universal taylor series on unbounded open sets

Analysis 26, 1001–1014 (2006)c© R. Oldenbourg Verlag, Munchen 2006

Universal Taylor series on unbounded open sets

E. Diamantopoulos, Ch. Mouratides, N. Tsirivas

Received: December 16, 2005; Revised: March 18, 2006

Summary: Universal Taylor series where the universal approximation is valid on the boundary ofthe open unit disc can not be smooth or bounded. If the universal approximation is not requestedto be valid on the boundary, then it is well known that the universal function can be bounded andsmooth. We extend this result to unbounded open sets Ω, such that ∞ ∪ (C− Ω) is connected.

1 IntroductionUniversal Taylor series in the open unit disc, where the universal approximation is validon the circle also ([Nes]) can not be smooth nor bounded. If the universal approximationis not requested to be valid on the boundary of the disc ([CP], [Luh1]) then the univer-sal function can be smooth on the closed unit disc and bounded ([MN]). If we replacethe open unit disc by an unbounded domain Ω then we can have smooth universal Tay-lor series in the sense of Luh and Chui–Parnes ([KKN]). However, such a function isnot automatically bounded. We obtain the boundedness of the universal function f byrequesting limz→∞,z∈Ω f (l)(z) = 0 for all derivatives f (l), l = 0, 1, . . .

Let Ω be an unbounded open set in the complex plane with ∞∪(C−Ω) connected.Every function f analytic on Ω has a Taylor series development with center ζ ∈ Ω,

f(z) =∞∑

k=0

f (k)(ζ)k!

(z − ζ)k,

defined in an open disc. We denote by

SN (f, ζ)(z) =N∑

k=0

f (k)(ζ)k!

(z − ζ)k,

the partial sums of the infinite Taylor series. In the case where ζ belongs to ∂Ω andf (l)(ζ) are continuously extendable at ζ we keep the last notation for the partial sums,although in this case we do not have a formal Taylor development.

AMS 2000 subject classification: 30B30Key words and phrases: Taylor series, overconvergence, generic property, smooth functions.The project is co-funded by the European Social Fund and National Resources – (EPEAEK II) PYTHAGORASII.

1002 Diamantopoulos – Mouratides – Tsirivas

Let Z be the subset of H(Ω) of all functions g continuous on Ω∪∞, analytic on Ωsuch that g(l) extends continuously on Ω ∪ ∞ and limz→∞ g(l)(z) = 0 for any l ∈ N.Let Y be the subset of Z which consist of all rational functions in Z with poles only inC− Ω.

For any g ∈ Z the family of seminorms ‖g‖l = supΩ∪∞ |g(l)(z)| is well definedand it induces a metric on the space Z. Z is a complete metric space.

In this article we consider universal properties of functions in

B∞0 (Ω) =f ∈ H(Ω) limit in the topology of space Z of a sequence Rn∞n=0

of rational functions with poles only in C− Ω such that

limz→∞,z∈Ω

R(l)n (z) = 0 for any n, l ∈ N = Y

Z.

B∞0 (Ω) is a complete metric space as a closed subspace of the complete space Z, thus

Baire’s theorem is at our disposal.We define the set

U =

f ∈ B∞0 (Ω) : for every K ⊂ C compact, K ∩ Ω = ∅,

and for every function h : K → C, h ∈ C(K) ∩H(Ko)

there exists λn∞n=0 ∈ NN such that

(a) For all compact sets L ⊂ Ω,

limn→+∞

supζ∈L

supz∈K

∣∣Sλn(f, ζ)(z)− h(z)∣∣ = 0,

(b) For all compact sets L, M ⊂ Ω and any l ∈ N,

limn→+∞

supζ∈L

supz∈M

∣∣∣∣∂l

∂zlSλn(f, ζ)(z)− f (l)(z)

∣∣∣∣ = 0.

