On uni e sal en i e unc ions wi h ze o- ee de i a i es
By
LUIS BERNAL–GONZ´
ALEZ*
Abs ac . We p o e in his no e a gene aliza ion o a heo em
due o G. He zog on ze o- ee uni e sal en i e unc ions. Speci ically,
i is shown ha , i a nonnega i e in ege qand a noncons an en i e
unc ion Φ o subexponen ial ype a e gi en, hen he e is a esidual
se in he class o en i e unc ions wi h ze o- ee de i a i es o o de s q
and q+1, such ha e e y membe o ha se is uni e sal wi h espec
o Φ(D), whe e Dis he diffe en ia ion ope a o .
1. In oduc ion and no a ion. We deno e by C he complex plane, by N
he se o posi i e in ege s and by N0 he se N∪ {0}. I > 0, B( ) (B( )) is he
euclidean open (closed, espec i ely) disk wi h cen e 0 and adius . We ag ee ha
B(+∞) = C.H(B( )) will s and, as usual, o he space o holomo phic unc ions
in B( ), endowed wi h he opology o uni o m con e gence on compac subse s. In
H(C), his opology is induced by he me ic
(1) d( , g) =
∞
∑
j=1
1
2j
|| −g||j
1 + || −g||j
,
whe e ||h|| = maxB( )|h|(∀ > 0). I is well known ha H(B( )) is a sepa able
F ´eche space, so i is a Polish space and also a Bai e space (see, e.g., [14, pp. 213-
214 and 238]). In a Bai e space X, a subse is esidual when i con ains a dense
Gδ-subse o Xo , equi alen ly, when i s complemen is o i s ca ego y. Such a
subse is “ e y la ge” in X.
*This wo k is suppo ed in pa by DGICYT g an PB93-0926.
1991 Ma hema ics Subjec Classi ica ion: P ima y 30E10. Seconda y 47B99,
47E05.
Key wo ds and ph ases: uni e sal en i e unc ion, ze o- ee de i a i e, subex-
ponen ial ype, MacLane’s heo em, esidual se .
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We use a e y gene al no ion o uni e sali y, which can be ound in [10], namely:
Le Xand Ybe nonemp y opological spaces and Lbe a amily o con inuous
mappings om Xin o Y. Then an elemen x∈Xis called uni e sal wi h espec
o Li he se {Lx :L∈ L} is dense in Y. As in [13], we deno e he se o
all uni e sal elemen s by U(L). Uni e sal elemen s a e usually called hype cyclic
in he case ha X=Yis a opological ec o space and Lis he sequence o
i e a es {Ln}∞
1o a single linea con inuous ope a o Lon X(see, o ins ance,
[9]). S. Rolewicz [17] was he i s o gi e an example o a hype cyclic elemen in
he Banach/Hilbe se ing.
In 1952, G. R. MacLane [15] s a ed ha he e exis en i e unc ions such ha
he se o de i a i es { (n):n∈N}is dense in H(C) o , equi alen ly, ∈U(L)
o L={Dn:n∈N}, whe e Dis he diffe en ia ion ope a o on H(C), ha
is, D = ′. The esul is also p o ed in [3] (see also [4]). S. M. Duyos Ruiz
[7] has shown ha , in ac , he e is a esidual se o such unc ions. Fu he mo e,
R. M. Ge hne and J. H. Shapi o [8] and K. G. G oße-E dmann [10, Sa z 2.2.8] ha e
de i ed he same esul o e e y simply connec ed domain. Fo addi ional esul s
abou he opic, he eade is e e ed o [2], [11] and [12] and many o he s in hei
e e ences. Fo ins ance, G oße-E dmann [11] p o ides a sha p esul on g ow h o
D-uni e sal unc ions.
