On Billard’s theorem for random Fourier series
Guy Cohen
Christophe Cuny
and
Abstract
In this note we show that Billard’s theorem on a.s. uniform convergence of random
Fourier series with independent symmetric coefficients is not true when the coefficients
are only assumed to be centered independent. We give some necessary or sufficient
conditions to ensure the validity of Billard’s theorem in the centered case.
1
Introduction
In this paper we deal with a.s. uniform convergence of random Fourier series. Let Γ be
the unit circle, and denote by C(Γ) the Banach space of continuous functions on Γ, with
the sup-norm that we denote by k · kC(Γ) . Given a sequence of independent centered
random variables {Xn } defined on
space (Ω, P), one wonders whether, for
Pa probability
n
P almost every x ∈ Ω, the series ∞
X
(x)λ
is
uniformly convergent on Γ.
n
n=1
Since the early study of Paley and Zygmund [14], this matter benefited from many
works by Salem and Zygmund [16], Billard [1], Kahane [9], Marcus [11], Cuzick and Lai
[2], Marcus and Pisier [12] [13], Talagrand [17], Fernique [4], and Weber [18].
Most of the time, the sequence {Xn } is first assumed
P∞ to be symmetric, so that, by a
theorem of Billard [1], it suffices to prove that, a.s., n=1 Xn (x)λn represents a continuous function. Then, one may hope to reach the general centered case by symmetrization.
Our purpose is to prove that this procedure may fail sometimes. We show that
Billard’s theorem is no longer true when the independent sequence {Xn } is only assumed
to be centered. To illustrate our result, we recall some more or less known moment
conditions which ensure the validity of Billard’s theorem, and give also necessary or
sufficient conditions.
2
Conditions for validity of Billard’s theorem
Let us first recall Billard’s theorem, as it appears in Kahane [9, Theorem 3, p. 58].
Theorem 2.1 (Billard). Let {Xn } be a sequence of independent symmetric complex
valued random P
variables. Then the following conditions are equivalent for the random
n
Fourier series ∞
n=1 Xn λ :
(i) Almost surely the series represents a bounded function.
(ii) Almost surely the series represents a continuous function.
(iii) Almost surely the series is convergent at every point.
(iv) Almost surely the series is uniformly convergent.
We would like to know what happens in the centered case. We discuss conditions
which allow to proceed by symmetrization. We denote by LR1 (P, C(Γ)) the Banach space
of all C(Γ)-valued random variables ξ on Ω with the norm Ω kξkC(Γ) dP. The following
result is a simple combination of a theorem of Itô and Nisio [7] (see also Ledoux and
Talagrand [6, Theorem 6.1]), and a theorem of Hoffman-Jørgensen [5] (see also Jain and
Marcus [8]).
0
AMS 2000 subject classifications. Primary: 42A61,60G17; Secondary: 60G50.
Key words and phrases. Banach valued random variables, random Fourier series, almost sure
uniform convergence, Billard’s theorem
1
Theorem 2.2. Let {Xn } ⊂ L1 (P) be a sequence of independent complex valued centered random variables on (Ω, P). Assume that E(supn≥1 |Xn |) < ∞. Let {Xn0 } be an
independent copy of {Xn }, defined on (Ω0 , P0 ). The following conditions are equivalent:
P∞
(i) Almost surely the series n=1 Xn λn converges in C(Γ).
P
0
n
(ii) Almost surely in Ω × Ω0 the series ∞
n=1 (Xn − Xn )λ converges in C(Γ).
P∞
0
n
(iii) The series n=1 (Xn − Xn )λ converges in L1 (P ⊗ P0 , C(Γ)).
P
n
(iv) The series ∞
n=1 Xn λ converges in L1 (P, C(Γ)).
Pn
Furthermore, if any of the above holds, then supn≥1 max|λ|=1 | k=1 (Xk − Xk0 )λk |
Pn
and supn≥1 max|λ|=1 | k=1 Xk λk | are integrable.