We notice that Ω is considered as a subset of C, it does not contain ∞.Our purpose is to prove the following

Theorem 1.1 U is a dense Gδ set in the space B∞0 (Ω), in particular U is non void.

This theorem appears as a natural generalization of older results ([CP, KKN, Luh1,Luh2, MN, Nes]). The tools for the proof of Theorem 1.1 are an extended versionof Runge’s Theorem and a topological lemma extending a previous result of Grosse-Erdmann ([Erd1]) and Costakis ([CG]). We do not use Arakeljian’s Theorem, and ourresult is valid for more open sets than those for which properties (K1) and (K2) of [Tsi]are satisfied (see also [Ga])). Certainly, our result is valid for all half-planes or strips. Forthe role of Baire’s Theorem in various branches of analysis we refer to [Erd2, Ka].

2 PreliminariesLemma 2.1 Let K ⊂ C be compact and P a rational function with poles z0, z1, . . . , zk

such that dist(zi,K) > diam(K) for all i = 0, 1, . . . , k. Then for every ε > 0 there is a

Universal Taylor series on unbounded open sets. 1003

n0 ∈ N such that for every n ≥ n0 and l ∈ N,

supz∈K

supζ∈K

∣∣∣∣∂l

∂zlSn(P, ζ)(z)− P (l)(z)

∣∣∣∣ < ε.

Proof: Since the derivative of a rational function is also a rational function with the samepoles and ∂

∂z Sn(P, ζ)(z) = Sn−1(P′, ζ)(z) it suffices to prove the claim only for l = 0.

Every rational function P with poles z0, z1, . . . , zr has decomposition to a linearcombination of a polynomial Q and finite fractions of the form 1/(z − zi)ρ where i ∈0, 1, . . . , k and 1 ≤ ρ ≤ multiplicity(zi). The partial sums are linear thus, it sufficesto prove the Lemma for every part of this decomposition, in particular for the simplefractions since for the polynomial it clearly holds. Without restriction of the generalitywe consider z0 = 0. Let

fρ(z) =1zρ

,

and auxiliary suppose ρ = 1. Then for every ζ ∈ K we compute the Taylor series of f1,

f1(z) =1ζ

∞∑

k=0

(−1)k

(z − ζ

ζ

)k

.

Since,|z − ζ| ≤ diam(K) < dist(0,K) ≤ |ζ|,

we get ∣∣∣∣z − ζ

ζ

∣∣∣∣ ≤ λ < 1,

and the convergence of the Taylor representation of f1 is uniformly for all z, ζ ∈ Kwhich implies that for every ε > 0 there is a n0 ∈ N such that for every n ≥ n0,

supz∈K

supζ∈K

|Sn(f1, ζ)(z)− f1(z)| < ε.

which is the desired result.For ρ = 2, it suffices to notice that

(f1(z))′= − 1

z2= −f2(z),

(Sn(f1, ζ)(z))′= −Sn−1(f2, ζ)(z),

and apply Weierstrass Theorem in the open set w ∈ C : |w| < 1 for w = (z− ζ)/ζ. Ingeneral, the same argument applies for all ρ ∈ N and finishes the proof of the lemma. 2

In the next lemma we realize the complement of a set as the complement in theextended complex plane.


Lemma 2.2 Let K ⊂ C∪∞ be a compact set and A ⊂ C∪∞ a set containing onepoint from every component of Kc. Let W ⊂ C∪∞ be open with K ⊂ W . Then thereis an open set V such that K ⊂ V ⊂ W , V c has finite components and every componenthas a non empty intersection with A.

Proof: We consider the metric in C ∪ ∞ induced by the stereographic projection andEuclidian metric in R3. All discs D(w, ρ) are understood in this metric. Furthermore,when a polygonal line ends at ∞, we mean that its last edge is a closed half line with thepoint at ∞.

The sets K and W c are compact sets with empty intersection thus dist(K, W c) =ρ > 0.

