Generalized universal series

0.32 MB·

by  Stéphane Charpentier

GENERALIZED UNIVERSAL SERIES
S. CHARPENTIER,A. MOUZE,V. MUNNIER
Abstract. We unify the recently developed abstract theories of universal series and extended
universalseriestoincludesumsoftheformPn a x forgivensequencesofvectors(x )
k=0 k n,k n,k n≥k≥0
inatopologicalvectorspaceX. Thealgebraicandtopologicalgenericityaswellasthespaceability
are discussed. Then we provide various examples of such generalized universal series which donot
proceed from the classical theory. In particular, we build universal series involving Bernstein’s
polynomials, we obtain a universal series version of MacLane’s Theorem, and we extend a result
of Tsirivas concerning universal Taylor series on simply connected domains, exploiting Bernstein-
4 Walsh quantitativeapproximation theorem.
1
0
2

n 1. Introduction
a
J In 1914, Fekete proved that there exists a formal Taylor series on [ 1,1] whose subsequences of
−
8 partial sums approximate uniformly any function continuous on [ 1,1] which vanishes at 0 [25].
Since then, several results of this type have appeared. In 1996, Ne−storidis provided renewed vigor
]
A to the research on this kind of maximal divergence in [23] by showing that there exists a power
F series k≥0akzk of radius of convergence 1 such that for any compact set K ⊂ C \ D (where
. D = z C; z < 1 ) with connected complement, and any function h continuous on K and
h
{P∈ | | }
at holomorphic in the interior of K, there exists an increasing sequence (λn)n ⊂ N such that
m λn
[ sup akzk h(z) 0, as n + .
1 z∈K(cid:12)(cid:12)Xk=0 − (cid:12)(cid:12) → → ∞
(cid:12) (cid:12)
v These approximating series are n(cid:12)ow referred to a(cid:12)s universal series. A lot of other universal series
4 appeared in different context (Fab(cid:12)er series, Laure(cid:12)nt series, Jacobi series...). We refer the reader to
9
the excellent surveys [8] and [10].
5
1 The theory of universal series is a part of the theory of universality in operator theory which
.
1 consists in the study of the orbit of a vector under the action of a sequence of operators. In
0 fact universal series and hypercyclic operators constitute for now the most important instances
4
of universality. The abstract theories of hypercyclicity and universal series respectively are now
1
: rather well-known (see [3, 2] and the references therein) and provide a very efficient tool to exhibit
v
new instances of hypercyclicity or universal series. But in a lot of situations we have to deal with
i
X objectswhichcannotbehandledbythesetheories. Forexamplewecanbeinterestedinstudyingthe
r universal properties of the Cesa`ro means of the sequence of the iterates of a hypercyclic operator T
a
([12])orinstudyingtheaveragedsums 1 n a zk .Yetthesenewobjectscannotbeconsidered
n k=0 k n
from the point of view of hypercyclicity or universal series any more. Furthermore the previous
(cid:0) P (cid:1)
abstracttheoryofuniversalseriescannotprovideatooltostudytheactionofsummabilityprocesses
on universal series. A very abstract theory of universality has been developed in [7] and in Part C
of [2] in order to provide criteria for universality and a description of the set of universal vectors.
The point of view is very general and, so far, it was only used in the context of both universality
and universal series in [20].
The purpose of this paper is to generalize the notion of classical universal series and to provide,
like[2]foruniversalseries, arathercomplete abstracttheory ofgeneralized universal series inorder
to significantly enlarge the class of universal objects which can be understood in a systematic way.
Our theory is based on the more general under-exploited result of [2, Part C]. We introduce the
following notion of generalized universal series. Given two convenient Fr´echet spaces E and X,
2010 Mathematics Subject Classification. 30K05, 40A05, 41A10, 47A16.
Key words and phrases. universal series, approximations by polynomials, Taylor series.
1
2 S.CHARPENTIER,A.MOUZE,V.MUNNIER
a sequence (e ) E and a double-indexed sequence (x ) X we say that f E is a
k k ⊂ n,k n≥k≥0 ⊂ ∈
generalized universal series if there exists (a ) KN (K = R or C) such that f = a e E
and the set n a x , n k 0 is denskekin⊂X. This definition differs from that ofka kclakss∈ical
{ k=0 k n,k ≥ ≥ } P
universal series by the only fact that a simple sequence (x ) is replaced by a double-indexed
P n n≥0
sequence (x ) but we will see that this modification is significant. When E = KN we will
n,k n≥k≥0
refer to a generalized universal series as a formal generalized universal series.
Uptonow,veryspecificresultshavebeenprovidedtowardsthisdirection. Hadjiloucas[9]defined
extended universal series by requiring the set 1 n a x , n 0 to be dense in X, where
φ(n) k=0 k k ≥
φ is an increasing function converging in (0,+n ]. He extended the abostract results obtained for
∞ P
classical universal series in [2] but all his examples settled in RN or CN and directly came from
classical universal series. In 2012, Tsirivas [26] continued the study of [9] by producing the first
examples of extended universal Taylor series in simply connected domains. In particular, he linked
the asymptotic behavior of φ to the existence of function f holomorphic on a simply connected
domain Ω such that any entire function is the uniform limit on every compact subset in Ωc, with
connectedcomplement, ofasubsequenceof 1 n a (z ξ)k ,wherea = f(k)(ξ)/k!,ξ Ω.
