Analysis 22, 57 - 66 (2002) sis
C Oldenbou g Ve lag Manchen 2002
UNIVERSAL FUNCTIONS WITH SMALL DERIVATIVES
AND EXTREMELY FAST GROWTH
M. C. Calde ón-Mo eno1
Recei ed: May 31, 2001
Abs ac . We p o e ha i α (E (0, and Τ is an in ini e o de di e en ial ope a o he e
exis s a dense linea mani old M. o en i e unc ions such ha
o e e y e M and any plane s ip S. Mo eo e , e e y non-null unc ion in M exhibi s
some ansla ion-uni e sali y p ope y wi h espec o Τ and i s g ow h index wi h espec
o any p e ixed sequence o non-cons an en i e unc ions is in ini e.
2000 Ma hema ics Subjec Classi ica ion: P ima y 30D15. Seconda y 30E10, 47A16, 47E05.
1 In oduc ion
Th oughou he las decades se e al au ho s ha e gi en "coun e examples" o he well-
known Liou ille's heo em. Fo ins ance, he e exis s a non-null en i e unc ion which ends
o ze o on e e y line (see [11, 15, 16]) o such ha i and e en all i s de i a i es ha e also
anishing in eg als on e e y line (see [2, 19]). In 1997 Be nal [5] (see also [4]) go many
unc ions which no only " iola ed" Liou ille's heo em in bo h senses bu also possessed
an ex emely as g ow h and "sha p" asymp o ic beha iou a in ini e. In o de o speci y
exac ly his esul , and wi h i he amewo k o his pape , le us in oduce some no a ion.
The symbol Σ will s and o he amily consis ing o all s ips in C (i.e., plane egions
be ween wo pa allel s aigh lines) and all sec o s
'The au ho has been pa ially suppo ed by DGES G an BFM2000-0514, DGES G an BFM2001-2717
and he Jun a de' Andalucía.
lim exp(|z|a)T/(z) = 0
s0:={z: 0 < a g 2 < β} (/?
<Ξ
(0, 2ττ))
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58 Calde ón
and L will s and o he amily o all s aigh lines. We deno e by H(C) he space o all
en i e unc ions endowed wi h he compac -open opology, so H{C) is a sepa able F éche
space. I > 0 and / G H(C) we deno e M ( ) := max{|/(z)| : |z| = } and o any
non-cons an unc ion h
G
H(C) he ela i e g ow h o de o / wi h espec o h, (see [3]) is
de ined as ... log M^(M ( ))
ph( ) = h nsup ,
—»oo log
Gi en any sequence Τ = {/ΐ·η}ϊ° o non-cons an en i e unc ions, he g ow h index o / wi h
espec o Τ is ip( ) = min{ c
G
Ν : phn{î) < °°}· We se ip( ) = oo i Phn( ) = 00 °
η. Obse e ha hese concep s ex end he olde one o ela i e g ow h o de [18].
Wi h his in mind, Bemal's esul [5, Theo em 3] eads as ollows:
THEOREM 1.1. (Be nal [5] j Assume ha a e (0,1/2) and ha φ : [0, +oo) (0, +oo)
is a con inuous unc ion which is in eg able on (l,+oo). Assum.e, in addi ion, ha F =
{Λ.η}^ is a sequence o noncons an en i e unc ions. Then he e is a linea mani old Λ4 =
Λ4(α, ψ, Τ) C H(C) sa is ying he ollowing se en p ope ies:
(a) M is dense in H{C).
(b) lim exp(|z|3/V(z))/(z) = 0 o all S 6 Σ and all e M.
z€S
(c) lim exp(|z|a)/ü)(2) = 0 o all S e Σ, all e M and all j > 0.
z€S
(d) W is bounded on S o all S
G
Σ, all e Μ and all j > 0.
(e) ) is in eg able on S wi h, espec o he plane Lebesgue measu e, o all S G Σ, all
e M and all j > 0.
( ) ^ in eg able on S wi h espec o he leng h measu e o all l G L, all e M and
all j > 0.
(g) = 0 ° al1 le L, all £ M and all j > 1.