From the open cover⋃

x∈W c D(x, ρ/3) of the compact set W c we get a finite sub-cover

D(x1, ρ/3) ∪D(x2, ρ/3) ∪ · · · ∪D(xk, ρ/3),

for x1, x2, . . . , xk in W c. Obviously, every xi, i = 1, . . . , k is possible to get connectedwith a point in A using a polygonal line Γi lying entirely in Kc.

The set

L =k⋃

i=1

(Γi ∪D(xi, ρ/3)

),

is compact andW c ⊂ L ⊂ Kc.

Let V = Lc. Then V is an open set and we notice that

K ⊂ V ⊂ W.

Furthermore, for all i = 1, 2, . . . , k the set Γi∪D(xi, ρ/3) is connected and intersects Aat ai. Thus, V c has finite components and all of them intersects A. The proof is complete.

2

The next lemma is necessary for the proof of our theorem. Since it appears in [Luh2,MN], we present it without proof.

Lemma 2.3 There is a sequence Km∞m=0 of compact sets Km ⊂ C − Ω with Kcm

connected such that for every compact K ⊂ C − Ω with Kc connected, there existm ∈ N such that K ⊂ Km.

3 Proof of the TheoremLet

Ln = z ∈ Ω : |z| ≤ n,


and fj∞j=0 be the sequence of all polynomials whose coefficients have rational coordi-nates. We define the following sets

E(ρ,m, n, j, s) =

f ∈ B∞

0 (Ω): supζ∈Lρ

supz∈Km

∣∣Sn(f, ζ)(z)−fj(z)∣∣ <

1s, l = 1, 2, . . . , s

,

Ξ(ρ, n, s) =

f ∈ B∞

0 (Ω): supζ∈Lρ

supz∈Lρ

∣∣∣ ∂l

∂zlSn(f, ζ)(z)−f (l)(z)

∣∣∣ <1s, l = 1, 2, . . . , s

.

Lemma 3.1

U =⋂ρ

⋂m

⋂

j

⋂s

∞⋃n=0

[E(ρ,m, n, j, s)

⋂Ξ(ρ, n, s)

].

Proof: It suffices to show

⋂ρ

⋂m

⋂

j

⋂s

∞⋃n=0

[E(ρ,m, n, j, s)

⋂Ξ(ρ, n, s)

]⊆ U,

since the other inclusion follows directly from the definitions of the sets U , E(ρ,m, n, j, s)and Ξ(ρ, n, s).

Let

f ∈⋂ρ

⋂m

⋂

j

⋂s

∞⋃n=0

[E(ρ,m, n, j, s)

⋂Ξ(ρ, n, s)

],

K, L, M be compact sets, K ⊂ C − Ω with connected complement, L, M ⊂ Ω andh : K → C be a function continuous in K and analytic in the interior of K.

Let l ∈ N and s > l, s ∈ N be fixed. By Mergelyan’s theorem there is a polynomialfjs whose coefficients have rational coordinates such that

supz∈K

|h(z)− fjs(z)| < 1s, (3.1)

for all l = 1, 2, . . . , s. We choose a member Km0 of the family Km∞m=0 such thatK ⊂ Km0 . Moreover if ρ0 ∈ N is such that ρ0 > maxmaxz∈L |z|,maxz∈K |z| weeasily notice that L, M ⊂ Lρ for all ρ ≥ ρ0.

By our assumption it follows that

f ∈∞⋃

n=0

[E(ρ0,m0, n, js, s) ∩ Ξ(ρ0, n, s)

],

Thus, there is at least one ns ∈ N such that f ∈ E(ρ0,m0, ns, js, s) and f ∈ Ξ(ρ0, ns, s).In particular,

supζ∈Lρ0

supz∈Km0

∣∣Sns(f, ζ)(z)− fjs(z)∣∣ <

1s


and

supζ∈Lρ0

supz∈Lρ0

∣∣∣∣∂l

∂zlSns(f, ζ)(z)− f (l)(z)

∣∣∣∣ <1s.