Proof. Clearly (i) ⇒ (ii). By assumption E(supn≥1 |Xn − Xn0 |) < ∞, hence (ii) ⇒ (iii)
by [5, Corollary 3.3]. The implication (iii) ⇒ (iv) follows by independence and convexity
(see [12, Lemma 4.2, p. 42]). The implication (iv) ⇒ (i) follows by Itô and Nisio theorem
[7]. If one of the conditions (i)−(iv) holds, then the integrability of the maximal functions
follow by [5, Corollary 3.3].
Remark. In the above theorem, if we assume that {Xn } ⊂ Lp (P) for some 1 ≤ p <
∞, and E(supn≥1 |Xn |p ) < ∞, then we have, in fact, convergence in Lp (P, C(Γ)) and
Lp (P, C(Γ))-integrability of the maximal functions (see [5, Corollary 3.3]).
The next lemma is a simple case of Lemma 3.1 in [8].
Lemma 2.3. Let {Xn } ⊂ L1 (P) bePa sequence of independent random variables. Then
∞
E(supn≥1 |Xn |) < ∞ if and only if n=1 E(|Xn |1{|Xn |≥a} ) < ∞ for some a > 0.
Corollary 2.4. Let 1 < p < 2, and let {an } be a sequence of complex numbers, such
p 1/p
P ∞ (P ∞
k=n |ak | )
< ∞. Let {Xn } ⊂ L1 (P) be a sequence of centered independent
n=1
n(log n)1/p
that
random variables. Assume that forPsome M > 0 we have P(|Xn | ≥ u) ≤ uMp , for every
n
u ≥ 1 and n ≥ 1. Then the series ∞
n=1 an Xn λ is a.s. uniformly convergent.
Proof. It follows from Fernique [4] (see also Marcus and Pisier [13], Theorem B and the
discussion in Remark 4.4) , that the result is true for symmetric independent {Xn }.
Now, by the condition on the tails of the variables {Xn }, we have, for an 6= 0,
Z ∞
p
E(|an Xn |1{|an Xn |≥1} ) = |an |
P(|Xn | > u)du + P(|an Xn | ≥ 1) ≤
M |an |p .
p−1
1/|an |
Hence, by Lemma 2.3 we have E(supn≥1 |an Xn |) < ∞. So, Theorem 2.2 yields the
result.
Remark. In a similar way, the results of Fernique for 2 ≤ p < ∞ (see [4, Examples
(d) and (e)]) can be extended to the centered case.
Corollary 2.5. Let 1 < p ≤ 2, and let {Xn } ⊂ Lp (P) be a sequence of centered in∞ P∞
X
( k=n E|Xk |p )1/p
dependent random variables. If
< ∞, in particular, if for some
n(log n)1/p
n=2
P∞
> 0 the
series n=2 E|Xn |p (log n)p−1 (log log n)p+ converges, then the random Fourier
P∞
series n=1 Xn λn is a.s. convergent in C(Γ).
Proof. First we prove the case p = 2. By Theorem 5.1.5 of Salem and Zygmund [16],
and by the principle of reduction [9, p. 9] (see also [12, Ch. VII, §1]), we obtain the
assertion for the case where {Xn } are symmetric random variables. Since the condition
of the theorem implies E(supn≥1 |Xn |p ) < ∞, the non-symmetric case follows from the
symmetrization argument of Theorem 2.2.
2
In the case 1 < p < 2 we do the following: for Xn 6= 0 put an = (E|Xn |p )1/p
and Yn = Xn /(E|Xn |p )1/p . Otherwise put an = 0 and Yn = 0. Clearly, we have
P(|Yn | > u) ≤ u1p . By application of Corollary 2.4 to {an } and {Yn }, we obtain the
result.
Remark. Under the condition of the corollary above, we have E(supn≥1 |Xn |p ) < ∞.
Hence, Theorem 2.2 yields also convergence in Lp (P, C(Γ)) and Lp -integrability of the
maximal function (see the remark after Theorem 2.2).
Corollary
2.6 (Cuzick and Lai, [2]). Let {Z
n } be centered i.i.d. random variables, with
R
P∞
+
Zn n
|Z
|
log
|Z
|
dP
<
∞.
Then
the
series
1
1
n=1 n λ is a.s. uniformly convergent.