φ(n) k=0 k − n k ∈
Now, this essentially reduces to study gener(cid:16)alized universal series(cid:17)where x = xk where (x ) is
P n,k φ(n) k k
a sequence in X. We also mention that Melas and Nestoridis exhibited large classes of power series
which are universal and whose Cesa`ro means are also universal [17] (more general summability
processes are considered). Notice that we unify all the previous results with our formalism.
In this paper we characterize the existence of generalized universal series in terms of easy-to-
check approximating lemmas and, using [2, Theorem 27] we obtain a description of the set of
generalized universal series in terms of genericity (G -dense and algebraic genericity). We also
δ
adapt a general criterion given in [5] for the spaceability of these sets, i.e. the existence of a closed
infinite dimensional subspace of E consisting of generalized universal series. When E = KN the
existenceofformalgeneralizeduniversalseriesreducestoaverysimplecriterionandthespaceability
canbecharacterized, extending[5,Theorem4.1](seebelowformoredetails). Theresultscontained
in [9] - in which the spaceability was not discussed - then appear as a very particular case. The
rest of the paper is dedicated to examples of generalized universal series which are connected to
classical phenomenaof convergence or universality butwhichfollow fromneither hypercyclicity nor
classical universal series.
The organization of the paper is as follows: Section 2 deals with the above mentioned abstract
theory of generalized universal series. In Section 3, we provide the first results about the genericity
of the set of extended universal series in the sense of [9], in settings different from RN or CN. A
simpleideatoproducepotentiallygeneralizeduniversalseriesconsistsinconsideringpartialsumsof
the form n a α x , where (α ) is a sequence in K = R or C, and where the sequence
k=0 k n,k k n,k n≥k≥0
(x ) induces classical universal series of the form n a x . In particular, it is motivated by
k k P k=0 k k
the natural question: how much or in which manner can we perturb a classical universal series in
P
order to keep a universal object (to get a generalized universal series)? In Section 3, we derive
from Fekete’s Theorem that, there exists a function f ∞ such that every continuous functions
∈ C
on R which vanishes at 0 can be uniformly approximated on each compact subset of R by some
partial sums of the form n f(k)(0)α xk, provided that the sequence (α ) is convergent for
k=0 k! n,k n,k n
every k 0. We prove in an abstract fashion how the algebraic and topological genericity, as well
≥ P
as the spaceability, are preserved under such a perturbation. In the same section, we also build
generalized universal series with respect to a sequence (x ) of Bernstein’s type polynomials
n,k n≥k≥0
(e.g. n xk(1 x)n−k, n k 0). We recall that this sequence is associated to any continuous
k − ≥ ≥
function g on [0,1] and the sums n n g(k/n)xk(1 x)n−k converge to g(x) uniformly on [0,1]
(cid:0) (cid:1) k=0 k −
as n + [11]. Thus, these Bernstein generalized universal series come in a natural way. Such a
→ ∞ P (cid:0) (cid:1)
simple example is again specific to the formalism of generalized universal series and do not proceed
from the classical theory.
Ineveryinstanceofclassicaluniversalseries,themainideaisthatwhatevercanbeapproximated
by polynomials can also be approximated by some partial sums of a series. Then the notion of
GENERALIZED UNIVERSAL SERIES 3
universal series cannot be but connected to approximation theory. Nevertheless, non-quantitative
approximation theorems such as Weierstrass Theorem or Mergelyan Theorem were sufficient to
construct classical universal series. In [26], the author used Bernstein’s Walsh quantitative approx-
imation theorem to build the extended universal series mentioned above. We improve his result by
showing the existence of generalized universal series induced by a sequence (x ) of the form
n,k n≥k≥0
α zk . With such a kind of “perturbations”, the situation is technically more involved.
n,k n≥k≥0
We obtain a result of the following type (see Section 4):
(cid:0) (cid:1)
Theorem. Let (α ) be non-zero complex numbers. Denote by (D) (where D stands for
n,k n≥k≥0 Uα
the open unit disk) the set of holomorphic functions f (D) such that, for every compact subset
∈ H
K C, K D = , Kc connected, every function h continuous on K and holomorphic in the
⊂ ∩ ∅
interior of K, there exists a sequence (λ ) such that λn α f(k)(0)zk h(z), uniformly on K,
n n k=0 n,k k! →
as n + . The following assertions hold:
→ ∞ P
(1) If the sequence (α ) is convergent in C for every k 0 and if we have
n,k n ≥
limsup min n α 1,
n,k
0≤k≤n | | ≥
(cid:18) (cid:26)q (cid:27)(cid:19)
then (D)= ;
α
U 6 ∅
(2) If we have (D) = then
α
U 6 ∅
limsup max n α 1.
n,k
0≤k≤n | | ≥
(cid:18) (cid:26)q (cid:27)(cid:19)
Inthecaseα = 1/φ(n)whereφ(n)convergesinC 0 ,theaboveresult,inanequivalent
n,k
\{ }∪{∞}
form, was proven by Tsirivas in [26]. Our result is more general and our proof is direct since we
are using a precise version of Bernstein-Walsh’s Theorem (see Theorem 4.7). The proof of this
result shows that having approximating polynomials is no more sufficient to construct generalized
universalseries,butwealsoneedacontrolonthedegreeoftheapproximatingpolynomialsinvolved
in the construction of the series. In addition, this control is also connected to the way in which the
perturbation φ grows along some of its subsequences. This connection clearly appears in the proof
of this theorem. This is the purpose of Section 4.