Re u ning now o he gene al case o a amily Lo con inuous mappings om X
in o Y, G. He zog [13] p oposed ecen ly he ollowing in e es ing ques ion: Which
addi ional p ope ies o elemen s o Xa e compa ible wi h uni e sali y? I U(L)
is esidual and A⊂Xis a Gδ-subse , he p o es ha unde ce ain condi ions on
Aand L(see Theo em 1 below) he se A∩U(L) is esidual in A. Then, by using
his heo em, he de i es he exis ence o ze o- ee uni e sal en i e unc ions ( o D)
ha ing e en a ze o- ee i s de i a i e. The basic ools employed by He zog a e
he heo y o uni e sali y de eloped by G oße-E dmann [10, specially Sa z 1.2.2],
Alexand off’s heo em on comple eness o Gδ-subse s (see, e.g., [16, pp. 47-48]) and
2
elemen a y esul s om Complex Analysis. He zog [13, Sec ion 3] himsel poin s
ou ha he e is no uni e sal en i e unc ion such ha · ′· ′′ is ze o- ee, since
{ ∈H(C) : · ′· ′′ is ze o- ee}={eαz+β:α, β ∈C, α = 0}(see [6, p. 433]
and [19]).
I q∈N0, le us deno e A(q) = { ∈H(C) : (q)(z) (q+1)(z)= 0 ∀z∈C}.
Since exp ∈∩q∈N0A(q), e e y A(q) is nonemp y. Ou aim in his no e is o u nish
a s ong gene aliza ion o He zog’s heo em on ze o- ee de i a i es. Speci ically, we
show in Theo em 5 ha , i a nonnega i e in ege qand a noncons an en i e unc ion
Φ o subexponen ial ype a e gi en, hen he e is a esidual subse in A(q) sa is ying
ha e e y membe o such a subse is uni e sal wi h espec o he ope a o Φ(D),
whe e Dis he diffe en ia ion ope a o . In e ms o g ow h o den and ype, ou
esul is bes possible.
2. P elimina y esul s. The echnique o p o ing Theo em 5 will be e y
simila o ha in [13], bu we need an addi ional elemen a y esul on an ide i a-
i es oge he wi h he “good beha iou ” o ce ain ela ed non-linea ope a o s el-
a i ely o con e gence ( his is Theo em 2; i s p oo is easy and le o he eade ), a
s ong asse ion due o Gode oy and Shapi o [9] (Theo em 3) and, inally, a esul
which asse s he con inui y o subexponen ial diffe en ial ope a o s on e e y space
H(B( )) ( > 0) (Theo em 4). Mo eo e , i is also employed he abo e men ioned
Theo em 1, which is exac ly Theo em 2.1 o [13].
Theo em 1. Assume ha Xis a Polish space and Yis a sepa able me izable
space. Le dX, dYbe me ics inducing he opologies o X, Y , espec i ely. Le
{Ak:k∈N}be a sequence o open subse s o Xwi h A≡∩∞
k=1 Ak=∅. Le
L={Ln:n∈N}be a sequence o con inuous mappings om Xin o Y, wi h
U(L) esidual in X. Deno e L|A={Ln|A:n∈N}, whe e Ln|Ais he es ic ion
o Ln o A. I
lim
k→∞ sup
n∈N
in
z∈A(dX(ak, z) + dY(Lnak, Lnz)) = 0
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o e e y sequence {ak}∞
1(ak∈Ak, k ∈N), hen U(L|A)is esidual in A.
Theo em 2. a) I q∈N0and ∈H(B( )) hen (q) (q+1) is ze o- ee i and
only i he e exis s g∈H(B( )) such ha
(z) = { (0) exp(∫z
0exp(g( )) d )i q= 0
∑q−1
ν=0
(ν)(0)
ν!zν+ (q)(0)
(q−1)! ∫z
0(z− )q−1exp(∫
0exp(g(u)) du)d i q≥1
o all z∈B( ).
b) Assume ha q∈N0and T: ∈H(B( )) 7→ T ∈H(B( )) is he mapping
gi en by
T (z) = {exp(∫z
0exp( ( )) d )i q= 0
∫z
0(z− )q−1exp(∫
0exp( (u)) du)d i q≥1.
Then Tis a well-de ined con inuous ope a o on H(B( )).