Ω
Proof. It was provedRin Cuzick and Lai [2] and Talagrand [17], that for a symmetric i.i.d.
sequence {Zn } with Ω |Z1 | log+ log+ |Z1 | dP < ∞, the conclusion of the corollary holds.
R
Now, since Ω |Z1 | log+ |Z1 | dP < ∞ we have,
Z
∞
∞
X
X
1{|Z1 |≥n}
|Zn |
E
1{|Zn |≥n} ≤
|Z1 |
dP ≤
n
n
Ω
n=1
n=1
Z
[|Z1 |]+1
Ω
|Z1 |
X
n=1
1
dP ≤
n
Z
Ω
|Z1 | log(|Z1 | + 1)dP < ∞ ,
where [|Z1 |] denotes the integral part of |Z1 |. Hence, Lemma 2.3 and Theorem 2.2 yield
the result.
Remarks. 1. Corollary 2.6 is already proven in [2]. We gave it to illustrate the
use of Theorem 2.2, and also because the statement of this corollary is very close to
the situation of the counterexample given in Theorem 3.1 below. Indeed, modulo a
rearrangement, the sequence {an } defined in the proof of Theorem 3.1 is essentially the
sequence { n1 }, and by Lemma 2.3, the condition E(sup |Znn | ) < ∞ is actually equivalent
R
R
to |Z1 | log+R |Z1 |dP < ∞. In Theorem 3.1 we have |Z1 |(log+ |Z1 |)1− dP < ∞ for every
> 0, while |Z1 | log+ |Z1 |dP = ∞.
R
2. The optimality of the condition |Z1 | log+ |Z1 |dP < ∞ is also justified by the
discussion in Marcinkiewicz and Zygmund [10, p. 78].
{Zn } are symmetric i.i.d., Talagrand [17] proved that the condition
R 3. When
|Z1 | log+ log+ |Z1 |dP < ∞ is necessary and sufficient for the a.s. uniform convergence
P∞
n
of n=1 Znnλ .
Using Billard’s theorem and the symmetrization argument of Theorem 2.2, we obtain
the following extension of Billard’s theorem in the centered case.
Theorem 2.7. Let {Xn } ⊂ L1 (P) be a complex sequence of centered independent random
variables. If E(supn≥1 P
|Xn |) < ∞, then the following conditions are equivalent for the
n
random Fourier series ∞
n=1 Xn λ :
(i) Almost surely the series represents a bounded function
(ii) Almost surely the series represents a continuous function
(iii) Almost surely the series converges everywhere.
(iv) Almost surely the series converges uniformly.
P
n
Definition 2.1. We say that the random Fourier series ∞
n=1 Xn λ has property B, if
conditions (i) − (iv) of Theorem 2.7 are all equivalent, i.e., either all of them hold or all
of them false.
3
Pn
Remark.
The condition supn≥1 k k=1 Xk λk kC(Γ) < ∞ a.s. yields that, almost
Pn
surely, k=1 Xn λn represents a bounded function (see [19, Theorem 4.2(ii), p. 136]).
Hence, under the assumption E(supn≥1 |Xn |) < ∞ the assertions (i) − (iv) are all equivP
alent to supn≥1 k nk=1 Xk λk kC(Γ) < ∞ a.s., for centered independent variables. In
the symmetric case, it remains true without the assumption E(supn≥1 |Xn |) < ∞, by
Billard’s theorem (see also the discusssions in [9, Theorem 1, p.13]).
Here we give a different type of moment condition.
Corollary 2.8. Let {Xn } ⊂ L4 (P) be a sequence of independent centered random
variP
n
ables, and assume that supn≥1 E(|Xn |4 )/(E|Xn |2 )2 < ∞. Then the series ∞
X
n=1 n λ
has property B.
Proof. Let {Xn0 } be an independent copy of {Xn }, and denote by E0 the corresponding P
expectation. If one of the above conditions (i) − (iv) holds,
also
P∞ then it holds
∞
0
n
0
n
(X
−
X
)λ
con(X
−
X
)λ
.