The last inequalities and the relation (3.1), implies that there is a ns ∈ N such that

supζ∈L

supz∈K

∣∣Sns(f, ζ)(z)− h(z)∣∣ <

2s

and supζ∈L

supz∈M

∣∣∣∣∂l

∂zlSns(f, ζ)(z)− f (l)(z)

∣∣∣∣ <1s.

By letting s →∞, we get a sequence ns of natural numbers such that

lims→∞

supζ∈L

supz∈K

∣∣Sns(f, ζ)(z)− h(z)

∣∣ = 0

and

lims→∞

supζ∈L

supz∈M

∣∣∣∣∂l

∂zlSλn

(f, ζ)(z)− f (l)(z)∣∣∣∣ = 0,

for the arbitrary l ∈ N we choose, which implies f ∈ U and finishes the proof of thelemma. 2

Lemma 3.2 The set E(ρ,m, n, j, s) is open in the topology of B∞0 (Ω) for all ρ,m, n, j ∈

N and s ∈ N∗.

Proof: Let ρ0, m0, n0, j0, s0 ∈ N and f ∈ E(ρ0,m0, n0, j0, s0).For any l ∈ N, we define

a =1s0− sup

ζ∈Lρ0

supz∈Km0

∣∣Sn0(f, ζ)(z)− fj0(z)∣∣ > 0,

and we notice that the inclusion hypotheses for f ensures a > 0. Let

M = supz,w∈Lρ0∪Km0

|w − z| < +∞,

and ε > 0 such thatε <

a∑n0i=0 M i

.

Let fn∞n=0 be a sequence of functions in B∞0 (Ω) converging to f in the topology of

that space. It suffices to prove that there is a N ∈ N such that fn ∈ E(ρ0,m0, n0, j0, s0)for n ≥ N .

Since fn → f , there is a N ∈ N such that for all n > N ,

supζ∈Lρ0

|f (d)n (ζ)− f (d)(ζ)| < ε,

for d = 0, 1, . . . , n0.


For all ζ ∈ Lρ0 , z ∈ Km0 we estimate

|Sn0(fn − f, ζ)(z)| =∣∣∣∣

n0∑

d=0

f (d)(ζ)− f(d)n (ζ)

d!(ζ − z)d

∣∣∣∣

≤n0∑

d=0

|f (d)(ζ)− f(d)n (ζ)|

d!|ζ − z|d

<

n0∑

d=0

ε

d!Md < a.

From triangular inequality we get that for n > N ,

supζ∈Lρ0

supz∈Km0

|Sn0(fn, ζ)(z)− fj0(z)|

≤ supζ∈Lρ0

supz∈Km0

|Sn0(fn − f, ζ)(z)|+ supζ∈Lρ0

supz∈Km0

|Sn0(f, ζ)− fj0(z)|

< a + supζ∈Lρ0

supz∈Km0

|Sn0(f, ζ)− fj0(z)| = 1s0

,

which implies fn ∈ E(ρ0,m0, n0, j0, s0) for all n > N and complete the proof oflemma. 2

In a similar way we prove that Ξ(ρ, n, s) is open in B∞0 (Ω).

Lemma 3.3 The set Ξ(ρ, n, s) is open in the topology of B∞0 (Ω) for all ρ, n ∈ N and

s ∈ N∗.

Proof: Let ρ0, n0, s0 ∈ N and f ∈ Ξ(ρ0, n0, s0).We define

a =1s0−max

l≤s0

sup

ζ∈Lρ0

supz∈Lρ0

∣∣∣∣∂l

∂zlSn0(f, ζ)(z)− f (l)(z)

∣∣∣∣

> 0,

and we notice that the inclusion hypotheses for f ensures a > 0. Let

M = supz,w∈Lρ0

|w − z| < +∞,

and ε > 0 such thatε <

a∑n0i=0 M i + 1

.

