NEGATIVE RESULTS CONCERNING FOURIER SERIES ON THE COMPLETE PRODUCT OF S3

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Negative Results Concerning Fourier Series

R. Toledo vol. 9, iss. 4, art. 99, 2008

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NEGATIVE RESULTS CONCERNING FOURIER SERIES ON THE COMPLETE PRODUCT OF S

₃

R. TOLEDO

Institute of Mathematics and Computer Science College of Nyíregyháza

P.O. Box 166, Nyíregyháza, H-4400 Hungary

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 inL^p-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 sequenceΨ. The most simple example of this groups is the complete product of S3. In this case we proved the existence of an1< p <2number for which exists anf∈L^psuch that its n-th partial sum of Fourier seriesSndo not converge to the functionfinL^p-norm. In this paper we extend this ”negative” result for all1< p <∞andp6= 2numbers.

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Introduction

In Section1we 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 ofS₃, i.e. the symmetric group on 3 elements (see [2]). Section2 extends the definition of the sequenceΨfor allp≥1. Finally, we use the results of Section2to study the convergence in theL^p-norm (p ≥ 1) of the Fourier series on bounded groups with unbounded sequenceΨ, supposing all the same finite groups appearing in the product of Ghave the same system ϕ at all of their occurrences.

These results appear in Section3and they complete the statement proved by G. Gát and the author of this paper in [2] for the complete product ofS₃. There have been similar results proved with respect to Walsh-like systems in [4] and [5].

Throughout this work denote byN,P,Cthe 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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1. Representative Product Systems

Letm := (m_k, k ∈ N) be a sequence of positive integers such that m_k ≥ 2 and G_k a finite group with order m_k, (k ∈ N). Suppose that each group has discrete topology and normalized Haar measureµ_k. LetGbe the compact group formed by the complete direct product ofGkwith the product of the topologies, operations and measures (µ). Thus each x ∈ G consists of sequences x := (x0, x1, . . .), where x_k ∈ G_k, (k ∈ N). We call this sequence the expansion ofx. The compact totally disconnected groupGis called a bounded group if the sequencemis bounded.

If M₀ := 1and M_k+1 := m_kM_k, k ∈ N, then every n ∈ N can be uniquely expressed asn=P∞

k=0n_kM_k,0≤n_k < m_k,n_k ∈N. This allows us to say that the sequence(n₀, n₁, . . .)is the expansion ofnwith respect tom.

Denote byΣ_kthe dual object of the finite groupG_k(k ∈N). Thus eachσ ∈Σ_kis a set of continuous irreducible unitary representations ofG_kwhich are equivalent to some fixed representationU^(σ). Letd_σ be the dimension of its representation space and let{ζ₁, ζ₂, . . . , ζ_d_σ}be a fixed but arbitrary orthonormal basis in the representation space. The functions

u^(σ)_i,j(x) :=hU_x^(σ)ζ_i, ζ_ji (i, j ∈ {1, . . . , d_σ}, x∈G_k)

are called the coordinate functions forU^(σ) and the basis {ζ₁, ζ₂, . . . , ζ_d_σ}. In this manner for eachσ ∈ Σ_k we obtaind²_σ number of coordinate functions, in totalm_k number of functions for the whole dual object ofG_k. TheL²-norm of these functions is1/√

d_σ.

Let{ϕ^s_k : 0 ≤ s < m_k}be the set of all normalized coordinate functions of the group G_k and suppose that ϕ⁰_k ≡ 1. Thus for every 0 ≤ s < m_k there exists a σ∈Σk,i, j ∈ {1, . . . , dσ}such that

ϕ^s_k(x) = p

dσu^(σ)_i,j (x) (x∈Gk).

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Letψ be the product system ofϕ^s_k, namely ψ_n(x) :=

∞

Y

k=0

ϕⁿ_k^k(x_k) (x∈G),

wherenis of the formn =P∞

k=0n_kM_kandx= (x₀, x₁, . . .). Thus we say thatψ is the representative product system ofϕ. The Weyl-Peter’s theorem (see [3]) ensures that the systemψis orthonormal and complete onL²(G).

The functionsψ_n(n∈N)are not necessarily uniformly bounded, so define Ψ_k:= max

n<Mk

kψ_nk₁kψ_nk∞ (k ∈N).