Billard’s
theorem
yields
that
a.s.
for
n
n
n
n
n=1
n=1
P
0 2
verges uniformly. By the Riesz-Fischer theorem we have that ∞
n=1 |Xn − Xn | con0
0 4
4
0
0 2
2
verges a.s. Since EE |Xn − Xn | ≤ 16E|Xn | and EE |Xn − Xn | = 2E|Xn | , we have
0 2 2
that supn≥1 EE0 (|Xn − Xn0 |4 )/(EE0 |Xn − X
| ) < ∞. Theorem 5 in [9, Ch. 3, p.
Pn∞
0 2
0
0 2
33], applied to {|XP
−
X
|
},
yields
that
n
n
n=1 EE |Xn − Xn | < ∞, and that yields
∞
2
2
E(supn≥1 |Xn | ) ≤ n=1 E|Xn | < ∞. Theorem 2.7 yields the result.
Corollary 2.9. Let {an } be a sequence of complex numbers, and let {Xn } ⊂ L1 (P) be
a sequenceR of centered independent random variables. Assume that there exists C > 0,
∞
such that u P(|Xn | >P
v)dv ≤ CuP(|Xn | > u), for every n ≥ 1 and u ≥ 1. Then the
n
random Fourier series ∞
n=1 an Xn λ has property B.
Remarks. 1. If there exists p > 1, such that for every n ≥ 1 the function u 7→
up P(|Xn | > u) is non-increasing, then the condition of the corollary is satisfied.
2. The condition is also satisfied in the case where the {Xn } are centered i.i.d., and
the tail P(|X1 | > u) is regularly varying, with exponent ρ < −1 (see the definition in
Feller [3, p. 276], and use [3, Theorem 1(a), p. 281]).
P∞
Proof. Assume that the series n=1 an Xn λn satisfies one of the conditions (i) − (iv)
above. Conditions (iii) or (iv) yield an Xn → 0 a.s. Conditions (i) or (ii) and the
Riemann-Lebesgue lemma yield that an Xn → 0 a.s. Hence, the events {|an Xn | ≥ 1}
take place P
a finite number of time a.s., and by the Borel-Cantelli lemma, we have that
the series ∞
n=1 P(|an Xn | ≥ 1) converges. By the assumption on the tail, applied with
u = 1/|an | (when an 6= 0), we obtain
|an |
Z
∞
1/|an |
E(|an Xn |1{|an Xn |≥1} ) =
P(|Xn | > v)dv + P(|an Xn | ≥ 1) ≤ (C + 1)P(|an Xn | ≥ 1).
Hence by Lemma 2.3 we have E(supn≥1 |an Xn |) < ∞, and the result follows by Theorem 2.7.
Theorem 2.10. LetP
{Xn } be a sequence of centered independent random variables. If the
∞
n
deterministic series
n=1 E(Xn 1{|Xn |≤1} )λ is uniformly convergent, then the random
P∞
n
Fourier series n=1 Xn λ has property B.
P
n
Proof. Assume the series ∞
n=1 Xn λ , satisfies one of the conditions (i) − (iv) above,
say (j) P
∈ {(i), .., (iv)}. As shown in the proof of Corollary
0
P∞2.9, we have Xn →
∞
n
E(X
1
)λ
is
condition
(j).
Since
a.s., so n=1 Xn 1{|Xn |≤1} λn satisfies
n
{|Xn |≤1}
n=1
P
n
uniformly convergent, the series ∞
n=1 [Xn 1{|Xn |≤1} − E(Xn 1{|Xn |≤1} )]λ satisfies condition (j). Theorem 2.7 yields that it is a.s. uniformly convergent, so, the series
P
∞
n
is a.s. uniformly convergent. Since Xn → 0 a.s., the series
n=1 Xn 1{|Xn |≤1} λ
P∞
n
X
λ
satisfies
(i)
− (iv).
n
n=1
4
Theorem 2.11. Let {Xn }P
be a sequence of centered independent random variables. If
∞
the random Fourier series n=1 X
λn satisfies one of the above conditions (i) − (iv),
Pn∞
then also the deterministic series n=1 E(Xn 1{|Xn |≤1} )λn satisfies this condition.
