Volume 9 (2008), Issue 4, Article 99, 7 pp.
NEGATIVE RESULTS CONCERNING FOURIER SERIES ON THE COMPLETE
PRODUCT OF S3
R. TOLEDO
I NSTITUTE OF M ATHEMATICS AND C OMPUTER S CIENCE
C OLLEGE OF N YÍREGYHÁZA
P.O. B OX 166, N YÍREGYHÁZA , H-4400 H UNGARY
toledo@nyf.hu
Received 26 November, 2007; accepted 13 October, 2008
Communicated by S.S. Dragomir
A BSTRACT. The aim of this paper is to continue the studies about convergence in Lp -norm
of the Fourier series based on representative product systems on the complete product of finite
groups. We restrict our attention to bounded groups with unbounded sequence Ψ. The most
simple example of this groups is the complete product of S3 . In this case we proved the existence
of an 1 < p < 2 number for which exists an f ∈ Lp such that its n-th partial sum of Fourier
series Sn do not converge to the function f in Lp -norm. In this paper we extend this ”negative”
result for all 1 < p < ∞ and p 6= 2 numbers.
Key words and phrases: Fourier series, Walsh-system, Vilenkin systems, Representative product systems.
2000 Mathematics Subject Classification. 42C10.
In Section 1 we introduce basic concepts in the study of representative product systems and
Fourier analysis. We also introduce the system with which we work on the complete product
of S3 , i.e. the symmetric group on 3 elements (see [2]). Section 2 extends the definition of the
sequence Ψ for all p ≥ 1. Finally, we use the results of Section 2 to study the convergence
in the Lp -norm (p ≥ 1) of the Fourier series on bounded groups with unbounded sequence Ψ,
supposing all the same finite groups appearing in the product of G have the same system ϕ at all
of their occurrences. These results appear in Section 3 and they complete the statement proved
by G. Gát and the author of this paper in [2] for the complete product of S3 . There have been
similar results proved with respect to Walsh-like systems in [4] and [5].
Throughout this work denote by N, P, C the set of nonnegative, positive integers and complex
numbers, respectively. The notation which we have used in this paper is similar to [3].
1. R EPRESENTATIVE P RODUCT S YSTEMS
Let m := (mk , k ∈ N) be a sequence of positive integers such that mk ≥ 2 and Gk a finite
group with order mk , (k ∈ N). Suppose that each group has discrete topology and normalized
Haar measure µk . Let G be the compact group formed by the complete direct product of Gk
with the product of the topologies, operations and measures (µ). Thus each x ∈ G consists of
sequences x := (x0 , x1 , . . .), where xk ∈ Gk , (k ∈ N). We call this sequence the expansion of
348-08
2
R. T OLEDO
x. The compact totally disconnected group G is called a bounded group if the sequence m is
bounded.
If P
M0 := 1 and Mk+1 := mk Mk , k ∈ N, then every n ∈ N can be uniquely expressed as
n= ∞
k=0 nk Mk , 0 ≤ nk < mk , nk ∈ N. This allows us to say that the sequence (n0 , n1 , . . . )
is the expansion of n with respect to m.
Denote by Σk the dual object of the finite group Gk (k ∈ N). Thus each σ ∈ Σk is a
set of continuous irreducible unitary representations of Gk which are equivalent to some fixed
representation U (σ) . Let dσ be the dimension of its representation space and let {ζ1 , ζ2 , . . . , ζdσ }
be a fixed but arbitrary orthonormal basis in the representation space. The functions
(σ)
ui,j (x) := hUx(σ) ζi , ζj i
(i, j ∈ {1, . . . , dσ }, x ∈ Gk )
are called the coordinate functions for U (σ) and the basis {ζ1 , ζ2 , . . . , ζdσ }. In this manner for
each σ ∈ Σk we obtain d2σ number of coordinate functions, in total√mk number of functions for
the whole dual object of Gk . The L2 -norm of these functions is 1/ dσ .
Let {ϕsk : 0 ≤ s < mk } be the set of all normalized coordinate functions of the group Gk and
suppose that ϕ0k ≡ 1. Thus for every 0 ≤ s < mk there exists a σ ∈ Σk , i, j ∈ {1, . . . , dσ } such
that
p
(σ)
ϕsk (x) = dσ ui,j (x)
(x ∈ Gk ).
s
Let ψ be the product system of ϕk , namely
ψn (x) :=
∞
Y
ϕnk k (xk )
(x ∈ G),
k=0
P∞
where n is of the form n =
k=0 nk Mk and x = (x0 , x1 , . . .). Thus we say that ψ is the
representative product system of ϕ. The Weyl-Peter’s theorem (see [3]) ensures that the system
ψ is orthonormal and complete on L2 (G).