Some examples of generalized universal series also highlight other strange behaviors. In Section
5, we construct functions f holomorphic around 0 which are universal in the sense that the set
α S f(n) , n N is dense in (C), where S (g) stands for the n-th partial sum of the Taylor
n n n
∈ H
expansionofg at0and(α ) isasequenceofnonzerocomplex numbers. Whenα = 1foreveryn,
n n n
(cid:8) (cid:0) (cid:1) (cid:9)
it reminds MacLane’s result concerning the hypercyclicity of the differentiation operator in (C).
H
In general, we relate in a precise way the radius of convergence of the Taylor expansion of f at 0
to the asymptotic behavior of the sequence (α ) .
n n
2. Abstract theory of generalized universal series
2.1. Background on universality. The first part of this subsection is mostly inspired by Part
C of [2] and Section 3 of [5]. We will recall the formalism introduced in [2].
Let K = R or C. Let Y be a separable complete metrizable topological vector space (over K)
and Z a metrizable topological vector space (over K), whose topologies are induced by translation-
invariant metrics d and ̺, respectively. Let M be a separable closed subspace of Z. Let L :Y
Y n
→
Z, n N, be continuous linear mappings.
∈
Definition 2.1. With the above notations, we say that y Y is universal with respect to (L )
∈ n n
and M if
M L y : n N .
n
⊂{ ∈ }
We denote by (L ,M) the set of such universal elements.
n
U
We make the following assumption.
Assumption (I). There exists a dense subset Y of Y such that, for any y Y , (L y) converges
0 ∈ 0 n n
to an element in M.
We recall the following result from [2].
4 S.CHARPENTIER,A.MOUZE,V.MUNNIER
Theorem 2.2. (1) ([2, Theorems 26 and 27]) Under Assumption (I), the following are equivalent.
(i) (L ,M) = ;
n
U 6 ∅
(ii) For any open subset U = of Y and any open subset V = of Z with V M = there is
6 ∅ 6 ∅ ∩ 6 ∅
some n N with L (U) V = ;
n
∈ ∩ 6 ∅
(iii) For every z M and ε> 0, there exist n 0 and y Y such that
∈ ≥ ∈
̺(L y,z) < ε and d (y,0) < ε;
n Y
(iv) (L ,M) is a dense G subset of Y.
n δ
U
(2) ([2, Theorem 28 (1)]) If, for every increasing sequence (µ ) N, (L ,M) is non-empty,
n n ⊂ U µn
then (L ,M) contains, apart from 0, a dense subspace of Y.
n
U
Remark 2.3. When the sequence (L ) satisfies Condition (ii) in the above theorem, we say that
n n
(L ) is topologically transitive for M.
n n
In the classical theory of universal series, it is useful to assume that (L ) satisfies the following
n n
additional assumptions.
Assumption (II). There exist continuous linear maps U : Y Y, n N, and a continuous map
n
→ ∈
P :Z Z with P = id such that
|M M
→
For any y Y, n N, L U y PL y as m + ;
m n n
• ∈ ∈ → → ∞
For any y Y, if (L ) converges to an element in M, then U y as k + .
• ∈ nk k nk → → ∞
We use the notations L = U = 0. Under these extra assumptions, Theorem 2.2 can be
−1 −1
improved as follows. We refer to [2, Theorem 29] and [2, Remark 27 (a)].
Theorem 2.4. Under Assumptions (I) and (II), the following assertions are equivalent.
(1) (L ,M) = ;
n
U 6 ∅
(2) (L ,M) is a dense G subset of Y;
n δ
U
(3) For every z M and ε > 0, there are n 0 and y Y such that
∈ ≥ ∈
̺(L y,z) < ε and d (U y,0) < ε;
n Y n
(4) For every z M, ε> 0, and p 1, there are n > p and y Y such that
∈ ≥ − ∈
̺(L y L y,z) < ε and d (U y U y,0) < ε;
n p Y n p
− −
(5) For every increasing sequence (µ ) N, (L ,M) is a dense G subset of Y;
n n ⊂ U µn δ
(6) For every increasing sequence (µ ) N, (L ,M) contains, apart from 0, a dense subspace
n n ⊂ U µn
of Y.
The question of the spaceability of the set of universal elements has always been considered
separately. Actually, it is more involved in general because the condition (L ,M) = does not
n
U 6 ∅
always imply that (L ,M) is spaceable. As far as we know, the most general criterion for the
n
U
spaceability of the universal set is Proposition 3.7 in [5], given when M = Z. This result naturally
extends to any subspace M Z. To state this latter, we introduce a condition.
⊂
Definition 2.5. With the above notations, assume that Y is a Fr´echet space. The sequence (L )
n n
satisfies Condition (C) for M if there exist an increasing sequence (n ) N and a dense subset
k k ⊂
Y Y such that
1
⊂
For all y Y , L y 0, as k + ;
• ∈ 1 nk → → ∞
For every continuous semi-norm p on Y, M T ( y Y : p(y) < 1 ).