P∞
Proof. Assume the series n=1 Xn λn , satisfies one of the
(i) − (iv) above,
Pconditions
∞
say (j) ∈ {(i), .., (iv)}. So, Xn → 0 a.s., and the series n=1 Xn 1{|Xn |≤1} λn satisfies
condition (j). Let {Xn0 } be an independent copy of {Xn }, and let E0 be the corresponding expectation in the probability space of {Xn0 }. For every n ≥ 1, define
P the symmetric
n
random variables Yn = Xn 1{|Xn |≤1} − Xn0 1{|Xn0 |≤1} . Now, the series ∞
n=1 Yn λ satisfies
condition (j), so by
theorem it is a.s. uniformly convergent. Hence, by TheoPBillard’s
∞
rem 2.2, the series n=1 [Xn 1{|Xn |≤1} − E(Xn 1{|Xn |≤1} )]λn is a.s. uniformly convergent,
and the result follows.
3
A counterexample in the centered case
Theorem 3.1. There exist a sequence of independent identically distributed centered
(hence in L1 ) real random variables {Zn }, and
P∞a sequence of complex numbers {an } ∈ `2
such that almost surely the Fourier seriesP n=1 an Zn λn converges at every λ ∈ Γ and
∞
n
represents a continuous function, while
n=1 an Zn λ is almost surely non uniformly
convergent. Moreover, almost surely, the partial sums of the series are uniformly bounded
on the circle.
Remarks. 1. Cuzick and Lai [2, p. 10] constructed an example of random Fourier
series, for which a.s. there is convergence at each point, except one for which, in fact,
the random Fourier series diverges to infinity. So, the limit function is not continuous.
2. The sequence {an Zn } in the theorem is complex. We do not know of any counterexample with {an Zn } being real valued.
Theorem 3.1 is based on the following basic result.
Proposition 3.2. There exists C > 0 such that for all l ≥ 2, we have
n
X
sup sup n≥1 x∈R
Pn
k=1
C
sin kx
.
≤
(k + l) log(k + l)
log l
Proof. Define Sn (x) = k=1 sinkkx , for every n ≥ 1. It is well known (see e.g. Zygmund
[19, Vol I, p. 61]), that {Sn } is uniformly bounded, i.e., K := supn≥1 supx∈R |Sn (x)| <
Pn
kx
∞. Define Rl,n (x) = k=1 (k+l)sinlog(k+l)
, for every n ≥ 1, and put S0 (x) ≡ 0. Also, for
l
1
.
any l ≥ 2 and k ≥ 1 define ul (k) = log(k+l) and vl (k) = (k+l) log(k+l)
By Abel’s summation by parts we have
Rl,n (x) =
n
X
k=1
=
n
X
k=1
n
[Sk (x) − Sk−1 (x)]
X
k
=
[Sk (x) − Sk−1 (x)][ul (k) − vl (k)]
(k + l) log(k + l)
k=1
Sk (x)[ul (k) − vl (k) − ul (k + 1) + vl (k + 1)] + Sn (x)[ul (n + 1) − vl (n + 1)].
Hence, by monotonicity (in k) of {ul (k)} and {vl (k)} we have
|Rl,n (x)| ≤
n
X
k=1
|Sk (x)|[ul (k) − ul (k + 1)] +
n
X
k=1
|Sk (x)|[vl (k) − vl (k + 1)]+
|Sn (x)||ul (n + 1) − vl (n + 1)| ≤
5
K[ul (1) − ul (n + 1)] + K[vl (1) − vl (n + 1)] + K[ul (n + 1) + vl (n + 1)] ≤
h
i
1
2K
l
K[ul (1) + vl (1)] ≤ K
≤
+
.
log(l + 1) (l + 1) log(l + 1)
log(l + 1)
n
n+1
. In addition put
Proof of Theorem 3.1. For any n ≥ 1 put un = 22 and vn = un +u
2
ik
ik
1
1
ak = un +vn −k e n if un +1 ≤ k ≤ vn −1, and ak = − un −vn +k e n if vn +1 ≤ k ≤ un+1 −1.
Otherwise put ak = 0.