It seems that the boundedness of the sequenceΨplays an important role in the norm convergence of Fourier series.

For an integrable complex functionf defined inGwe define the Fourier coeffi- cients and partial sums by

fb_k:=

Z

Gm

f ψ_kdµ (k ∈N), S_nf :=

n−1

X

k=0

fb_kψ_k (n ∈P).

According to the theorem of Banach-Steinhauss,S_nf → f asn → ∞in theL^p norm forf ∈L^p(G)if and only if there exists aC_p >0such that

kS_nfk_p ≤C_pkfk_p (f ∈L^p(G)).

Thus, we say that the operator S_n is of type (p, p). Since the system ψ forms an orthonormal base in the Hilbert spaceL²(G), it is obvious thatS_nis 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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m_k = 2 and G_k := Z₂, the cyclic group of order 2 for allk ∈ N. Moreover, we obtain the Vilenkin systems if the sequencem is an arbitrary sequence of integers greater than 1 andG_k :=Z_m_k, the cyclic group of orderm_kfor allk ∈N.

Letm_k = 6for allk ∈NandS₃be the symmetric group on 3 elements. LetG_k :=

S₃ for allk ∈ 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 functionsϕ^s_k. In the notation the indexk is omitted because all of the groupsG_kare the same.

e (12) (13) (23) (123) (132) kϕ^sk₁ kϕ^sk∞

ϕ⁰ 1 1 1 1 1 1 1 1

ϕ¹ 1 −1 −1 −1 1 1 1 1

ϕ² √

2 −√ 2

√ 2 2

√ 2

2 −

√ 2

2 −

√ 2 2

2√ 2 3

√2 ϕ³ √

2 √

2 −

√ 2

2 −

√ 2

2 −

√ 2

2 −

√ 2 2

2√ 2 3

√2

ϕ⁴ 0 0 −

√ 6 2

√ 6

2 −

√ 6 2

√ 6 3

√ 6 2

ϕ⁵ 0 0 −

√6 2

√6

2 −

√6 2

√6 3

√6 2

Notice that the functions ϕ^s_k 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

0≤s<6kϕ^sk₁kϕ^sk∞ = ⁴₃, thus Ψ_k = ⁴₃k

→ ∞if k → ∞. More examples of representative product systems have appeared in [2] and [7].

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2. The Sequence of Functions Ψ

_k

(p)

We extend the definition of the sequenceΨfor allp≥1as follows:

Ψ_k(p) := max

n<Mk

kψ_nk_pkψ_nk_q

p≥1, 1 p +1

q = 1, k ∈N

(ifp= 1thenq=∞). Notice thatΨ_k = Ψ_k(1)for allk ∈N. Clearly, the functions Ψ_k(p)can be written in the form

Ψ_k(p) =

k−1

Y

i=0

maxs<mi

kϕ^s_ik_pkϕ^s_ik_q

=:

k−1

Y

i=0

Υ_i(p)

p≥1, 1 p +1

q = 1, k ∈N

.

Therefore, we study the productkfk_pkfk_qfor 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 Gbe a finite group with discrete topology and normalized Haar measure µ, and let f be a normalized complex valued function on G (kfk₂ = 1).

Thus,

1. ifkfk1kfk∞= 1, thenkfkpkfkq = 1for allp≥1and ¹_p +¹_q = 1.

2. ifkfk₁kfk∞>1, thenkfk_pkfk_q >1for allp≥1,p6= 2and ¹_p + ¹_q = 1.

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Proof.

1. The conditions imply the equality Z

G

|f|dµ· kfk∞= 1 = Z

G

|f|² dµ.

Letf₀ := _kfk^f

∞. Then

(2.1) |f₀(x)| ≤1 (x∈G)

and (2.2)

Z

G

|f₀|dµ= Z

G

|f₀|² dµ.

Thus by (2.1) we obtain|f0(x)| − |f0(x)|² ≥0 (x∈G)and by (2.2) we have Z

G

|f₀| − |f₀|² dµ= 0.

Hence|f₀(x)|=|f₀(x)|²for allx∈G. Thus, we have|f₀(x)|= 1or|f₀(x)|= 0 for allx ∈ G, therefore |f(x)| = kfk∞ or|f(x)| = 0for all x ