively. If A⊂Cthen A◦,¯ A, and ∂A will stand (respectively) for the interior, the closure, and the boundary of Ain C.We use C∞to denote the extended complex plane. Recall that a domain is a nonempty connected open subset of C. Let H() be the linear space of holomorphic functions on a domain . In particular, H(C)is the space of entire functions. Consider the metric d(f,h) := ∞ j=1 1 2j·f−hCj 1+f−hCj (f,h∈H()), where f−hM:=sup z∈M |f(z)−h(z)|. Here {Cj:j≥1}is a fixed exhaustive sequence of compact subsets of ;that is, Cj⊂C◦ j+1(j ≥1)and =∞ j=1Cj.It is possible to select {Cj:j≥1}so that each connected component of C∞\Cjcontains some connected component of C∞\;in particular, if is simply connected (i.e., if C∞\is connected) then we can choose every Cjwithout “holes”. The aforementioned metric dgenerates on H() the topology of uniform convergence on compact subsets of ;see [5]. In the sequel, we will always consider the complete metric space (H(),d). Received April 7, 2008. Revision received February 24, 2009. The first author has been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by MEC Grant MTM2006-13997-C02-01. The second author has been partially supported by MEC and FEDER MTM2008-05891. Both authors have been partially supported by MEC Acción Especial MTM2006-26627-E. 627
628 L. Bernal-González, A. Bonilla, & M. Nieß According to Baire’s category theorem, every complete metric space Xis a Baire space; in other words, the intersection of countably many open dense subsets of Xis also dense in X. In a Baire space X, a subset is residual when it contains a dense Gδ-subset of X. In particular, this applies to (H(),d). Throughout this paper, we will use the following notion of universality. Definition 1. Let (X,dX)and (Y,dY)be metric spaces and let L=(Lj)j∈Jbe a family of continuous mappings Lj:X→Y. (i) An element x∈Xis called L-universal if Y={Ljx:j∈J}. The set of all such elements x∈Xis denoted by U(L). The family Lis called universal if U(L)= ∅. (ii) Lis called topologically transitive if it has the following property: For every x∈X,y∈Y, and ε>0, there is a z∈Xand a j∈Jsuch that dX(x,z)<ε and dY(y,Ljz)<ε. In fact, the last definition can be easily extended to the setting of topological spaces, but such a generality will not be needed here. If X=Yand L:X→X is a continuous self-map, then Lis said to be universal (topologically transitive, resp.) if the family L={Ln:n≥1}of its iterates is universal (topologically transitive, resp.). If X,Yare topological vector spaces and the Lj(or L,ifweare dealing with self-maps) are linear, then it is customary to say hypercyclic instead of universal. Readers interested in these concepts are referred to the surveys [10] and [13]. In 1952, MacLane [14] stated that there exist entire functions ϕsuch that the set of derivatives {ϕ(n) :n∈N}is dense in (H(C),d) or, equivalently, ϕ∈U(D) for D={Dn:n∈N}, where Dis the differentiation operator on H(C)given by Df =f.In 1994, Herzog [12] posed the following question: Which additional properties of elements of Xare compatible with universality? For U(L)residual and A⊂XaGδ-subset, he proved that under certain conditions on Aand L(see Section 2) the set A∩U(L)is residual in A. By using his theorem, Herzog derived the existence of D-universal functions having a zero-free qth and (q +1)th derivative (q ∈N0). This result was extended by the first author (see [1] and [2, Thm. 12]) for infinite-order differential operators (D) =∞ n=0anDn, where Dis again the differentiation operator (with D0=I, the identity operator) and (z) =∞ n=0anznis an entire function of subexponential type; that is, given ε>0, there is a positive constant A=A(ε) with |(z)|≤Aeε|z|for all z∈C.Recall that an entire function is said to be of exponential type if there are positive constants Aand Bwith |(z)|≤AeB|z|for all z∈C.Of course, every entire function of subexponential type is of exponential type. By an operator we mean a continuous linear self-map on a topological vector space. It is not difficult to see that if is of subexponential type then (D) defines an operator on H() (and on H(C), assuming only that is of exponential type).