(h.) ij ( ) = oo o all
G
M {0}.
Ou aim in his pape is o show ha no only he de i a i e ope a o o o de j, D3 =
can be eplaced by in ini e o de di e en ial ope a o s in he se en abo e p ope ies,
bu also ha we can p o ide an eigh h p ope y whose ea u e is o ally di e en om he
o he s, namely, a p ope y abou wild beha iou nea he in ini y poin .
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Uni e sal unc ions wi h as g ow h 59
2 De ini ions and s a emen o he main esul
An en i e unc ion φ(ζ) = akZk is said o be o exponen ial ype i he e a e cons an s
A,B>0 such ha φ(ζ) < AeB'2' o all ζ (E C. The unc ion φ is o subexponen ial
ype i gi en ε > 0, hen he e is a cons an A = Α(ε) > 0 such ha φ{ζ) < Aee^
o all ζ e C. This happens i and only i limsup^^ clla cl)1/* = 0 (see, e.g., [8, 2.2.9-
11]). Each en i e unc ion o subexponen ial ype is also o exponen ial ype and e e y
en i e unc ion φ o exponen ial ype de ines a (linea , con inuous) in ini e o de di e en ial
ope a o φ{Ό) = Σ,Ζ,ο0·^ on #(c)> which is on o (see [10, 14]). He e D° = I = he
iden i y ope a o .
On he o he hand, gi en a con inuous sel mapping Τ on H(C), we say ha an en i e
unc ion / is T-uni e sal whene e o each g € H(C) he e exis s a sequence (a,n) C C
sa is ying
(T )(z + an)^g(z) (n
—>
oo) in H(C).
Now, we a e able o es ablish he main esul , which will be p o ed in he nex sec ion.
THEOREM 2.1. Le be gi en an α e (0, a con inuous unc ion ψ : [0, +oo) (0, +oo)
which, is in eg able on (1, +oo) and a sequence. Τ = {h.n} o non-cons an , en i e unc ions.
Assume, in addi ion, ha , {ipi,m(z)}m=o i ~ 1,2^ a e wo sequences o en i e unc ions o
subexponen ial ype. Then he e is a linea mani old Μ = Μ(α,φ,^,(φι:τη),(·ψ2,τη)) o
en i e unc ions sa is ying he ollowing p ope ies:
(a) M is dense in H(C).
(b) lim exp( z 3/2<p(z)) {z) = 0 o all S e Σ and all e M.
(c) lim exp( z a)( phm(D) )(z) = 0 o all S 6 Σ, all e M and all m. > 0.
(d) Ψ ,m(D) /s bounded on S o all S 6 Σ, all (ï M and ali m > 0.
(e) iPi,m{D) is in eg able on S wi h, espec o he plane Lebesgue measu e o all S e Σ,
all € A4 and all m > 0.
( ) i>l,m(D) is in eg able on S wi h espec o he leng h. m.easu e o all l G L, all e M
and all
m,
> 0.
(9) ΙίΨιΑ0) = 0, hen j^l¡m{D) ds = 0 o all
I.
€ L and all e M.
(h) iA ) =
00
o all e M {0}.
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60 Calde ón
(i) E e y non-null unc ion in M is ip2,m{D)-uni e sal o all m > 0.
Obse e ha any polynomial p(z) is an en i e unc ion o subexponen ial ype. In pa -
icula , aking pi¡m(z) = zm o all m > 0, we ob ain he condi ions (a)-(h) o Theo em 1.1
oge he wi h an addi ional uni e sal p ope y o a sequence o in ini e o de di e en ial
ope a o s. I is no ewo hy he case in which also ip2,m{z) = zm o all τη > 0. Then Theo em
2.1 p o ides a linea dense mani old o en i e unc ions such ha each o hem and all i s
de i a i es a e uni e sal unc ions in Bi kho 's sense [7] (see also [6, 12, 13] o he ela ed
concep o holomo phic mons e in C) wi h g ow h condi ions.