Now, let fn∞n=0 be a sequence of functions in B∞0 (Ω) converging to f in the topology

of that space. It suffices to prove that there is a N ∈ N such that fn ∈ E(ρ0,m0, n0, j0, s0)for n ≥ N .


Since fn → f , there is a N ∈ N such that for all n > N ,

supζ∈Lρ0

|f (l)n (ζ)− f (l)(ζ)| < ε,

for l = 0, 1, . . . , maxn0, s0.For all ζ, z ∈ Lρ0 and n > N we get

∣∣∣ ∂l

∂zlSn0(fn, ζ)(z)− f (l)

n (z)∣∣∣ ≤

∣∣∣ ∂l

∂zlSn0(fn − f, ζ)(z)

∣∣∣ +∣∣∣ ∂l

∂zlSn0(f, ζ)− f (l)(z)

∣∣∣+ |f (l)(z)− f (l)

n (z)|.

Now, we estimate

∣∣∣ ∂l


∣∣∣ =∣∣∣ ∂l

∂zl

n0∑n=0

(fn − f)(n)(ζ)n!

(z − ζ)n∣∣∣

=∣∣∣

n0−l∑n=0

(fn − f)(n+l)(ζ)n!

(z − ζ)n∣∣∣

≤n0−l∑n=0

εMn ≤ ε

n0∑n=0

Mn.

From preceding inequalities we take∣∣∣∣

∂l


∣∣∣∣ +∣∣f (l)(z)− f (l)

n (z)∣∣ < a,

and we conclude

supζ∈Lρ0

supz∈Lρ0

∣∣∣∣∂l

∂zlSn0(fn, ζ)(z)− f (l)

n (z)∣∣∣∣

≤ a + supζ∈Lρ0

supz∈Lρ0

∣∣∣∣∂l

∂zlSn0(f, ζ)− f (l)(z)

∣∣∣∣ ≤1s0

,

which implies that fn ∈ Ξ(ρ0, n0, s0) for all n > N and complete the proof of lemma.2

Runge’s Theorem provides a sequence of rational functions approximating functionsanalytic on an open set Ω. In our article we use the following well known extendedversion of this theorem.

Theorem 3.4 (Extended Runge’s Theorem) Let Ω be an open set in the extended plane,let A be a set which has one point in each component of S2−Ω, and assume f ∈ H(Ω).Then there is a sequence Rn of rational functions, with poles only in A, such that Rn → funiformly on compact subsets of Ω.


Proposition 3.5 The set⋃∞

n=0[E(ρ,m, n, j, s)∩Ξ(ρ, n, s)] is dense in the space B∞0 (Ω)

for any ρ,m, j, n,∈ N, s ∈ N∗.

Proof: Clearly, it suffices to prove that⋃∞

n=0[E(ρ,m, n, j, s)∩Ξ(ρ, n, s)] is dense in thespace Y .

Let f ∈ Y and ε > 0. It suffices to find a function P ∈ Y such that

d(f, P ) =∞∑

n=0

12n

‖f − P‖n

1 + ‖f − P‖n< ε, (3.2)

supζ∈Lρ

supz∈Km

|Sn(P, ζ)(z)− fj(z)| < 1s, (3.3)

and

supζ∈Lρ

supz∈Lρ

| ∂l

∂zlSn(P, ζ)(z)− P (l)(z)| < 1

s, for l = 0, 1, . . . , s. (3.4)

Since by assumption ∞∪(C−Ω) is connected it follows that Ω is simply connectedand all components of C − Ω are unbounded. Let Tii∈I , I ⊂ N be the connectedcomponents of C − Ω. The class Tii∈I consist an open cover of the compact set Km

so we can choose a finite set I′ ⊂ I ⊂ N such that Km ⊂ ∪i∈I′Ti.

