NEGATIVE RESULTS CONCERNING FOURIER
SERIES ON THE COMPLETE PRODUCT OF S3
Negative Results Concerning
Fourier Series
R. TOLEDO
Institute of Mathematics and Computer Science
College of Nyíregyháza
P.O. Box 166, Nyíregyháza,
H-4400 Hungary
R. Toledo
vol. 9, iss. 4, art. 99, 2008
Title Page
EMail: toledo@nyf.hu
Received:
26 November, 2007
Accepted:
13 October, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
42C10.
Key words:
Fourier series, Walsh-system, Vilenkin systems, Representative product systems.
Abstract:
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.
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Contents
1
Representative Product Systems
4
2
The Sequence of Functions Ψk (p)
7
3
Negative Results
13
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vol. 9, iss. 4, art. 99, 2008
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Introduction
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].
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vol. 9, iss. 4, art. 99, 2008
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1.
Representative Product Systems
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 x. The compact totally
disconnected group G is called a bounded group if the sequence m is bounded.
If M0 := 1 and
PMk+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
2
number
√ of functions for the whole dual object of Gk . The L -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 ).
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vol. 9, iss. 4, art. 99, 2008
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Let ψ be the product system of ϕsk , 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
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vol. 9, iss. 4, art. 99, 2008
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
f ψ k dµ (k ∈ N),
fbk :=
Gm
Sn f :=
n−1
X
fbk ψk
(n ∈ P).
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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
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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 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.
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ϕ0
ϕ1
ϕ2
ϕ3
s
s
e
(12)
(13)
(23)
(123)
(132)
kϕ k1
kϕ k∞
1
1
1
1
1
1
1
1
−1
1
1
1
√
√
√
√
1
√
2
√
2
1
−1
−1
√
√
√
2
2 − 2
2
√
√
√
2
2 − 22
√
ϕ4
0
0
ϕ5
0
0
6
2
√
− 26
−
2
2
√
−
√
2
2
6
2
√
6
2
2
2
√
− 22
√
6
2
√
− 26
−
2
2
√
− 22
√
− 26
√
6
2
−
2 2
3
√
2 2
3
√
6
3
√
6
3
√
6
2
√
6
2
vol. 9, iss. 4, art. 99, 2008
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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
k
the other hand, max kϕs k1 kϕs k∞ = 34 , thus Ψk = 43 → ∞ if k → ∞. More
0≤s<6
examples of representative product systems have appeared in [2] and [7].
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2.
The Sequence of Functions 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
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vol. 9, iss. 4, art. 99, 2008
max kϕsi kp kϕsi kq
s<mi
i=0
k−1
Y
i=0
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Υi (p)
p ≥ 1,
1 1
+ = 1, k ∈ N .
p q
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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
1
p
+
1
q
2. if kf k1 kf k∞ > 1, then kf kp kf kq > 1 for all p ≥ 1, p 6= 2 and
+
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= 1.
1
p
JJ
1
q
= 1.
Proof.
1. The conditions imply the equality
Z
Z
|f | dµ · kf k∞ = 1 =
|f |2 dµ.
G
Let f0 :=
f
.
kf k∞
G
Then
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|f0 (x)| ≤ 1
(2.1)
(x ∈ G)
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vol. 9, iss. 4, art. 99, 2008
and
Z
Z
|f0 | dµ =
(2.2)
G
|f0 |2 dµ.
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G
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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.
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Hence |f0 (x)| = |f0 (x)|2 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 < ∞,
1
+ 1q = 1 and the equality
p
Z
1=
|f |2 dµ = kf kp kf kq
G
holds.
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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
p
q
A|f (x)| = B|f (x)|
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(x ∈ G).
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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.
vol. 9, iss. 4, art. 99, 2008
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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
Ψ(p) =
kf k2∞
m
m
X
i=1
! p1
api
m
X
i=1
! 1q
aqi
.
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Since q =
p
,
p−1
we have
∂q
1
q2
=−
=
−
.
∂p
(p − 1)2
p2
Therefore,
"
#
Pm p
∂Ψ
1
1 i=1 ai log ai
Pm p
= Ψ(p) − 2 log
api +
∂p
p
p
i=1 ai
i=1
"
!
#
Pm q
m
X
1
1 i=1 ai log ai
q2
q
Pm q
ai +
+ Ψ(p) − 2 log
− 2 .
q
q
p
i=1 ai
i=1
m
X
!
The condition 1 < p < 2 ensures that
R. Toledo
vol. 9, iss. 4, art. 99, 2008
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1 q2
1
1
<− ,
− · 2 =−
q p
p(p − 1)
p
from which we have
"
!
!#
m
m
X
X
1 ∂Ψ
1
≤ 2 log
aqi − log
api
Ψ(p) ∂p
p
i=1
i=1
Pm q
Pm p
ai log ai
ai log ai
1
i=1
i=1
Pm p
Pm q
+
−
.
p
i=1 ai
i=1 ai
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
q
p
ai ≤ 0.
(2.3)
log
ai − log
i=1
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i=1
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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)
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vol. 9, iss. 4, art. 99, 2008
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Pm p
Pm q
ai log ai
a log a
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
of the lemma.
∂Ψ
∂p
Contents
≤ 0 for all 1 < p < 2, which completes the proof
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:
kϕsk k1
Theorem 2.3. Let Gk be a coordinate group of G such that
= 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].
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It is important to remark that the functions Υk (p) and Ψk (p)
aremonotone inp
creasing if 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 .
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1.3
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1.1
1.05
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2
4
6
8
p
10
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Figure 1: Values of Υ(p) for the group S3
3.
Negative Results
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).
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 q
k p
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) = |ϕikk (x)|q ∈ R+ if ϕikk (x) 6= 0. Hence
both equalities hold in Hölder’s inequality. For this reason
Z
ik
(3.1)
fk ϕk dµk ϕikk p = kfk kp ϕikk q ϕikk p .
Gk
If k is an arbitrary positive integer and n :=
Fk (x) :=
k−1
Y
Pk−1
j=0 ij Mj ,
(3.2)
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fj (xj )
(x = (x0 , x1 , . . . ) ∈ G).
j=0
Since kFk kp =
then define Fk ∈ Lp (G) by
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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
Close
=
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
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.
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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
s
s
Υ(1) := max kϕ k1 kϕ k∞ > 1.
s<|F |
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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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References
[1] G. BENKE, Trigonometric approximation theory in compact totally disconnected groups, Pacific J. of Math., 77(1) (1978), 23–32.
[2] G. GÁT AND R. TOLEDO, Lp -norm convergence of series in compact totally
disconected groups, Anal. Math., 22 (1996), 13–24.
[3] E. HEWITT AND K. ROSS Abstract Harmonic Analysis I, Springer-Verlag, Heidelberg, 1963.
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vol. 9, iss. 4, art. 99, 2008
[4] F. SCHIPP, On Walsh function with respect to weights, Math. Balkanica, 16
(2002), 169–173.
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[5] P. SIMON, On the divergence of Fourier series with respect to weighted Walsh
systems, East Journal on Approximations, 9(1) (2003), 21–30.
[6] R. TOLEDO, On Hardy-norm of operators with property ∆, Acta Math. Hungar., 80(3) (1998), 157–168.
[7] R. TOLEDO, Representation of product systems on the interval [0, 1], Acta Acad.
Paed. Nyíregyháza, 19(1) (2003), 43–50.
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