The functions ψn (n ∈ N) are not necessarily uniformly bounded, so define
Ψk := max kψn k1 kψn k∞
n<Mk
(k ∈ N).
It seems that the boundedness of the sequence Ψ plays an important role in the norm convergence of Fourier series.
For an integrable complex function f defined in G we define the Fourier coefficients and
partial sums by
Z
n−1
X
b
fk :=
f ψ k dµ (k ∈ N),
Sn f :=
fbk ψk (n ∈ P).
Gm
k=0
According to the theorem of Banach-Steinhauss, Sn f → f as n → ∞ in the Lp norm for
f ∈ Lp (G) if and only if there exists a Cp > 0 such that
kSn f kp ≤ Cp kf kp
(f ∈ Lp (G)).
Thus, we say that the operator Sn is of type (p, p). Since the system ψ forms an orthonormal
base in the Hilbert space L2 (G), it is obvious that Sn is of type (2, 2).
The representative product systems are the generalization of the well known Walsh-Paley
and Vilenkin systems. Indeed, we obtain the Walsh-Paley system if mk = 2 and Gk := Z2 , the
cyclic group of order 2 for all k ∈ N. Moreover, we obtain the Vilenkin systems if the sequence
m is an arbitrary sequence of integers greater than 1 and Gk := Zmk , the cyclic group of order
mk for all k ∈ N.
Let mk = 6 for all k ∈ N and S3 be the symmetric group on 3 elements. Let Gk := S3 for
all k ∈ N. S3 has two characters and a 2-dimensional representation. Using a calculation of the
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 99, 7 pp.
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N EGATIVE R ESULTS C ONCERNING F OURIER S ERIES
3
matrices corresponding to the 2-dimensional representation we construct the functions ϕsk . In
the notation the index k is omitted because all of the groups Gk are the same.
ϕ0
ϕ1
ϕ2
ϕ3
e
(12)
(13)
1
1
1
(23) (123) (132)
1
1
−1
−1 −1
√
√
√
√
2
2
2 − 2
2
2
√
√
√
√
2
2 − 22 − 22
√
ϕ4
0
0
ϕ5
0
0
6
2
√
− 26
−
√
6
2
√
6
2
1
1
1
√
2
2
√
− 22
√
6
2
√
− 26
−
kϕs k1 kϕs k∞
1
1
1
1
√
√
1
√
2
√
2
2
2
√
− 22
√
− 26
√
6
2
−
2 2
3
√
2 2
3
√
6
3
√
6
3
√
6
2
√
6
2
Notice that the functions ϕsk can take the value 0, and the product system of ϕ is not uniformly
bounded. These facts encumber the study of these systems. On the other hand, max kϕs k1 kϕs k∞
0≤s<6
4
4 k
= 3 , thus Ψk = 3 → ∞ if k → ∞. More examples of representative product systems have
appeared in [2] and [7].
2. T HE S EQUENCE OF F UNCTIONS Ψk (p)
We extend the definition of the sequence Ψ for all p ≥ 1 as follows:
1 1
Ψk (p) := max kψn kp kψn kq
p ≥ 1, + = 1, k ∈ N
n<Mk
p q
(if p = 1 then q = ∞). Notice that Ψk = Ψk (1) for all k ∈ N. Clearly, the functions Ψk (p) can
be written in the form
Ψk (p) =
k−1
Y
i=0
=:
max kϕsi kp kϕsi kq
s<mi
k−1
Y
Υi (p)
i=0
1 1
p ≥ 1, + = 1, k ∈ N .
p q
Therefore, we study the product kf kp kf kq for normalized functions on finite groups. In this
regard we use the Hölder inequality (see [3, p. 137]). First, we prove the following lemma.
Lemma 2.1. Let G be a finite group with discrete topology and normalized Haar measure µ,
and let f be a normalized complex valued function on G (kf k2 = 1). Thus,
(1) if kf k1 kf k∞ = 1, then kf kp kf kq = 1 for all p ≥ 1 and p1 + 1q = 1.
(2) if kf k1 kf k∞ > 1, then kf kp kf kq > 1 for all p ≥ 1, p 6= 2 and p1 + 1q = 1.
Proof.
(1) The conditions imply the equality
Z
Z
|f | dµ · kf k∞ = 1 =
|f |2 dµ.