Finally, we men ion ha in 2000 A. Bonilla [9] s udied an analogous p oblem in he space
o ha monic unc ions in RN, p o iding simila condi ions (c)-( ) o any de i a i e ope a o
Da and he uni e sal condi ion (i) o he iden i y ope a o .
3 An auxilia y esul and p oo o he main esul
We will use he ollowing heo em abou angen ial app oxima ion due o A akelian [1,
p. 1189], F om now on C^ is he ex ended plane. I F C C is a closed se , hen A(F) is he
space o all con inuous unc ions on F which a e holomo phic in he in e io o F. A closed
subse F C C is said o be an A akelian se [17] whene e Coo F is bo h connec ed and
locally connec ed a in ini y.
THEOREM 3.1. ('A akelian [1],) Assume ha F C C is an A akelian se , and ha e( )
is con inuous and posi i e o > 0. In addi ion, suppose ha
/
oo
3/2log£(i)di > -00. (1)
Then o e e y g € A(K) he e exis s an en i e unc ion such, ha ,
|/(*)-ί,(ζ)|<ε(|ζ|) (ζ e F).
The s a emen does no emain alid o e e y F is (1) is iola ed.
P oo , (o Theo em 2.1) Suppose ha α, φ, Τ = and
oo
= (i = 1,2)
k=0
a e as in he hypo heses. Thus o each m > 0, i = 1,2, he e exis s a posi i e cons an Ai¡m
such ha
Ι&ΓΙ <
aJ^-
(V c > 0).
(2)
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Uni e sal unc ions wi h as g ow h 61
Fix a sequence {pn)S=i which is dense in H(C) and a numbe β e (α, Fo e e y η € Ν,
he unc ion
e( ) = e„( ) := minji · γ^,βχρ(-Í3/VM - *")}
is posi i e and con inuous o > 0 and sa is ies (1). Le Ρ be he pa abolic cu e
Ρ = {x- ix1'2 :
a;
> 0}.
Fo each η € Ν, we de ine he se s
En = {ze C : z >n+l and dis (z, Ρ) > 1 + |z|},
Bn = {z
:
Μ < η}·
Conside a sequence o closed balls Dj = B(a,j, 1 + 2·') such ha
DjC{z : dis (z, Ρ) <
1
+ |z|} Ρ
and
Kl + 2J+2 < |oj+i| (i>l).
In pa icula he balls Dj a e paj wise disjoin and
|z|
> 2j (z € Dj). (3)
Le H = {zk}kLi be a sequence o pai wise di e en complex numbe s in Ρ wi h zk —» oo
(k.
—>
oo). Fo each η e Ν we de ine
Fn = Bn U En
U
H U ( (J Dj
J
,
j> jo /
whe e jo = jo(n) is he la ges index such ha Dj Π Bn+ φ 0. Then Fn is an A akelian se .
Di ide {üj} in o in ini ely many disjoin subsequences {o.¿(m,/j)} by se ing
, [(m + l){m + I + 1) + 2j][(m + l)(m, +
/
+ !) + 2(j + 1)] .
i(m, i,j)— g +
J
o all m > 0, all / > 1 and all j > 1. De ine induc i ely a sequence {/n}£Li o en i e
unc ions as ollows. Deno e k = zk o all k > 1 and o each m > 0 conside a sequence
{lm,n}n ° en i e unc ions such ha
1p2,m{D)qm,n = Pn ("· > 1)·
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62 Calde ón
Recall ha φ2,M(D) is an on o ope a o on H(C). Le g : F
—»
C deno e he unc ion
Pi(-z) {zeB,)
0 (2
G
£1)
51(2) = 1 + max^x c Mhj(exp k) (z = zk and zk > 1)
qm,j(z — 1i(m,lj)) (Z e A(m,lj))
Then gj e and by Theo em 3.1 he e exis s an en i e unc ion i such ha
l(z)-g1(z) <£l( z ) {z € Fi).