Let Z be a real random variable with distribution law given by:
Z ∞
f (x)dx,
P(Z = −1) = α, and for t ≥ 2 P(Z ≥ t) = β
t
1
x2 (log x)2 ,
where f (x) =
variable.
and α and β are chosen such that Z be a centered random
P∞Let {Zn } nbe a sequence of independent copies of Z. We want to prove that the series
n=1 an Zn λ satisfies the assertions of the theorem.
As suggested by Theorem 2.10 and Theorem 2.11, we will proceed by truncation in
order to reduce our proof to the case of a deterministic Fourier series.
n −vn +k
n +vn −k
if un ≤ k ≤ vn − 1, and put bk = logu(u
For any n ≥ 1 put bk = logu(u
n +vn −k)
n −vn +k)
if
v
≤
k
≤
u
−
1.
Now
we
define
Y
=
Z
1
,
and
we
show
that
the
series
n
n+1
n
n
{Z
≤b
}
n
n
P∞
|a
||Z
−
Y
|
converges
a.s.
n
n
n
n=1
(i) Since E|Z| < ∞,
∞
X
k=5
∞ vX
n −1
X
n=1
un+1 −1
k=un +1
P(|Zk − Yk | ≥ un + vn − k) +
X
k=vn +1
un+1 −1
∞ vX
n −1
X
n=1
P(|ak ||Zk − Yk | ≥ 1) =
k=un +1
2
P(Zk ≥ un + vn − k) +
vX
∞
n −1
X
n=1 k=un +1
X
k=vn +1
P(Z ≥ k) ≤ 2
∞
X
n=5
P(|Zk − Yk | ≥ un − vn + k) =
P(Zk ≥ un − vn + k) =
P(Z ≥ n) < ∞
(ii) In a similar way of breaking the sums we have,
∞
X
k=5
=
∞ vX
n −1
X
n=1
k=un +1
=
∞
X
un+1 −1
X
|ak |E Zk 1{bk <Zk ≤1/|ak |} +
|ak |E Zk 1{bk <Zk ≤1/|ak |}
k=vn +1
vX
n −1
n=1 k=un +1
∞
X
E |ak ||Zk − Yk |1{|ak ||Zk −Yk |≤1}
un+1 −1
X
n=1 k=vn +1
1
E Z1{ un +vn −k ≤Z≤un +vn −k} +
log
(u
+v
−k
n
n
un + vn − k
1
E Z1{ un −vn +k ≤Z≤un −vn +k} =
log(un −vn +k)
un − vn + k
6
2
vX
n −1
∞
X
n=1 k=un +1
= 2β
∞
X
1
n
n=5
Z
∞
X
1
1
E Z1{ k ≤Z≤k} ≤ 2
E Z1{ logn n ≤Z≤n}
log k
k
n
n=5
∞
∞
X
X
log log n
1
1 1
< ∞.
−
≤C
n
log
n
−
log
log
n
log
n
n(log
n)2
n=5
n=5
n
xf (x)dx = 2β
n
log n
Combining (i) and (ii) together yields that
We have
∞
X
un+1
X
2n/2
n=1
∞
√ X
2n/2
2
n=1
k=un +1
X
∞
k=un +1
E|ak Yk |2
1
(α +
k2
1/2
k
log k
Z
∞
X
≤
2n/2 2
n=1
∞
X
C0
n=1
n=1
|an ||Zn − Yn | converges a.s.
vX
n −1
k=un +1
x2 f (x)dx
2
P∞
1/2
≤C
1/2
1
E(Z 2 1{|Z|≤ k } )
≤
2
log
k
k
∞
X
2n/2
n=1
X
k≥un +1
1/2
1
≤
3
k(log k)
2−n/2 < ∞ ,
0
for some positive constants C and C .
Let {Yn0 } be an independent copy of {Yn } and write Ŷn = Yn − Yn0 . Then
un+1
X
2
n/2
n≥1
X
k=un +1
E|ak Ŷk |2
1/2
< ∞.