G
Let f0 :=
(2.1)
f
.
kf k∞
G
Then
|f0 (x)| ≤ 1
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 99, 7 pp.
(x ∈ G)
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4
R. T OLEDO
and
Z
Z
|f0 |2 dµ.
|f0 | dµ =
(2.2)
G
G
2
Thus by (2.1) we obtain |f0 (x)| − |f0 (x)| ≥ 0 (x ∈ G) and by (2.2) we have
Z
|f0 | − |f0 |2 dµ = 0.
G
2
Hence |f0 (x)| = |f0 (x)| for all x ∈ G. Thus, we have |f0 (x)| = 1 or |f0 (x)| = 0 for all
x ∈ G, therefore |f (x)| = kf k∞ or |f (x)| = 0 for all x ∈ G. For this reason we obtain
an equality in the Hölder inequality for all 1 < p < ∞, p1 + 1q = 1 and the equality
Z
1=
|f |2 dµ = kf kp kf kq
G
holds.
(2) Suppose there is a 1 < p < 2 such that
Z
kf kp kf kq = 1 =
|f |2 dµ.
G
Then the equality in the Hölder inequality holds. For this reason there are nonnegative
numbers A and B not both 0 such that
A|f (x)|p = B|f (x)|q
(x ∈ G).
Thus, there is a c > 0 such that |f | = c or |f | = 0 for all x ∈ G (c = kf k∞ ). Then
|f | · kf k∞ = |f |2 . Integrating boths part of the last equation we have kf k1 kf k∞ = 1.
We obtain a contradiction.
However, the following lemma states much more.
Lemma 2.2. Let G be a finite group with discrete topology and normalized Haar measure µ,
and let f be a complex valued function on G. Thus, the function Ψ(p) := kf kp kf kq ( p1 + 1q = 1)
is a monotone decreasing function on the interval [1, 2].
Proof. Let f0 := kffk∞ . Then Ψ(p) = kf k2∞ kf0 kp kf0 kq . Let m be the order of the group G. We
take the elements of G in the order, G = {g1 , g2 , . . . , gm }, to obtain the numbers
ai := |f0 (gi )| ≤ 1
(i = 1, . . . , m),
with which we write
kf k2∞
Ψ(p) =
m
Since q =
p
,
p−1
m
X
i=1
! p1
api
m
X
! 1q
aqi
.
i=1
we have
∂q
1
q2
=−
=
−
.
∂p
(p − 1)2
p2
Therefore,
"
!
#
P
m
p
X
a
log
a
∂Ψ
1
1 m
i
p
i=1
Pm i p
= Ψ(p) − 2 log
ai +
∂p
p
p
i=1 ai
i=1
"
!
#
Pm q
m
X
a
log
a
1
1
q2
i
q
i
i=1
Pm q
+ Ψ(p) − 2 log
ai +
− 2 .
q
q
p
i=1 ai
i=1
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N EGATIVE R ESULTS C ONCERNING F OURIER S ERIES
5
The condition 1 < p < 2 ensures that
1 q2
1
1
− · 2 =−
<− ,
q p
p(p − 1)
p
from which we have
"
!
!#
Pm q
Pm p
m
m
X
X
ai log ai
ai log ai
1 ∂Ψ
1
1
q
p
i=1
i=1
Pm p
Pm q
≤ 2 log
ai − log
ai
+
−
.
Ψ(p) ∂p
p
p
a
a
i
i
i=1
i=1
i=1
i=1
Both addends in the sum above are not positive. Indeed, the facts ai ≤ 1 for all 1 ≤ i ≤ m and
p < q imply that aqi ≤ api for all 1 ≤ i ≤ m, from which it is clear that
!
!
m
m
X
X
(2.3)
log
aqi − log
api ≤ 0.
i=1
i=1
Secondly,
Pm x
a log a
Pm i x i
h(x) := i=1
i=1 ai
is a monotone increasing function. Indeed,
Pm x
Pm x 2 Pm x
2
i=1 ai log ai
i=1 ai − (
i=1 ai log ai )
0
h (x) =
P
x 2
( m
i=1 ai )
Pm
x x
2
i,j=1 ai aj (log ai − log aj )
=
≥ 0.
Pm x 2
( i=1 ai )
Consequently, we have
(2.4)
Pm p
Pm q
a log a
ai log ai
i=1
Pm p
Pm i q i ≤ 0.
− i=1
i=1 ai
i=1 ai
By (2.3) and (2.4) we obtain
lemma.