Be o e s a ing he nex wo heo ems, we ecall ha an en i e unc ion Φ(z) =
∑∞
j=0 ajzjis said o be o exponen ial ype whene e he e exis posi i e cons an s
Aand Bsuch ha |Φ(z)| ≤ AeB|z| o all z∈C. Cauchy’s inequali ies show ha
his happens i and only i lim supj→∞(j!|aj|)1/j is ini e (c . [18, Chap. VII]). I
is shown in [9, Sec ion 5] ha i Φ is o exponen ial ype and L= Φ(D) ( ha is,
L=∑∞
j=0 ajDj, whe e D0=I= he iden i y ope a o ), hen Lis a well-de ined
con inuous linea ope a o on H(C).
By analogy, we adop he nex e minology. We say ha an en i e unc ion
Φ(z) = ∑∞
j=0 ajzjis o subexponen ial ype whene e he ollowing p ope y holds:
Gi en ε > 0, he e is a posi i e cons an A=A(ε) such ha
|Φ(z)| ≤ Aeε|z|∀z∈C,
ha is, Φ is ei he o g ow h o de less han one o o g ow h o de one and minimal
ype. E e y en i e unc ion o subexponen ial ype is i ially o exponen ial ype.
As be o e, Cauchy’s inequali ies show ha Φ is o subexponen ial ype i and only
i limj→∞(j!|aj|)1/j = 0 (c ., e.g., [5, 2.2.9-11]).
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Theo em 3. Suppose ha Lis he con inuous linea ope a o on H(C)gi en
by L= Φ(D), whe e Φis a noncons an en i e unc ion o exponen ial ype. Then
he e is a dense, in a ian submani old o H(C)each o whose non-ze o elemen s
is hype cyclic o L.
I should be poin ed ou he e ha a con inuous linea ope a o Lon H(C)
is o he o m L= Φ(D), whe e Φ is an en i e unc ion o exponen ial ype, i
and only i Lcommu es wi h each o he ansla ion ope a o s τa(a∈C), whe e
τa (z) = (z+a) ( ∈H(C), z ∈C) (see [9, Theo em 5.1, P oposi ion 5.2] o he
p oo o Theo em 3 and his no e; we jus use de case CN=Co [9, Sec ion 5]).
Thus he ope a o s Lon H(C) commu ing wi h ansla ions a e a special class o
“in ini e o de ” linea diffe en ial ope a o s wi h cons an coefficien s.
Unde he hypo hesis o Theo em 3, U(L) is no emp y o L={Ln:n∈N}.
Then U(L) is esidual in H(C) (see [8, P oposi ion 2.1]).
Theo em 4. Le Φ(z) = ∑∞
j=0 ajzjbe an en i e unc ion o subexponen-
ial ype and L= Φ(D). Then Lis a well-de ined con inuous linea ope a o on
H(B( )).
P o o . Fix ∈(0, ) and choose any s∈( , ). Le ∈H(B( )). Cauchy’s
inequali ies gua an ee ha ||Dj || ≤j!|| ||s
(s− )j o e e y j≥0. Le ε=s−
2. By
hypo hesis, he e is a posi i e cons an Asuch ha |aj| ≤ A·εj
j! o e e y j≥0.
Then we ha e ha ∑∞
j=0 ||ajDj || =∑∞
j=0 |aj| · ||Dj || ≤∑∞
j=0 A·εj
j!·j!|| ||s
(s− )j=
A|| ||s∑∞
j=0(1/2)j= 2A|| ||s<+∞. The e o e ∑∞
j=0 ajDj con e ges uni o mly
on e e y closed disk B( ) (0 < < ) and Lde ines a mapping om H(B( )) in o
i sel . The linea i y is i ial and, since ||L || ≤2A|| ||s, we ha e also ob ained
ha Lis con inuous on H(B( )). ////
3. The main esul . We a e now eady o s a e ou heo em on uni e sali y.
He zog’s esul is he special case q= 0, L=D.