Assume ha ae {2,3,...} and ha we ha e cons uc ed 2 η
—
2 unc ions /1,...,
/„_ 1 in such a way ha </¿ 6 A(FÌ), FI € #(C) and
o all i 6 {1, 2,..., η
—
1}. Now, we de ine he unc ion gn : Fn
—»
C by
Pn(z) (z € Bn)
0 (z e En)
Sn(2) = 1 + maxi^-a Mhj{exp k) + c^X"/ (2 = and M > "-)
0 (z G A(m,ij) I ¿ η).
T i ially g„ € A(Fn) and by Theo em 3.1 he e exis s an en i e unc ion n such ha
n(z)-gn(z) <en( z ) (zeFn).
Thus, o all η e Ν,
„(z) - pn{z) < - (ζ e ¿U
n(z) <exp(- z VMz)- z ß) (ζ e EN),
n{z) - (1 + max Mh (exp k) + S(n,k)) < - < 1
i<j<k η
o all k such ha zk > n, whe e 5(1, k) = 0 and 5(n, k) = k ^"j/ M i( k) i η > 2,
I n(z) - qm,j(z -
Oi(m,nj))|
< - ·
η
1
+ z
o all z G A(m,nj), all j > 1 and all m > 0, and
1/nWI < Γ 1
η
1
+ 1*1
(4)
(5)
(6)
(7)
(8)
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Uni e sal unc ions wi h as g ow h 63
o all ζ e i3¿(m,¿j), all j > 1, all m > 0 and all I > 1 wi h
Ι,
φ η. Al hough we do no
men ion i explici ly, i is clea ha we ha e (8) whene e D¿(m,í,;) ΠΒ„ = 0.
By (4), he sequence is dense in //(C). Le us de ine M. as he linea span o
{/n}n- E iden ly, M. is a linea dense mani old o //(C): his p o es (a). In o de o e i y
ha (b), (c) hold o e e y / 6 ΛΊ, i su ices o check ha bo h p ope ies a e sa is ied o
e e y unc ion / = n. P om (5),
exp(|z|3^(*))l/»(*)l < exp(-M") (ζ^οο,ζε En).
Fo any sec o o s ip S G Σ, we ha e ha S En is a bounded se , hus
eM z 3/Mz)) n(z) - 0 (z^oo,zeS).
This p o es (b).
Now, we de ine he se E* as
E*n = {z e C : z >n + 2 and dis (z, P) > 2 + |z|}.
Then, one uses he Cauchy es ima es and (5) o in e ha
l/ Ml^lmaxilAHl:
w
-
z
= 1} <
< c!max{exp(-M") :
|ω|
>
z
- 1} < c!exp(-(|z| - i )
o all ζ e E* and all k. > 0 ( emembe ha ψ is posi i e). Hence, o each m. > 0,
|exp { z a)^m{D) n{z) = <
< expdzD · Alim ¿ M . c!exp(-(M - 1)") <
< 2Ai¡mexp(|z|a -
( z
- 1)") ^ 0 (z-*oo,z€ E*n).
I S e Σ, we ha e again ha S E* is bounded, so
limejcp(|2|e^,m(£>)/n(z)) = 0,
VeS
which p o es (c).
The p oo s o (d)-( ) and (h) a e analogous o hose we may ind in [5]. In o de o p o e
(g), we ix / 6 M. Suppose ha Vi.m(0) = 0, hen φ1 ¡m(D) = £(ΣΓ=ι ^Ύ*"1'). and by
he undamen al calculus heo em
/
OO 00
(A,m(D) )ds = limC^b' ^Hb)) - limC£bl'm ^(a)).
b~i
k= 1 aZl k=1
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64 Calde ón
Now we ha e only o obse e ha he same way ollowed o ob ain (c) leads us o "(c)" o
T.ZibkmDik~l)·
I emains only o ge (i). Le be / G M and ix τη, > 0. Since e e y non-ze o scala
mul iple o a ^2,m(ö)-uni e sal unc ion is again ^2,m(ß)-uni e sal, we may suppose ha
/ = Σ]ζΐ
aj j
wi h an = 1 and I =
{ji,...,
jT} ini e. In o de o p o e ha / is ip2,m(D)-
uni e sal i is enough o check ha
lim ((Tp2,m{D) )(z + <lj(mji,n)) ~ Pn(*)) = 0 (9)
uni o mly on compac subse s.