∂Ψ
∂p
≤ 0 for all 1 < p < 2, which completes the proof of the
We can apply Lemma 2.1 and Lemma 2.2 to obtain similar properties for Υk (p) and Ψk (p)
because these functions are the maximum value and the product of finite functions satisfying
the conditions of the two lemmas. Consequently, we obtain:
Theorem 2.3. Let Gk be a coordinate group of G such that kϕsk k1 = 1 for all s < mk . Then
Υk (p) ≡ 1. Otherwise, the function Υk (p) is a strictly monotone decreasing function on the
interval [1, 2].
The function Ψk (p) ≡ 1 if kϕsi k1 = 1 for all s < mi and i ≤ k. Otherwise, the function
Ψk (p) is a strictly monotone decreasing function on the interval [1, 2].
It is important to remark that the functions Υk (p) and Ψk (p) are monotone increasing if
p
p > 2. It follows from the property Υk (p) = Υk p−1
. In order to illustrate these properties
we plot the values of Υ(p) for the group S3 .
3. N EGATIVE R ESULTS
Theorem 3.1. Let p be a fixed number on the interval (1, 2) and p1 + 1q = 1. If G is a group with
unbounded sequence Ψk (p), then the operator Sn is not of type (p, p) or (q, q).
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 99, 7 pp.
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6
R. T OLEDO
1.3
1.25
1.2
1.15
1.1
1.05
1
2
4
6
8
10
p
Figure 2.1: Values of Υ(p) for the group S3
Proof. To prove this theorem, choose ik < mk the index for which the normalized coordinate
function ϕikk of the finite group Gk satisfies
i i ϕ k ϕ k = max kϕsk k kϕsk k .
k p
k q
p
q
s<mk
Define
q−2
fk (x) := ϕikk (x) ϕikk (x)
(x ∈ Gk ).
Thus, |fk (x)|p = |ϕikk (x)|q and fk (x)ϕikk (x) = |ϕkik (x)|q ∈ R+ if ϕikk (x) 6= 0. Hence both
equalities hold in Hölder’s inequality. For this reason
Z
i i
k
ϕ k = kfk kp ϕik ϕik .
ϕ
dµ
(3.1)
f
k
k
k
k q
k p
k p
Gk
If k is an arbitrary positive integer and n :=
Fk (x) :=
k−1
Y
Pk−1
j=0 ij Mj ,
then define Fk ∈ Lp (G) by
(x = (x0 , x1 , . . . ) ∈ G).
fj (xj )
j=0
Since kFk kp =
(3.2)
Qk−1
kfj kp , it follows from (3.1) that
Z
kSn+1 Fk − Sn Fk kp = Fk ψ n dµ kψn kp
j=0
=
G
k−1
Y Z
j=0
G
fj ϕsj dµj kϕsj kp ≥ Ψk (p)kFk kp .
On the other hand, if Sn is of type (p, p), then there exists a Cp > 0 such that
kSn+1 Fk − Sn Fk kp ≤ kSn+1 Fk kp + kSn Fk kp ≤ 2Cp kFk kp
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N EGATIVE R ESULTS C ONCERNING F OURIER S ERIES
7
for each k > 0, which contradicts (3.2) because the sequence Ψk (p) is not bounded. For this
reason, the operators Sn are not uniformly of type (p, p). By a duality argument (see [6]) the
operators Sn cannot be uniformly of type (q, q). This completes the proof of the theorem. By Theorem 3.1 we obtain:
Theorem 3.2. Let G be a bounded group and suppose that all the same finite groups appearing
in the product of G have the same system ϕ at all of their occurrences. If the sequence Ψ is
unbounded, then the operator Sn is not of type (p, p) for all p 6= 2.
Proof. If the sequence Ψk = Ψk (1) is not bounded, there exists a finite group F with system
{ϕs : 0 ≤ s < |F |} (|F | is the order of the group F ) which appears infinitely many times in the
product of G and
Υ(1) := max kϕs k1 kϕs k∞ > 1.
s<|F |
Hence by Theorem 2.3 we have
Υ(p) := max kϕs kp kϕs kq > 1
s<|F |
for all p 6= 2. Denote by l(k) the number of times the group F appears in the first k coordinates
of G. Thus l(k) → ∞ if k → ∞ and
Ψk (p) ≥
l(k)
Y
Υ(p) → ∞
if k → ∞,
i=1
for all p 6= 2. Consequently, the group G satisfies the conditions of Theorem 3.1 for all 1 < p <
2. This completes the proof of the theorem.
Corollary 3.3. If G is the complete product of S3 with the system ϕ appearing in Section 2,
then the operator Sn is not of type (p, p) for all p 6= 2.
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