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Theo em 5. Assume ha Lis he con inuous linea ope a o on H(C)gi en
by L= Φ(D), whe e Φis a noncons an en i e unc ion o subexponen ial ype. Fix
q∈N0and se A=A(q),L={Ln:n∈N}. Then he se U(L|A)is esidual in
A.
P o o . Fi s ly, no e ha A=∩∞
k=1 Akwhe e Ak={ ∈H(C) :
minB(k)| (q)· (q+1)|>0}. Pu X=Y=H(C). I is e iden ha e e y Ak
is open in H(C), so Ais a nonemp y Gδ-subse o X. F om Theo em 3, U(L) is
esidual in X. In o de o apply Theo em 1, we should demons a e ha , o e e y
ixed sequence { k}∞
1( k∈Ak, k ∈N), i holds ha
(2) lim
k→∞ sup
n∈N
in
h∈A(d( k, h) + d(Ln k, Lnh)) = 0,
dbeing de ined by (1). Fix k∈Nand a unc ion ∈Ak. The e is an ε > 0 such
ha (q)(z) (q+1)(z)= 0 o all z∈B(k+2ε). F om Theo em 2, he e is a unc ion
g∈H(B(k+ 2ε)) such ha is gi en on B(k+ 2ε) by he o mula gi en in ha
heo em. The e exis s a sequence o polynomials {Pm}∞
1sa is ying ||Pm−g||k+ε→0
(m→ ∞). Wi h he no a ion o Theo em 2, we ha e (z) = (0) ·Tg(z) i q= 0
and (z) = ∑q−1
ν=0
(ν)(0)
ν!zν+ (q)(0)
(q−1)! ·Tg(z) i q≥1 o all z∈B(k+ 2ε). Le us
de ine
hm(z) = { (0) ·TPm(z) i q= 0
∑q−1
ν=0
(ν)(0)
ν!zν+ (q)(0)
(q−1)! ·TPm(z) i q≥1
o all m∈Nand o all z∈C. No e ha each hm∈A. F om Theo em 2,
TPm→T g (m→ ∞) uni o mly on compac subse s o B(k+ε), so hm→
(m→ ∞) in he opology o H(B(k+ε)). By Theo em 4, Lnhm→Ln (m→ ∞)
uni o mly on compac subse s o B(k+ε), o e e y n∈N. In pa icula , we ob ain
ha limm→∞ ||Lnhm−Ln ||k= 0 ∀n∈N. Hence limn→∞(||hm− ||j+||Lnhm−
Ln ||j) = 0 o e e y n∈Nand e e y j∈ {1,2, ..., k}. Gi en δ > 0, a posi i e
in ege m=m(δ, n, k) can be ound in such a way ha ||hm− ||j+||Lnhm−Ln ||j<
δ o all j∈ {1, ..., k}, so d( , hm) + d(Ln , Lnhm)<∑k
j=1 δ
2j+∑∞
j=k+1 1
2j+
∑∞
j=k+1 1
2j=δ+ 21−k. Then in h∈A(d( , h) + d(Ln , Lnh)) < δ + 21−k∀δ > 0 and
6
∀n∈N. The e o e we ge
sup
n∈N
in
h∈A(d( k, h) + d(Ln k, Lnh)) ≤21−k→0 (k→ ∞)
i k∈Ak(k∈N). Consequen ly, (2) is ul illed and he p oo is comple e. ////
Nex , we show he op imali y o ou heo em. The esul canno be ex ended
o all en i e unc ions Φ o exponen ial ype. In ac , o e e y ype τ > 0 he e
exis s an en i e unc ion Φ o o de 1 and ype τsuch ha he esul ails o Φ.
One need only conside Φ(ζ) = eτζ. Then Φ(D) (z) = ∑∞
n=0
(n)(z)
n!τn= (z+τ),
so ha any Φ(D)-hype cyclic elemen is uni e sal wi h espec o ansla es. Bu i
ollows om a classical heo em o Hu wi z (see, o ins ance, [1, p. 178]) ha he
ansla es o a ze o- ee en i e unc ion canno app oxima e a non-cons an en i e
unc ion wi h ze os. Hence he e canno exis a Φ(D)-hype cyclic unc ion.