Fix η 6 N. We ha e
{ΦίΑϋ)!){ζ + Η^,ή,η)) ~PÁZ) =
Σ o j • { p2,m{D) j){z + ai (.D)qm¡n(z)
je/
< i o,m(D)( h(z + ai(mJun)) - 9m,„(z))| + ^
aj j(z
+ ai{mJl¡n)) .
i i
ίΦή
Now, o any ζ e ß(0,2n) C B{0,2¿(m^'n>) we ha e z + aiimj1¡n) e DiimJun), hus by (3) and
(8),
j(z + ai(m,jun))I < - · n p- < - (10)
j (1 + |z + ai(mjlin)|) 2n
o all j ξ. I wi h j φ .
On he o he hand, because o Cauchy's o mula o de i a i es applied o he cu e
7 ξ {|ω| = 2n + ì}, we ha e by (7) ha
1
I(/i, (2 + Ηη,ή,η)) - qmM)WI < Μ-ΐρ (k ^ °)·
The e o e, by (2),
li e iD)( j Áz + a (2))l <
(1/2)* klJ_='^n J_
k ' 'il2" j! 2n
< ΣΜ,πι-
k—0 (11)
Joining (10) and (11) we ob ain
I(ip2,m(D) )(z + ai{mJiìn})-pn(z) < I ^ +
/ 2"
ϊΦή
o all ζ € B{0,2"). I is clea ha his implies (9). Consequen ly, / is ^2,m(O)-uni e sal
and we ha e (i). •
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Uni e sal unc ions wi h as g ow h 65
Re e ences
[1] N. U. A akelian, Uni o m app oxima ion on closed se s by en i e unc ions (in Russian),
Iz . Akad. Nauk SSSR Se . Ma . 28, 1187-1206 (1964).
[2] D. H. A mi age, A non-cons an con inuous unc ion on he plane whose in eg al on
e e y line is ze o, Ame .. Ma h. Mon hly 101, 892-894 (1994).
[3] L. Be nal-González, O den ela i o de c ecimien o de unciones en e as, Collec . Ma h.
39, 209-229 (1988).
[4] L. Be nal-González, A lo o "coun e examples" o Liou ille's heo em, J. Ma h. Anal.
Appi. 201, 1002-1009 (1996).
[5] L. Be nal-González, Small en i e unc ions wi h ex emely as g ow h, J. Ma h. Anal.
Appi. 207, 541-548 (1997).
[6] L. Be nal-González and M. C. Calde ón-Mo eno, Holomo phic T-mons e s and
s ongly omnip esen ope a o s, J. App ox. Theo y 104, 204-219 (2000).
[7] C. D. Bi kho , Démons a ion d'un héo ème élémen ai e su les onc ions en iè es,
C. R. Acad. Sci. Pa is 189, 473-475 (1929).
[8] R. P. Boas, En i e Func ions, Academic P ess, New Yo k, 1954.
[9] A. Bonilla, "Coun e examples" o he ha monic Liou ille heo em and ha monic unc-
ions wi h ze o non angen ial limi s, Colloq. Ma h. 83, 155-160 (2000).
[10] L. Eh enp eis, Mean pe iodic unc ions I, Ame . J. Ma h. 77, 293-328 (1995).
[11] P. M. Gau hie , Uni o m app oxima ion, in Complex Po en ial Theo y, pp. 1-37, Kluwe
Academic, Do d ech /No well, MA, 1994.
[12] K.-G. G osse-E dmann, Holomo phe Mons e und uni e selle Funk ionen, Mi . Ma h.
Sem. Glessen 176 (1987).
[13] W. Lüh, Holomo phic mons e s, J. App ox. Theo y 53, 128-144 (1988).
[14] B. Malg an^e Exis ence e app oxima ion des solu ions des équa ions aux dé i ées
pa ielles e des équa ions de con olu ion, Ann. Ins i u Fou ie (G enoble) 6, 271-355
(1955/1956).
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