Universal Functions with Prescribed Zeros and Interpolation Properties 629 In [15,Thm. 3.3], the third author showed the existence of D-universal functions that solve a given interpolation problem in C.Independently and with a different approach, Costakis and Vlachou [7] arrived at the same conclusion for any simply connected domain. Moreover, in [15, Thm. 2.3] it is proved that there are MacLane-universal entire functions having zeros at prescribed points with prescribed orders. On the other hand, a celebrated result due to Godefroy and Shapiro (see Section 2) asserts the universality on H(C)of every differential operator (D) as described here that is not a multiple of the identity. Recently, the first author [3] demonstrated the existence of (D)-universal holomorphic functions with given interpolation properties. Our aim in this paper is to prove the existence of holomorphic functions fon a simply connected domain that simultaneously satisfy the following conditions: (a) fis (D)-universal; (b) fhas zeros (only) at the points of a given subset of , with preassigned orders; and (c) fassumes prescribed values at prescribed points. This will be accomplished in Section 3. The combination of universal Taylor series with the property (b) or (c) has been considered by Costakis [6]. His improvement on Herzog’s theorem is also one of our auxiliary results (see Theorem 3). 2. Preliminary Results This section is devoted to establishing a number of statements that will be needed in the proof of our main result. We begin by presenting the following version, due to Grosse-Erdmann [11], of the well-known Birkhoff transitivity theorem. Theorem 2. Let Abe a nonempty Gδ-subset of a complete metric space, let Y be a separable metric space, and let L=(Lj)j∈Jbe a family of continuous mappings Lj:A→Y. Then the following assertions are equivalent: (i) there is a dense set of elements of Athat are L-universal; (ii) Lis topologically transitive. If either condition holds then the set U(L)is a dense Gδ-set, and so is residual, in A. Recall that a Polish space is a separable complete metric space. Next, we state the Herzog criterion [12] concerning inherited universality—more precisely, the (slightly improved) version due to Costakis [6]. Theorem 3. Assume that Xis a Polish space and that Yis a separable metric space. Also let dX,dYbe the corresponding metrics. Let Ln:X→Ybe a sequence of continuous functions with U({Ln})residual in X. For any B⊂X,let Ln|Bbe the restriction of Lnto B. Consider a sequence {Bk}k∈Nof Baire spaces that are subsets of Xsatisfying
630 L. Bernal-González, A. Bonilla, & M. Nieß A:= k∈N Bk= ∅, (1) Bk∩U({Ln})is residual in Bk.(2) If, in addition, lim k→∞ sup n∈N inf h∈A(dX(bk,h) +dY(Lnbk,Lnh)) =0 (3) holds for every sequence {bk}k∈Nwith bk∈Bk,then the set U({Ln|A})is residual in A. We also present the following result about the “internal control” property of differential operators (see [4]). We omit its easy proof, which is based on the Cauchy integral formula for derivatives. Theorem 4. Let ⊂Cbe a domain and let be an entire function of subexponential type. Assume that K,Lare compact sets in Cwith L⊂K◦.Then there exists a constant C=C(K,L) ∈(0, +∞)such that (D)f L≤CfKfor all f∈H(). The previously mentioned theorem of Godefroy and Shapiro states that if is an entire function of exponential type then the differential operator (D) is universal on H(C)[8, Sec. 5]. By restricting the