• ⊂ ∪k≥0 nk { ∈ }
Condition (C) has been introduced in [13] for sequence of operators between Banach spaces, and
extended in [19] in the case of Fr´echet spaces, when M = Z. In practice, Condition (C) is not quite
easy to handle. In fact, it is closely related to a more common notion, that of mixing sequence of
operators for a subspace.
Definition 2.6. With the above notations, we say that the sequence (L ) is mixing for M if the
n n
following condition is satisfied: for any open subset U = of Y and any open subset V = of Z
6 ∅ 6 ∅
with V M = there is some n N such that L (U) V = for any n N.
n
∩ 6 ∅ ∈ ∩ 6 ∅ ≥
GENERALIZED UNIVERSAL SERIES 5
Thefollowingimmediateproposition(seeforexample[2,Remark27(b)])reformulatesthenotion
of mixing sequence of operators for M.
Proposition 2.7. Withtheabovenotations, ifforeveryincreasing sequence(µ ) N, (L ,M)
n n ⊂ U µn
is non-empty, then the sequence (L ) is mixing for M.
n n
A straightforward adaptation of [5, Proposition 3.5] yields the following chains of implications:
Mixing for M = Condition (C) for M = Topologically transitive for M.
⇒ ⇒
Now, [19, Theorem 1.11 and Remark 1.12] extend to universality for asubspace, usingCondition
(C) for a subspace, as follows. The proof is, up to very slight changes, quite identical to that of
Menet’s result, and so it is omitted.
Proposition 2.8. With the above notations, we assume that Y is a Fr´echet space with continuous
norm. Let (p ) be a non-decreasing sequence of norms and (q ) a non-decreasing sequence of
n n n n
semi-normsdefiningthetopologies ofY andZ respectively. Itthe sequence(L ) satisfiesCondition
n n
(C) for M and if there exists a non-increasing sequence of infinite dimensional closed subspaces
(M ) of Y such that for every continuous semi-norm q on Z, there exists a positive number C, an
j j
integer k 1 and a continuous norm p on Y such that we have, for any j k and any x M
j
≥ ≥ ∈
q L (x) Cp(x),
nj ≤
then (L ,M) is spaceable.
n (cid:0) (cid:1)
U
2.2. Generalized universal series. In this paper, we are interested in generalizing the classical
notion of universal series studied in [2]. Let E, A KN and X be three Fr´echet spaces whose
⊂
topologies are defined by translation-invariant metrics d , d and ̺, respectively. Let (e ) denote
E A k k
the canonical sequence in KN. For p a polynomial in KN, i.e. a finite linear combination of the e ’s,
k
we denote by v(p) its valuation (i.e. v(p) = min(k;a = 0) where p = a e ) and by d(p) its
k 6 k≥0 k k
degree (i.e. d(p) = max(k;a = 0) where p = a e ). Let = (f ) be sequence in E and
k 6 k≥0 k k F kPk≥0
let = (x ) be a sequence in X. We make the following assumptions.
X n,k n≥k≥0 P
(i) The set G KN of all polynomials is included in A and is dense in A;
⊂
(ii) The coordinate projections A K, a = (a ) a , k 0, are continuous;
→ n n 7−→ k ≥
(iii) Theset G E of all polynomials in E (i.e. the finitelinear combinations of the f ’s) is dense
E k
⊂
in E;
(iv) There exists a linear continuous map T : E A such that T (f ) = e for every k 0;
0 0 k k
→ ≥
(v) For every k 0, the sequence (x ) is convergent in X.
≥ n,k n≥k
We also denote by SF : A E (resp. SX : A X) the map which takes (a ) to n a f
(resp. n a x ). n → n → n n k=0 k k
k=0 k n,k P
RemaPrk 2.9. One can observe that (iv) implies that the family (fk)k is linearly independent.
Definition 2.10. Wekeeptheaboveassumptions. Letµ = (µ ) Nbeanincreasingsequence.
n n≥0 ⊂
(1) An element f E is a µ-generalized universal series (with respect to (x )) if
n,k
∈
X = SX T (f): n 0 .
µn ◦ 0 ≥
We denote by µ( ) E the set of(cid:8)such elements. (cid:9)
U X ∩
(2) An element f E is a µ-restricted generalized universal series (with respect to (f ), (x )) if
k n,k
∈
X 0 SX T (f),f SF T (f) : n 0 .
×{ } ⊂ µn ◦ 0 − µn ◦ 0 ≥
µ
We denote by ( ) the set o(cid:8)f(cid:0)such elements. (cid:1) (cid:9)
UE X
Remark 2.11. (1) Observe that the indices n and k in (x ) do not play the same role in
n,k n≥k≥0
the definition of generalized universal series.
(2) When x = x for every n 0 and every 0 k n, we recover the standard notion of
n,k k
≥ ≤ ≤
universal (resp. restricted universal) series with respect to (x ) .
k k
(3) When (f ) is a Schauder basis of E, we automatically have µ E( )= µ( ).
k k U ∩ X UE X
6 S.CHARPENTIER,A.MOUZE,V.MUNNIER
Notation 2.12. When µ = N, we simply denote µ( ) E (resp. µ( )) by ( ) E (resp.