Let d = max|z − w| : z, w ∈ Lρ ∪Km, Af = w1, w2, . . . , wk be the poles ofthe function f and I

′′ ⊂ I be a finite set of integers such that Af ⊂ ∪i∈I′′Ti. We setI0 = I

′ ∪ I′′

.We consider the family zii∈I0 with zi ∈ Ti for all i ∈ I0 and

|zi| = max maxLρ∪Km

|z|+ d + 1,mini∈I0

|zi|+ 1 : zi ∈ Ti.

Letε1 = mindist(z, Ω) : z ∈ Af ∪ zii∈I0 and ε2 = distKm, Ω.

We define

ε0 =12

minε1, ε2 and M = 2max|z| : z ∈ Af ∪ zii∈I0,

and we consider the following sets

W1 = z ∈ C : dist(z, Ω) < ε0 ∪ z ∈ C : |z| > M ∪i∈I−I0 Ti,

W2 = z ∈ C : dist(z,Km) < min1/2, ε0,and

L = Ω ∪ ∞ ∪ [∪i∈I−I0Ti] ∪Km.

Clearly W1, W2 are open sets and we verify that L is a compact subset of C∪∞ eitherby proving directly that Lc is open or by noticing that L is identical to

[∪i∈I0Ti]c ∪Km,


where the complement is understood in C ∪ ∞.Furthermore, we notice that L ⊂ W1∪W2, Lc has exactly |I0| connected components

and A = zii∈I0 has one point from every component of Lc.From Lemma (2.2) we get that there is an open set V such that L ⊂ V ⊂ W1 ∪W2

and V c has finite components, each one of which intersects A.Since W1 ∩W2 = ∅, Ω ∪ ∞ ⊂ W1 and Km ⊂ W2 we get a decomposition of the

set V into V1 ∪ V2 where V1, V2 are open, V1 ∩ V2 = ∅, Ω ∪ ∞ ⊂ V1 and Km ⊂ V2.The set U = (V1 ∪ V2)c has exactly |I0| connected components and the finite set

zii∈I0 has exactly one point in every component.We consider the function

H(z) =

f(z), z ∈ V1

fj(z), z ∈ V2.

H is analytic in V1 ∪ V2 thus, from extended Runge’s Theorem there is a sequence Rn

of rational functions with poles in the set zii∈I0 such that

limn→∞

Rn = H

uniformly on compact subsets of V1 ∪ V2. In particular limn→∞Rn(∞) = H(∞) =0. We define the family of rational functions Pn = Rn − Rn(∞) and we notice thatlimn→∞ Pn = H uniformly on compact subsets of V1 ∪ V2 and Pn(∞) = Rn(∞) −Rn(∞) = 0 thus, Pn ∈ Y for all n ∈ N.

Let n0 ∈ N such that∞∑

n=n0+1

12n

<ε

2.

Weierstrass Theorem implies that for all l = 0, 1, . . . , maxn0, s there is a N0 ∈ Nsuch that for every N ≥ N0,

‖f − PN‖n = supz∈Ω

|P (l)N (z)− f (l)(z)| < ε

4.

Then, we calculaten0∑

n=0

12n

‖f − PN‖n

1 + ‖f − PN‖n<

ε

2,

and

d(f, PN ) =n0∑

n=0

12n

‖f − PN‖n

1 + ‖f − PN‖n+

∞∑n=n0+1

12n

‖f − PN‖n

1 + ‖f − PN‖n

<ε

2+

ε

2= ε,

thus (3.2) is fulfilled.Since Pn → H uniformly on Km, there is a N1 ∈ N such that for every n ≥ N1,

supz∈Km

|Pn(z)− fj(z)| < 12s

. (3.5)


We notice that

supζ∈Lρ

supz∈Km

|Sn(PN1 , ζ)(z)− PN1(z)| ≤ supζ∈Lρ∪Km

supz∈Lρ∪Km

|Sn(PN1 , ζ)(z)− PN1(z)|.(3.6)