By the proof of Theorem 1P
in [9, Ch. 7, p. 84] (see also [18, Theorem 10]), almost surely,
∞
the random Fourier series n=1 an Ŷn λn is uniformly convergent. Since by construction
we have
supn≥1 |an Yn | < ∞ everywhere, Theorem 2.2 yields that the random Fourier
P∞
n
series n=1
Pa∞n (Yn − EYn )λ isn a.s. uniformly convergent. Since by (i) and (ii) above
the series
n=1 an (Yn − Zn )λ is a.s. absolutely convergent, the random Fourier seP
n
ries ∞
n=1 an Zn λ a.s. converges everywhere (or
P∞uniformly), or represents a continuous
function if and only if the deterministic series n=1 an EYn λn converges everywhere (or
uniformly), or represents a continuous function.
For any k ≥ 5 we have
Z ∞
β
E Z1{Z≤ k } = −E Z1{Z> k } = −β
.
xf (x)dx = −
log k
log k
k
log
k
−
log log k
log k
Hence,
EYk = −
and we obtain
EYk = −
β
,
log |ak | − log log |ak |
log log |a | β
k
+O
.
log |ak |
log2 |ak |
(∗)
P∞
k
λk in L2 (dλ). By (*)
Using the convention 0/ log 0 = 0, define T (λ) = k=1 loga|a
k|
P∞
we have k=1 |ak EYk + logβa|akk | | < ∞. Hence, it is enough for our purpose to study the
behavior of T (λ).
For any n ≥ 1, define
Qn (λ) =
vX
n −1
k=un +1
λk − λ2vn −k
.
(un + vn − k)log(un + vn − k)
7
We have
T (λ) =
un+1 −1
∞
X
X
n=1 k=un +1
i
i
vX
∞
∞
n −1
X
X
i
(e n λ)k − (e n λ)2vn −k
ak
λk =
=
Qn (e n λ).
log |ak |
(u
+
v
−
k)
log(u
+
v
−
k)
n
n
n
n
n=1
n=1
k=un +1
Since
vX
n −1
Qn (λ) =
k=un +1
λk − λ2vn −k
= λvn
(un + vn − k) log(un + vn − k)
vn −u
Xn −1
k=1
λ−k − λk
,
(k + un )log(k + un )
Proposition 3.2 yields that
kQn kC(Γ) ≤
2C
2C
≤ n.
log un
2
Pn
Pun+1 ak λk
i/k
λ) converge uniformly , necessarily
Hence the partial sums k=1
k=1 Qk (e
log |ak | =
to T (λ), so T represents a continuous function.
P
ak λk
Now we show that ∞
k=1 log |ak | converges pointwise to T (λ). For any un < m ≤ un+1
we have
un
m
m
X
X
X
ak λk
ak λk ak λk
− T (λ) ≤ − T (λ) + .
log |ak |
log |ak |
log |ak |
k=1
k=un +1
k=1
As we already showed, the first term on the right hand
Pside converges
uniformly to zero,
m
ak λk so it is enough to show that pointwise
max k=un +1 log |ak | →n 0. Since we
un <m≤un+1
clearly have
max
un <m≤un+1
m
X
k=un +1
m
X
ak λk ≤ max un <m≤vn
log |ak |
k=un +1
m
X
ak λk max +
vn <m≤un+1
log |ak |
k=vn +1
ak λk ,
log |ak |
it is enough toP
show that each of the terms on the right hand side goes to zero pointwise.
Put Dn (t) := nk=1 eint . Simple computation yields that |Dn (t)| ≤ 4/|t| for 0 < |t| ≤ π.
Fix λ0 = eit0 . Using Abel’s summation by parts we have
m
X
k=un +1
ak λk0
=
log |ak |
m
X
k=un +1
|ak |eik(1/n+t0 )
=
log |ak |
m
X
k=un +1
|ak |[Dk (1/n + t0 ) − Dk−1 (1/n + t0 )]
=
log |ak |
m
|a |
X
|au +1 |Dun (1/n + t0 ) |am+1 |Dm (1/n + t0 )
|ak+1 | k
Dk (1/n+t0 )− n
−
+
.
log |ak | log |ak+1 |
log |aun +1 |
log |am+1 |
k=un +1
For any t0 we have eventually |1/n + t0 | > 0. Using monotonicity of {−|ak |/ log |ak |} we
have
m
X
ak λk0 1
1
8
.