U X ∩ UE X U X ∩
( )). When x = x for every n 0 and every 0 k n, we use the standard notations
E n,k k
Uµ XE and µ todenote thesetof classi≥cal universalserie≤s and≤that ofclassical restricted universal
U ∩ UE
series respectively [2].
The next result characterizes the existence of formal generalized universal series (i.e. when E =
A = KN).
Proposition 2.13. Let µ = (µ ) N be an increasing sequence. Under the previous notations
n n ⊂
and assumptions, the following assertions are equivalent.
(1) µ( ) KN = ;
U X ∩ 6 ∅
(2) For all K 0, span(x , K k n) is dense in X;
n,k
≥ ≤ ≤
n∈µ
[
(3) For all N K 0, span(x , K k n) is dense in X.
n,k
≥ ≥ ≤ ≤
n≥N
[
n∈µ
Proof. (2) (3): The implication (3) (2) is obvious. Then, we assume that (2) is satisfied. Let
⇐⇒ ⇒
(K,N) N2, K N, be fixed. By (2),
∈ ≤
X = span(x , N k n)
n,k
≤ ≤
n∈µ
[
= span(x , N k n)
n,k
≤ ≤
n≥N
[
n∈µ
span(x , K k n),
n,k
⊂ ≤ ≤
n≥N
[
n∈µ
which gives the conclusion.
(1) (2): Let(U ) beadenumerablebasisofopensetsinX andletusfixj . LetalsoK 0be
⇒ j j≥0 0 ≥
fixed. Let a = (a ) µ( ). Let p = (a ,a ,...,a ,0,...) and p = K a x . By assump-
n n ∈ U X 0 1 K n k=0 k n,k
tion (v) above, p has a limit in X when n tends to + . Hence a p= (0,...,0,a ,...,a ,...)
n K n
µ( ) also. Therefore, there exists n K, n µ, su∞ch that n− a xP U , which gives (2)∈.
U X ≥ ∈ k=K k n,k ∈ j0
(2) (1): The assumption allows to build a sequence of polynomials (p ) with d(p ) µ for
⇒ P j j≥0 j ∈
any j 0, v(p ) > d(p ), and such that SX (p ) U j−1SX (p ). Then the (block)
≥ j j−1 d(pj) j ∈ j − i=0 d(pi) i
sequence (p ) KN belongs to µ( ) KN. (cid:3)
j j ∈ U X ∩ P
Remark 2.14. When x = x for every n 0 and every 0 k n, we recover that KN is
n,k k
≥ ≤ ≤ U ∩
non-empty if and only if span(x , k n) is dense in X for every n 0.
k
≥ ≥
We are interested in a description of the set of generalized universal series. To this purpose, the
results of Subsection 2.1 are very useful. With the notations of this latter, we will consider the two
following settings, depending on what we are considering: generalized universal series or restricted
generalized universal series. We fix an increasing sequence µ = (µ ) N.
n n ⊂
(A) For generalized universal series:
A-(a) Y = E, Z = M = X;
A-(b) L = SX T , n N;
n µn ◦ 0 ∈
A-(c) Y = G , where we recall that G is the set of all finite combinations of the f ’s.
0 E E k
(cid:0) (cid:1)
(B) For restricted generalized universal series:
B-(a) Y = E, Z = X E and M = X 0
× ×{ }
B-(b) L = SX T ,id SF T , n N;
n µn ◦ 0 E − µn ◦ 0 ∈
B-(c) Y = G .
0 E
(cid:0) (cid:1)
It is easily checked that Assumption (I) (see above) is satisfied under (A) or (B). The main result
of this section states as follows.
Theorem 2.15. Let µ = (µ ) N be an increasing sequence.
n n ⊂
GENERALIZED UNIVERSAL SERIES 7
(A) For generalized universal series, we have:
(A)-(1) The following assertions are equivalent.
(i) µ( ) E = ;
U X ∩ 6 ∅
(ii) For every x X and every ε > 0, there exist n 0 and f E such that
∈ ≥ ∈
̺ SX T (f),x < ε and d (f,0) <ε;
µn ◦ 0 E
(cid:0) (cid:1)
(iii) µ( ) E is a dense G subset of E.
δ
U X ∩
(A)-(2) If, for every increasing sequence ν = (ν ) µ, ν( ) E is non-empty, then
n n ⊂ U X ∩
µ( ) E contains, apart from 0, a dense subspace of E.
U X ∩
(A)-(3) If E admits a continuous norm and if, for every increasing sequence ν = (ν ) µ,
n n ⊂
ν( ) E is non-empty, then µ( ) E is spaceable.
U X ∩ U X ∩
(B) For restricted generalized universal series, we have:
(B)-(1) The following assertions are equivalent.
µ
(i) ( )= ;
UE X 6 ∅
(ii) For every x X and every ε> 0, there exist n 0 and f E such that
∈ ≥ ∈
̺ SX T (f),x < ε and d SF T (f),0 < ε;
µn ◦ 0 E µn ◦ 0
(iii) µ( ) (cid:0)is a dense G s(cid:1)ubset of E. (cid:0) (cid:1)
(B)-(2) If, foUrEevXery increasing seδquence ν = (ν ) µ, ν ( ) is non-empty, then µ( )
n n ⊂ UE X UE X
contains, apart from 0, a dense subspace of E.