To inish, we poin ou ha i q∈N0,{Φk:k∈N}is a sequence o non-
cons an en i e unc ions o subexponen ial ype and Lk= Φk(D) (k∈N), hen
he e is an en i e unc ion wi h ze o- ee de i a i es o o de s qand q+ 1 which is
uni e sal wi h espec o e e y amily {Ln
k:n∈N}(k∈N). This is a i ial con-
sequence o he ac ha in e e y Bai e space he coun able in e sec ion o esidual
se s is esidual.
The au ho would like o hank he e e ee o help ul commen s and sugges-
ions.
Re e ences
[1] L.V. AHLFORS, Complex Va iables (3 d ed.). McG aw-Hill, London 1979.
[2] L. BERNAL-GONZ´
ALEZ, De i a i e and an ide i a i e ope a o s and he size
o complex domains. Annales Pol. Ma h. 59, 267-274 (1994).
[3] C. BLAIR and L. A. RUBEL, A uni e sal en i e unc ion. Ame . Ma h. Mon h-
ly 90, 331-332 (1983).
[4] C. BLAIR and L. A. RUBEL, A iply uni e sal en i e unc ion. Enseign. Ma h
30, 269-274 (1984).
[5] R.P. BOAS, En i e unc ions. Academic P ess, New Yo k 1954.
[6] R. B. BURCKEL, An In oduc ion o Classical Complex Analysis. Vol. 1.
Basel-Bos on 1979.
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[7] S. M. DUYOS RUIZ, Uni e sal unc ions and he s uc u e o he space o
en i e unc ions. So ie Ma h. Dokl. 30, 713-716 (1984).
[8] R. M. GETHNER and J. H. SHAPIRO, Uni e sal ec o s o ope a o s on
spaces o holomo phic unc ions. P oc. Ame . Ma h. Soc. 100, 281-288 (1987).
[9] G. GODEFROY and J. H. SHAPIRO, Ope a o s wi h dense, in a ian , cyclic
ec o mani olds. J. Func . Anal. 98, 229-269 (1991).
[10] K. G. GROSSE-ERDMANN, Holomo phe Mons e und uni e selle Funk ionen.
Mi . Ma h. Sem. Giessen 176 (1987).
[11] K. G. GROSSE-ERDMANN, On he uni e sal unc ions o G. R. MacLane.
Complex Va iables 15, 193-196 (1990).
[12] G. HERZOG, Uni e selle Funk ionen. Diploma bei , Uni e si ¨a Ka ls uhe
1988.
[13] G. HERZOG, On ze o- ee uni e sal en i e unc ions. A ch. Ma h. 63, 329-332
(1994).
[14] J. HORV´
ATH, Topological Vec o Spaces and Dis ibu ions. Vol. 1. Addison-
Wesley, Reading 1966.
[15] G. R. MACLANE, Sequences o de i a i es and no mal amilies. J. Analyse
Ma h. 2, 72-87 (1952).
[16] J. C. OXTOBY, Measu e and Ca ego y. Sp inge -Ve lag, Be lin-New Yo k-
Heidelbe g 1980.
[17] S. ROLEWICZ, On o bi s o elemen s. S udia Ma h. 32, 17-22 (1969).
[18] S. SAKS and A. ZYGMUND, Analy ic Func ions (2nd ed.). Polish Scien i ic
Publishe s, Wa saw 1965.
[19] W. SAXER, ¨
Ube die Pica dschen Ausnahmewe e sukzessi e De i ie en.
Ma h. Zei . 17, 206-227 (1923).
Luis Be nal-Gonz´alez
Depa amen o de An´alisis Ma em´a ico
Facul ad de Ma em´a icas
A enida Reina Me cedes. Apa ado 1160
41080 Se illa (S p a i n)
E-mail: lb[email p o ec ed]
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