−
≤
log |ak |
|1/n + t0 | un log un
vn log vn
k=un +1
P
P
m
ak λk
ak λk 0 Hence max m
→
0.
Similarly
max
n
k=un +1 log |ak |
k=vn +1 log |ak | →n 0
un <m≤vn
vn <m≤un+1
pointwise.
P∞ ak λk
Now, by evaluating certain blocks of the series
k=1 log |ak | along the sequence
Pn
ak λk
−i/n
{e
}, we show that k=1 log |ak | is not Cauchy in C(Γ), hence does not uniformly
converge. This is established by the following:
n −1
vX
ak e−ik/n =
log |ak |
k=un +1
vX
n −1
k=un +1
1
=
(un + vn − k) log(un + vn − k)
8
vX
n −1
k=un +1
1
≥
k log k
log log(vn − 1) − log log(un + 1) ≥ log log(un+1 /2) − log log(2un ) ≥
log((2n+1 − 1) log 2) − log((2n + 1) log 2) ≥ log 2 + o(1/2n ).
On the other hand, since
un
m
n+1
X
X
n uX
ak λk ak λk |ak | o
sup max ≤
≤ sup max + sup
log |ak |
log |ak |
− log |ak |
m≥1 |λ|=1
n≥1 |λ|=1
n≥1
k=1
∞
X
k=1
k=1
kQk (ei/k λ)kC(Γ) + 2 sup
n −1
n vX
n≥1
1 o
≤ 4C + 2 sup(log log(vn ) − log log(un )) ≤
k log k
n≥1
k=un +1
n+1
4C + 2 sup(log log(22
n≥1
the partial sums
ak λk
k=1 log |ak |
Pn
k=un +1
n
) − log log(22 )) = 4C + 2 log 2,
are uniformly bounded.
Remark. The construction in Theorem 3.1 is inspired by the counterexample of
Fejer as presented in Zygmund [19, Vol. I, p. 299].
Corollary 3.3. There exist a sequence of independent identically distributed centered real
random variables {Zn } and a sequence of complex numbersP
{an } ∈ `2 , such that almost
∞
n
surely, for any contraction T on a Hilbert space
H
the
series
n=1 an Zn T h converges in
P∞
norm for every h ∈ H, but the Fourier series n=1 an Zn λn does not converge uniformly.
Proof.PFor the sequence which was constructed in the previous theorem, a.s. the partial
n
sums k=1 ak Zk λk are uniformly bounded. We use the spectral theorem and the dominated convergence theorem to establish the corollary for unitary operators. Then we
apply the unitary dilation theorem to pass to the contraction case (see [15, Appendix 4,
§153]).
Remark. Let {bn } be a series of complex numbers, and let {γn } be a dense sequence
in Γ. On `2 we define a unitary operator U by the following rule:PU c = {γn cn } for every
n
c := {cn } ∈ `2 . It is easy to checkP
that uniform convergence of ∞
n=1 bn λ is equivalent
∞
n
to operator norm convergence of n=1 bn U . Hence, in the above corollary we cannot
have operator norm convergence for all contractions T (even only all unitary operators)
on some Hilbert space.
ACKNOWLEDGEMENTS. 1. Both authors were partially supported by Skirball
postdoctoral fellowships of the Center of Advanced Studies in Mathematics of BenGurion University.
2. The first author was also partially supported by grant 235/01 of the Israel Science
Foundation.
3. The manuscript was completed during the first author’s postdoctoral fellowship
at the E. Schröedinger Institute, Vienna, supported by the FWF Project P16004–N05.
4. The first author is grateful to Paul Abraham Fuhrmann for his advice and encouragement, and both authors thank Michael Lin for his careful reading of the manuscript
and for his helpful comments.
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Guy Cohen
Ben-Gurion University
Beer Sheva, Israel
Current address:
Erwing Schrödinger Institute
Boltzmanngasse 9
A-1090 Vienna
Austria
e-mail: [email protected], [email protected]
Christophe Cuny
University of New Caledonia
10
Department of Mathematics
Equipe ERIM
B.P. 4477
F.98847 NOUMEA cedex
e-mail: [email protected]
11