(B)-(3) If E admits a continuous norm and if, for every increasing sequence ν = (ν ) µ,
ν ( ) is non-empty, then µ( ) is spaceable. n n ⊂
UE X UE X
Proof. Parts (A)-(1), (A)-(2), (B)-(1) and (B)-(2) are direct applications of Theorem 2.2. We just
mention that, to get (B)-(1)-(ii), we use the density of polynomials (with respect to the f ’s) in E
k
and the continuity of the maps SF T , n 0.
µn ◦ 0 ≥
It remains to prove (A)-(3) and (B)-(3). Both of these assertions will proceed from Proposition
2.8. By Proposition 2.7, (L ) is mixing for M in both cases (in case (A), M = X, in case (B),
n n
M = X 0 ). In particular, it satisfies Condition (C) for some increasing sequence (n ) N.
For j ×0,{let}M = nj kerSX T . Every M is an infinite dimensional closed subspakcek o⊂f E,
≥ j ∩k=0 k ◦ 0 j
and (A)-(3) immediately comes from an application of Proposition 2.8. For (B)-(3), we just need
to observe that for any seminorm q of the Fr´echet space X E, there actually exists a continuous
×
norm p of E such that for any f M ,
j
∈
q L (f) = q 0,f SF(f) Cp(f),
nj − nj ≤
(cid:16) (cid:17)
for any j 0, by continuity of(cid:0)SF. (cid:1) (cid:3)
≥ nj
Remark 2.16. (1) When the sequence (f ) is a Schauder basis of E, we shall say that Part A
k k
and Part B of the previous theorem coincide.
(2) Using that the set of all finite combinations of the f ’s is dense in E, it easily stems that
k
Assertion (A)-(1) (ii) can be rephrased as follows, using the fact that the generalized universal
property is preserved under translation by polynomials:
(A)-(1) (ii)’ For every x X and every ε> 0, there exist m n 0 such that
∈ ≥ ≥
̺ SX T (f),x < ε and d SF T (f),0 < ε.
µn ◦ 0 E µm ◦ 0
On the one hand, if x(cid:0) = x for ev(cid:1)ery n 0 a(cid:0)nd every 0 k(cid:1) n, then we observe that
n,k k
≥ ≤ ≤
Assumption (II) in Subsection 2.1 is satisfied in the setting of restricted universal series, that is
under (B). Therefore, Theorem 2.4 immediately yields [2, Theorem 1] which asserts, in particular,
that if is non-empty, then it automatically contains, apart from 0, a dense subspace. In this
E
U
context, [18] also ensures that if E admits a continuous norm and if is non-empty, then is
E E
U U
spaceable. On the other hand, the formalism of generalized universal series is too much general to
proceed from Theorem 2.4 and [18] (or [4]).
8 S.CHARPENTIER,A.MOUZE,V.MUNNIER
2.3. Spaceability of generalized universal series in KN. Let us return in this section to the
space KN = RN or CN endowed with the Cartesian topology. It is well known that KN does not
admit a continuous norm, so the spaceability of the set of generalized universal series in KN cannot
proceed from Theorem 2.15. We keep the previous notations: let X be a metrizable vector space
over the field K = R or C. Let us denote = (x ) a fixed sequence of elements in X. In
X n,k n≥k≥0
this context we always have KN( ) = ( ) KN. We are interested in the spaceability of the set
U X U X ∩
( ) KN. In the case of classical universal series, a complete answer is given by Theorem 4.1
U X ∩
of [5]. The general case seems much more complicated. In the next proposition, we give sufficient
conditions for the spaceability (resp. the non spaceability) of ( ) KN, which will cover the new
U X ∩
examples of generalized universal sets. First, for every l 1, let us introduce the sets
≥
E = b X; (n,m) N2 with n > m l and b ,b ,...,b ,...,b K such that
l l l+1 m n
{ ∈ ∃ ∈ ≥ ∈
m m n
b = b x and b x = b x
j m,j j n,j j n,j
}
j=l j=l j=m+1
X X X
and
T = b X; (n,m) N2 with n > m l and b ,b ,...,b ,...,b ,b , K such that
l l l+1 m n n+1
{ ∈ ∃ ∈ ≥ ··· ∈
m m k
b = b x and for every k n, b x = b x
j m,j j k,j j k,j
≥ }
j=l j=l j=m+1
X X X
We are ready to state the following propositions.
Proposition 2.17. If for every l 1 the set T is dense in X, then ( ) KN is spaceable.
l
≥ U X ∩
Proof. Assume that, for every l 1, the set T is dense in X. In particular, X is separable and we
l
≥
can consider a countable basis of open sets (U ) in X. We want to construct a sequence (u )
p p p≥0
in KN such that the sequence of valuations (v(u )) is strictly increasing and all the non-zero
p p≥0
elements α u are universal. Using the hypothesis, for any p 0, any l 0, there exist
p≥0 p p ≥ ≥
b KN, m l, n m and b ,...,b ,b , K such that
∈ P≥ ≥ l n n+1 ··· ∈
m k
l v(b), SX(b) U and k n b x = b x .