class of operators, we may extend the result to all domains without holes. Specifically, we have the following assertion, which can be found in [2, Thm. 8]. Theorem 5. Let be a nonconstant entire function of subexponential type and consider the operator S=(D):H() →H(),where is a simply connected domain of C.Then Sis universal. In fact, U(S)is residual in (H(),d) for S={Sn:n∈N}. The final lemma in this section combines interpolation and approximation. It is a kind of Hermite interpolation using exponential functions instead of polynomials. The result improves [3, Lemma 2.1]. By ea(a ∈C)we denote the function ea(z) :=exp(az), and span X0will stand for the linear span of a subset X0of a vector space. Lemma 6. Assume that L,Kare compact subsets of a domain ⊂Cwith L⊂K◦,that a1,...,anare different points in L,that mis a natural number, and that Gis a nonempty open subset of C.Then there exist a positive constant M= M(L,K,a1,...,an,G,m) and a finite set of functions {αj,k:j=1, ...,n;k=0, ...,m−1}⊂span{ea:a∈G}, depending only on G,m,and the points a1,...,an,that satisfy the following property. For each pair of functions f,h∈H(),the function ϕdefined by ϕ(z) =h(z) + n j=1 m−1 k=0 (f (k)(aj)−h(k)(aj))αj,k(z)
Universal Functions with Prescribed Zeros and Interpolation Properties 631 satisfies: (a) ϕ∈H(); (b) ϕ(σ)(aj)=f(σ)(aj)(j =1, ...,n;σ=0, ...,m−1); (c) ϕ−fL≤Mh−fK. Proof. We can assume that G= C, so we choose a point c∈Gand select a positive number dsatisfying d< 1 m(n +1)inf{|z−c|:z∈C\G}(4) and d< min j,l∈{1,...,n} j=l 1 |aj−al|.(5) We define ,j(z) := n l=1 l=j (ed(z −al)−1)m(j =1, ...,n), βj,k(z) :=ec(z −aj)(ed(z −aj)−1)k k!dk ,j(z) ,j(aj) (j =1, ...,n;k=0, ...,m−1). From (5), it follows that 0<d|aj−al|<1<2π for all j,l∈{1, ...,n}with j= l,so,j(aj)= 0 for all j∈{1, ...,n}.Also, an easy calculation shows that β(σ) j,k(at)=1ift=jand σ=k, 0ift= jor σ<k, where σ∈{0,1, ...,m−1}is always assumed. We have no information about the values of β(σ) j,k(at)for t=jand σ>k.Hence, for each j∈{1, ...,n}we set αj,m−1(z) :=βj,m−1(z) and αj,k(z) :=βj,k(z) − m−1 ν=k+1 β(ν) j,k(aj)αj,ν(z) (k =0,1, ...,m−2), where the last expression makes sense only if m≥2.By induction we obtain α(σ) j,k(at)=1ift=jand σ=k, 0ift= jor σ= k. Observe that each function βj,k, and so each function αj,k, is a finite linear combination of functions of the form ec+sd with 0 ≤s < m(n +1). But each point c+sd is in Gbecause of (4). Hence, the functions αj,kare in span{ea:a∈G}. Let L,Kbe compact subsets as in the statement. By using Theorem 4 (with (z) =zk)or simply the Cauchy estimates, we obtain
632 L. Bernal-González, A. Bonilla, & M. Nieß g(k)L≤MkgK(k ∈N0,g∈H()) for some positive constant Mkthat is independent of g. Now we set M:=1+ n j=1 m−1 k=0 Mkαj,kL. Finally, if we fix holomorphic functions f,hon and define the corresponding function ϕas in the statement, then properties (a), (b), and (c) are obvious. 3. Main Result Suppose that is a domain in C, and let w={wk}k∈N,γ={γk}k∈N,m= {mk}k∈N, and β={βk}k∈Nbe sequences satisfying: w⊂,γ⊂,β⊂C\{0}, m⊂N,w∩γ=∅;the points wk,k∈N(as well as the points γk,k∈N)are pairwise distinct; and neither wnor γhave accumulation points in . To each such set of sequences we can associate the set A=A(w,m;γ,β) defined by A:={f∈H() :f(wk)=0 of order mk,f(γk)=βk(k ∈N), and f(z) = 0ifz∈\w}.