From Lemma (2.1) we find a sufficient large ns such that for every n ≥ ns,

supζ∈Lρ∪Km

supz∈Lρ∪Km

|Sn(PN1 , ζ)(z)− PN1(z)| < 12s

, (3.7)

and

supζ∈Lρ

supz∈Lρ

|Sn−l(P(l)N1

, ζ)(z)− P(l)N1

(z)| < 1s. (3.8)

From equations (3.5), (3.6) and (3.7) we deduce that there is a N1 ∈ N such that forevery n ≥ ns

supζ∈Lρ

supz∈Km

|Sn(PN1 , ζ)(z)− fj(z)| < 1s,

that is PN1 ∈ E(ρ,m, n, j, s) for n ≥ ns.Finally, since

Sn−l(P(l)N1

, ζ)(z) =∂l

∂zlSn(PN1 , ζ)(z),

relation (3.8) implies

supζ∈Lρ

supz∈Lρ

| ∂l

∂zlSn(PN1 , ζ)(z)− P

(l)N1

(z)| < 1s,

for every n ≥ ns and for all l = 0, 1, . . . , s, that is PN1 ∈ Ξ(ρ, n, s) for n ≥ ns. Theproof is completed. 2

Proof of theorem 1.1The space B∞

0 (Ω) is a complete metric space thus Baire’s Theorem is at our disposal.From Lemma 3.1 and Proposition 3.5 we take the desired result. 2

Remarks. 1. Let f ∈ U , ζ ∈ Ω and K ⊂ C compact with Kc connected. Since, forany function h analytic in an open simply connected set V ⊃ K there is an appropriatesequence of natural numbers λn such that the partial sums Sλn(f, ζ) converges uni-formly on K to h and this holds for all compact subsets of the open set V , WeierstrassTheorem implies the uniform convergence on compacta of the derivatives ∂l

∂zl Sn(f, ζ) of


every order l ∈ N to the derivatives h(l). In particular, if we define

U′=

f ∈ B∞

0 (Ω) : for every K ⊂ C compact, K ∩ Ω = ∅, Kc connected

and for every function h analytic in an open set V ⊃ K

there exists λn∞n=0 ∈ NN such that

(a) For all compact sets L ⊂ Ω and any l ∈ N,

limn→+∞

supζ∈L

supz∈K

∣∣∣∣∂l

∂zlSλn(f, ζ)(z)− h(l)(z)

∣∣∣∣ = 0,

(b) For all compact sets L,M ⊂ Ω and any l ∈ N,

limn→+∞

supζ∈L

supz∈M

∣∣∣∣∂l

∂zlSλn(f, ζ)(z)− f (l)(z)

∣∣∣∣ = 0.

,

then U′= U (see [CG]) and we immediately obtain the following

Theorem 3.6 U′

is a dense Gδ set in the space B∞0 (Ω), in particular U

′is non void.

We also notice that actually it suffices to suppose that the partial sums of a functionf in U

′approximate the polynomials. Indeed, for any appropriate compact set K and for

every function h analytic in an open set V ⊃ K Runge’s Theorem ensures the approxi-mation of h and all its derivatives using polynomials and certifies this assertion. Finally,using an argument similar to the proof of Proposition 3.5 of [CG] one can easily showthat if f ∈ U

′then f

′ ∈ U′.

2. Let Y be a hemicompact metric space, that is Y is a metric space for which there existan increasing sequence of compact subsets Yρ ⊂ Y , ρ ∈ N, such that every compactset J ⊂ Y is contained in some Yρ. We consider the space F of complex continuousfunctions f in Y × Ω

cfor which there is an open set V , depending on f such that Ω

c ⊂V ⊂ C and for every fixed x ∈ Y the function z → f(x, z) extends holomorphicallyin V and f is continuous on Y × V . F is endowed with the topology defined by theseminorms

supx∈J

supz∈K

∣∣∣∣∂l

∂zlf(x, z)

∣∣∣∣,

where l ∈ N, J = Yρ, and K = Ωc ∩ z ∈ C : |z| ≤ ρ for a ρ ∈ N.