≤ m ∈ p ∀ ≥ j k,j j k,j
j=l j=m+1
X X
Define the sequence a = (0,...,0,b ,...,b , b ,..., b , b ,...). Therefore we have
l m m+1 n n+1
− − −
l v(a), SX(a) U and k n SX(a) = 0.
≤ m ∈ k ∀ ≥ k
We finish the proof along the same lines as that of (2) (1) of [5, Theorem 4.1]. (cid:3)
⇒
Proposition 2.18. Assume that X admits a continuous norm . Then, if there exists l 1 such
k·k ≥
that E is not dense in X, then ( ) KN is not spaceable.
l
U X ∩
Proof. First assume that there exists l 0 such that the set E is not dense in X. Therefore
l
≥
there exists an open set U X such that U E = . Suppose that there exists an infinite
l
dimensional closed subspace ⊂F in (X) KN. O∩ne can fi∅nd a sequence (u(n)) in F 0 such
n≥0
U ∩ \{ }
that (v(u(n))) is strictly increasing and v(u(0)) l [4, Lemma 5.1]. Since for every n 0, u(n)
n≥0
≥ ≥
is a universal element, there exists m v u(n) such that SX (u(n)) = mn u(n)x U. If
n ≥ mn j=l j mn,j ∈
there exists k > m such that S (u(n)) = 0, then we have mn u(n)x = k u(n)x ,
n k 6 (cid:0) (cid:1) j=l j k,j P− j=mn+1 j mk,j
i.e. u(n) E , which is impossible. Thus we have SX(u(n)) = 0 for all k > m . We finish the proof
along the∈saml e lines as that of (1) (2) of [5, Thekorem 4.16P]. n P (cid:3)
⇒
Example 2.19. (1) When x = (1/ϕ(n))x for every n 0, where ϕ is an increasing function
n,k k
≥
N K 0 converging in K 0 , i.e. in the case of extended universal series [9] (see
→ \ { } \ { } ∪{∞}
Example 2.24 (1)), observe that the set T is dense in X if and only of the set E is dense in X.
l l
In particular, we obtain a characterization of the spaceability of sets of extended universal series
in KN. For example, Theorem 2.15 ensures that there exists a universal sequence a = (a )
n n≥0
GENERALIZED UNIVERSAL SERIES 9
of real numbers such that the set of all partial sums 1 n a is dense in R and that the set of
n j=0 j
such sequences is spaceable. Moreover it is easy to check that all the examples of sets of extended
P
universal series given in [9] are not spaceable.
(2)Furtherusingthefactthatthereexists auniversalsequencea = (a ) ofrealnumberssuch
n n≥0
that the set of all partial sums 1 p−1a , when p is a prime number, is dense in R, observe that
p j=0 j
we can construct a Lebesgue-measPurable function f such that its Riemann sums n1 jn=−01f nj
n
are dense in R. To do this, it suffices to set f j = a , for j = 0,...,p 1, if p i(cid:16)s aPprime nu(cid:16)m(cid:17)b(cid:17)er,
p j −
and f(t)= 0 otherwise. Observe that f = 0 L(cid:16)eb(cid:17)esgue almost-everywhere.
(3) Theorem 2.15 ensures that there exists a universal sequence a = (a ) of real numbers
n n≥0
such that the set of all partial sums n aj is dense in R and Proposition 2.17 ensures that the
j=0 n+j
set of such sequences is spaceable. This example is not covered by the extended abstract theory.
P
Furthernotice that it is possibleto argueas in theabove assertion (2) to findaLebesgue-integrable
function t f(t) such that the sequence of all its Riemann sums n−1 f(nj) is dense in R.
7→ 1+t j=0 j+n
(cid:18) (cid:19)n
(4) If for every integer n the family (xn,k)n≥k≥0 is a free family,Pthen Proposition 2.18 ensures
that the set ( ) KN is not spaceable.
U X ∩
2.4. A large class of generalized universal series. The easier way to producegeneralized uni-
versal series is the following. Let = (x ) be a sequence in X and let α = (α )
X k k≥0 n,k n≥k≥0
and β = (β ) be two sequences of non-zero elements of K, such that for any k 0,
n,k n≥k≥0 ≥
the sequences (α ) and (β ) are convergent in K. Roughly speaking, the next proposi-
n,k n n,k n
tion asserts that, whenever there exists a formal generalized universal series with respect to
α = (x ) := (α x ) , there also exists a formal generalized universal series with
X n,k n≥k≥0 n,k k n≥k≥0
respect to β = (x ) := (β x ) .
X n,k n≥k≥0 n,k k n≥k≥0
Proposition 2.20. Let µ N and ν N be two increasing sequences of natural numbers. The set
µ(α ) KN is non-empt⊂y if and on⊂ly if the set ν(β ) KN is non-empty.
U X ∩ U X ∩
Proof. Without loss of generality, it suffices to prove that µ(α ) KN = implies that ν(β )
U X ∩ 6 ∅ U X ∩
KN = . Let (U ) be a denumerable basis of open sets in X and let us fix j . Let also K 0.
6 ∅ j j≥0 0 ≥
According to Proposition 2.13, there exists n K, n µ, such that n a α x U . Let
m be an integer in ν such that m n. Then w≥e have ∈ m a αn,kβ xk=K Uk ,n,wkitkh∈a =j0 0 for
≥ k=K kβm,k mP,k k ∈ j0 k
every n <k m. By Proposition 2.13, we deduce that ν(β ) KN = . (cid:3)
P
≤ U X ∩ 6 ∅
Denoting by µ the set of formal classical universal series (i.e. µ = µ(β ) with β = 1 for
n,k
U U U X
every n k 0), we immediately deduce the following corollary, using Theorem 2.15.