(6) In other words, Ais the set of holomorphic functions in with prescribed zeros and interpolation conditions (corresponding to w,m,γ,β). First, we note in the following proposition that Apossesses good topological properties. Proposition 7. The set Adefined in (6) is a nonempty Gδ-subset of H();in addition, it is a Baire space when endowed with the compact-open topology inherited from H(). Proof. By the Weierstraß factorization theorem (see e.g. [18,Thm.15.9]), we know that there exists a function h∈H() such that h(wk)=0 of order mk(k ∈N)and h(z) = 0ifz= wk(k ∈N). Each γkis different from all the wk,soβk/h(γk)is well-defined. For each k∈N, let αkbe a fixed logarithm of βk/h(γk). By [18,Thm. 15.13] there exists a function g∈H() with g(γk)=αk.Then the function f(z) :=eg(z) ·h(z) is an element of A, hence A= ∅. Now consider the exhaustive sequence {Cn:n≥1}given in Section 1. Setting Mn:={z∈Cn:|z−wk|≥1/n for all k∈N}, we obtain that Ais the intersection of the open sets An:={f∈H() :|f(ν)(wk)|<1 nfor 0 ≤ν<m k,f(mk)(wk)= 0, |f(γk)−βk|<1 nfor 1 ≤k≤n, minz∈Mn|f(z)|>0},
Universal Functions with Prescribed Zeros and Interpolation Properties 633 where n≥1.So, Ais a Gδ-subset of (H(),d). Finally, since Ais a Gδ-subset in a complete metric space, Alexandroff’s theorem (see [17]) guarantees that the topological space Ais completely metrizable. Hence, it is a Baire space. Now, we suppose that is a simply connected domain. Before establishing the promised result on interpolation in its full strength, we present the following “discrete” version of it, which will be used in the proof of Theorem 9. Lemma 8. Let Lbe a compact subset of ,let w1,...,wn1and γ1,...,γn2be pairwise distinct points in L◦,and let m1,...,mn1∈Nand β1,...,βn2∈C\{0}, where n1,n2∈N.We define B:={f∈H() :f(wk)=0of order mkif k=1, ...,n1, f(γk)=βkif k=1, ...,n2, f(z) = 0if z∈L\{wk:k=1, ...,n1}}. Endow Bwith the compact-open topology inherited from H(). Let be a nonconstant entire function of subexponential type, and let S=(D) and S= {Sn:n∈N}.Then U(S)∩Bis a dense Gδ-subset of B. Proof. In a similar manner as for the set Ain Proposition 7, we deduce that Bis also a Gδ-subset of H(). According to Theorem 2, it suffices to show that Sis topologically transitive. We therefore fix f∈B,g∈H(), a compact set K⊂, and a number ε>0.We have to show the existence of some ϕ∈Band some N∈Nwith sup z∈K |ϕ(z) −f(z)|<ε (7) and sup z∈K |SNϕ(z) −g(z)|<ε. (8) Since f≡ 0 and is simply connected, we can find a compact set L1⊂ satisfying the following properties: •C\L1is connected; •K∪L⊂L◦ 1; •∂L1is a regular Jordan curve; and •fis zero-free on ∂L1. Thus d:=min z∈∂L1 |f(z)|>0. If fhas further zeros on L◦ 1(apart from w1,...,wn1), we denote them by ζ1,...,ζn3. Denote by r1,...,rn3their respective orders. By hypothesis, the entire function is nonconstant, so the open set G:={z∈C:|(z)|<1} is nonempty. Choose a compact subset L2⊂with L◦ 2⊃L1, and let
634 L. Bernal-González, A. Bonilla, & M. Nieß M=M(L1,L2,w1,...,wn1,γ1,...,γn2,ζ1,...,ζn3,G, max{m1,...,mn1,r1,...,rn3}) be the positive constant provided by Lemma 6. According to Theorem 5, there exists an S-universal function h∈H() with h−fL2<1 Mmin{d,ε}. By Lemma 6 one can find a function ϕ=h+α∈H(), with α∈span{ea: a∈G}, that satisfies ϕ(σ)(wk)=f(σ)(wk)=0(σ =0, ...,mk−1;k=1, ...,n1), ϕ(σ)(ζk)=f(σ)(ζk)=0(σ =0, ...,rk−1;k=1, ...,n3), ϕ(γk)=f(γk)=βk(k =1, ...,n2), and ϕ−fL1≤Mh−fL2<min{d,ε}.(9) Condition (7) follows directly from (9) and from the fact K⊂L1.By (9) we have |ϕ(z) −f(z)|<d≤|f(z)|for all z∈∂L1, so we conclude from Rouché’s theorem [18, Chap. 10] that the functions fand ϕ have the same number of zeros, counting multiplicities, in L◦ 1.Owing to the interpolation properties of ϕ(observe that the order of the zero wk,ζkfor ϕis at least mk,rk, respectively), it follows that ϕ∈B. It remains to show the existence of an N∈Nsuch that (8) is fulfilled. By definition, |(a)|<1 for every a∈G. Since (D)ec=(c)ecfor all c∈C,wehave (D)nec=(c)necfor every n∈N.Hence (D)nea→0(n →∞)compactly on Cwhenever a∈Gand, as a result, (D)nα→0(n →∞)compactly on Cfor every α∈span{ea:a∈G}. In particular, there is an N1∈Nsuch that SnαK<ε/2(n ≥N1). Because his S-universal, there is an N≥N1with SNh−gK<ε/2.Finally, since ϕ=h+αwith α∈span{ea:a∈G}, the linearity of Stogether with the triangle inequality drives us to (8), as required. We are now ready to state our main result. Theorem 9. Assume that ⊂Cis a simply connected domain and that is a nonconstant entire function of subexponential type. Let S=(D) and S= {Sn:n∈N}.Suppose that w={wk},γ={γk},m={mk},and β={βk}are sequences as in the beginning of this section, and suppose the set A=A(w,m;γ,β) is defined as in (6). Then the set A∩U(S)is residual in A. In particular, in A there is a dense Gδ-subset all of whose functions are (D)-universal.
Universal Functions with Prescribed Zeros and Interpolation Properties 635 Proof. The essential tool for the proof of this theorem will be Theorem 3. Therefore, let X:=(H(),d) =:Y. Recall that (H(),d) is a Polish space, so it is a separable metric space as well. Let Ln=Sn(n ∈N). From Theorem 5 we know that U(S)is residual in (H(),d). Without loss of generality, we can assume that the exhaustive sequence {Ck: k∈N}of compact sets defining the metric dof H() (see the Introduction) satisfies that there are two strictly increasing sequences {j1(k)}∞ 1,{j2(k)}∞ 1⊂Nsuch that Ck∩w={w1,...,wj1(k)}=C◦ k∩wand Ck∩γ={γ1,...,γj2(k)}=C◦ k∩γ. We define Bk:={f∈H() :f(w j)=0 of order mjif j=1, ...,j1(k), f(γ j)=βjif j=1, ...,j2(k), f(z) = 0ifz∈Ck\{wj:j=1, ...,j1(k)}}. Obviously, the intersection of the sets Bkis exactly the set A, which is nonempty by Proposition 7. An argument similar to the one given in the proof of that proposition shows that each Bkis a Baire space. Now, Lemma 8 shows the correctness of condition (2) in Theorem 3. In order to apply Theorem 3, it remains to prove that for every sequence {bk}k∈N with bk∈Bkwe have lim k→∞ sup n∈N inf ϕ∈A(d(bk,ϕ) +d(Snbk,Snϕ)) =0.(10) Let k∈Nand f∈Bkbe fixed. Hence, the zeros of fin Ckare exactly given by the points wj(of orders mj)with j∈{1, ...,j1(k)}.Choose a Jordan subdomain Uk⊂such that Uk⊃Ckand w∩(Uk\Ck)=∅=γ∩(Uk\Ck), which is possible because neither wnor γhave accumulation points in . Let g∈A. Since f/g is holomorphic and zero-free in Uk, there exists a function φholomorphic in Uksuch that f(z) g(z) =eφ(z) (z∈Uk). (11) For z=γjwith j≤j2(k), the left-hand side of (11) is 1 and so φ(γ j)=µj·2πi for some µj∈Zif j≤j2(k). Next, let h∈H() with zeros exactly at the points γjwith j>j 2(k) and with h(γj)=1ifj≤j2(k). That such a function exists can be proved by an argument similar to the one used in the proof of Proposition 7. Furthermore, we set c1:=h¯ Vk,c2:=g¯ Vk,c3:=exp(φ¯ Vk+1), where Vkis again a fixed Jordan domain, this time satisfying Ck⊂Vk⊂¯ Vk⊂Uk. Suppose ε>0.By Walsh’s theorem on simultaneous approximation and interpolation [19], there exists a polynomial psatisfying φ h−p ¯ Vk <minε c1·c2·c3 ,1, 1 c1