Let Tn : B∞0 (Ω) → F be a sequence of continuous maps. For any compact sets K ⊂

Ωc, J ⊂ Y and any rational function P with poles z0, z1, . . . , zk such that dist(zi,K) >

diam(K), i = 0, 1, . . . , k we assume that for every ε > 0 and l ∈ N there is a n0 ∈ Nsuch that for every n ≥ n0,

supz∈K

supx∈J

∣∣∣∣∂l

∂zlTn(P )(x, z)− P (l)(z)

∣∣∣∣ < ε.


We define the set

U′1 =

f ∈ B∞

0 (Ω) : for every K ⊂ C compact, K ∩ Ω = ∅, Kc connected

and for every function h analytic in an open set V ⊃ K

there exists λn∞n=0 ∈ NN such that(a) For all compact sets J ⊂ Y and any l ∈ N,

limn→+∞

supx∈J

supz∈K

∣∣∣∣∂l

∂zlTλn(f)(x, z)− h(l)(z)

∣∣∣∣ = 0,

(b) For all compact sets L,M ⊂ Ω and any l ∈ N,

limn→+∞

supζ∈L

supz∈M

∣∣∣∣∂l

∂zlSλn

(f, ζ)(z)− f (l)(z)∣∣∣∣ = 0.

Using the same arguments which led to the proof of Theorem 1.1 with some modifica-tions we can prove the following more general result

Theorem 3.7 U′1 is a dense Gδ set in the space B∞

0 (Ω), in particular U′1 is non void.

It is clear that Theorem 1.1 is a special case of Theorem 3.7 since the operators Sn ofthe partial sums obviously satisfy the assumption on the family of the operators Tn. Theoperators Tn could also be averages of the partial sums (see [KKN]).

Acknowledgments. We would like to thank V. Nestoridis for helpful suggestions anddiscussions during the preparation of this article.

References[CG] G. Costakis, Some remarks on universal functions and Taylor series, Math. Proc.

Cambridge Philos. Soc. 128 (2000), 157–175.

[CP] C. Chui and M. N. Parnes, Approximation by overconvergence of power series,J. Math. Anal. Appl. 36 (1971), 693–696.

[Erd1] K. G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt.Math. Sem. Giessen 176 (1987), 1–84.

[Erd2] K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull.Amer. Math. Soc. (N.S.) 36 (3)(1999), 345–381.

[Ga] Dieter Gaier, Lectures on complex approximation, Birkhauser Boston Inc., 1987.

[Ka] J. P. Kahane, Baire’s category theorem and trigonometric series, J. Anal. Math.80 (2000), 143–182.


[KKN] Ch. Kariofilis, Ch. Konstandilaki and V. Nestoridis, Smooth Universal Taylorseries, Monatshefte fur Mathematik, to appear.

[Luh1] W. Luh, Approximation analytischer Funktionen durch uberkonvergente Potenz-reihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen 88 (1970),1–56.

[Luh2] W. Luh, Universal approximation properties of overconvergent power series onopen sets, Analysis 6 (1986), 191–207.

[MN] A. Melas and V. Nestoridis, On various types of universal Taylor series, ComplexVariables Theory Appl. 44 (2001), 245–258.

[Nes] V. Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (5)(1996), 1293–1306.

[Tsi] N. Tsirivas, Boundedness, regularity and smoothness of Universal Taylor series,submitted.

E. Diamantopoulos8 Taskou Papageorgiou54631 [email protected]

Ch. MouratidesTechnological Institute of West MacedoniaKoila50100 [email protected]

N. TsirivasDepartment of MathematicsUniversity of AthensPanepistimiopolis15784, [email protected]

universal taylor series on unbounded open sets

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