≥ ≥
Corollary 2.21. Let µ N be an increasing sequence. The following assertions are equivalent.
⊂
(1) µ KN is non-empty;
U ∩
(2) For every increasing sequence ν N, ν(α ) KN is non-empty;
⊂ U X ∩
(3) For everyincreasing sequence ν N, ν(α ) KN isadense G -subsetof KN and contains,
δ
⊂ U X ∩
apart from 0, a dense subspace.
Remark 2.22. In view of Proposition 2.20, given two sequences α and β, we can wonder whether
the following equality holds:
µ(α ) KN = µ(β ) KN?
U X ∩ U X ∩
Weshowwithanexamplethattheanswertothisquestionisnegative. Withthepreviousnotations,
we take X = R and we consider a sequence = (x ) R 0 . Let q :N Q a one-to-one
X n,k n≥k≥0 ⊂ \{ } →
and onto sequence such that q = 0. We define a sequence (a ) R as follows: a = q and if
0 6 k k≥0 ⊂ 0 0
we assume that a ,...,a have been built, we define inductively a so that
0 n n+1
n+1
q = a x .
n+1 k n+1,k
k=0
X
10 S.CHARPENTIER,A.MOUZE,V.MUNNIER
Hence the numbers a , k 0, are defined in order to have
k
≥
n
q = a x for every n= 0,1,...,
n k n,k
k=0
X
so that the sequence ( n a x ) is dense in R, and then (a ) ( ) RN.
k=0 k n,k n n n ∈U X ∩
Notice that there are infinitely many non-zero a . We can define the sequence β = (β )
P k n,k n≥k≥0
in R as follows: For every n N and 0 k n, we set β = 0 if a = 0 and β = 1/a
n,k k n,k k
otherwise. Therefore n a β∈ N for≤every≤n N so, taking x = 1 for every 0 k n, we
k=0 k n,k ∈ ∈ n,k ≤ ≤
get ( ) RN = (β ) RN.
U X ∩ 6 U PX ∩
In the case E = A = KN we also have the following.
Proposition 2.23. Letµ Nbe anincreasing sequenceand let := (x ) and := (y )
⊂ X n,k n≥k≥0 Y k k≥0
be such that x y as n + for every k 0. If µ( ) KN = then µ( ) KN = but
n,k k
→ → ∞ ≥ U Y ∩ 6 ∅ U X ∩ 6 ∅
the converse does not hold.
Proof. The first assertion is obvious by property (2) of Proposition 2.13, the fact that x y as
n,k k
→
n + for every k 0 and the triangle inequality.
→We n∞ow prove that≥ µ( ) KN = does not necessarily imply µ( ) KN = . Observe that
U X ∩ 6 ∅ U Y ∩ 6 ∅
if y = 0 for every k 0 except a finite set of natural numbers then by property (2) of Proposition
k
≥
2.13 we have µ( ) KN = . To obtain the desired conclusion, it suffices to construct a sequence
U Y ∩ ∅
= (x ) where we have µ( ) KN = . So, using the previous notations, we take X = R,
X n,k n≥k≥0 U X ∩ 6 ∅
endowed with the usual topology, and K = R. Let q : N Q be a sequence that is one-t-one and
→
onto. We inductively define the sequence λ := (λ ) R as follows:
n n
⊂
λ +λ + +λ
0 1 n
λ = q and, for n= 1,2,..., q = ··· .
0 0 n
n
For every n = 1,2,..., we set
1
x = , k = 0,1,2,...,n.
n,k
n
Observe that we have
n
λ x = q ,
j n,j n
j=0
X
which implies that the sequence ( n λ x ) is dense in R. It follows that ( ) RN = and
j=0 j n,j n U X ∩ 6 ∅
since x 0, as n + , for every k N, the proof is complete.
n,k
→ → ∞ P ∈ (cid:3)
Example 2.24. (1) α = 1/ϕ(n) for every n 0, where ϕ is an increasing function N K 0
n,k
≥ → \{ }
converging in K 0 . As we already said, this class of examples has been studied in [9].
\{ }∪{∞}
His main result [9, Theorem 3.1] is a consequence of the more general Theorem 2.15 and Corollary
2.21. Also observe that an application of Proposition 2.20 to this case yields [9, Theorem 5.1].
n k+1
(2) α = − : it corresponds to consider the sequence of Cesa`ro means (M ) of the
n,k n n n≥0
partial sum operators S :
n
S +...+S
1 n
M = .
n
n
In particular, if the sequence of operators (S ) is universal, then the sequence (M ) is also
n n n n
universal. This result is a version for generalized universal series of Cesa`ro hypercyclicity (see [6]).
The examples given in this subsection are quite natural, but they live in KN (except Example
2.19 (2)), which is a strong limitation to produce interesting examples. In the next sections, we
will exhibit examples of more sophisticated (restricted) generalized universal series, some living in
spaces different from KN, and inducing surprising properties (see Sections 4 and 5 especially).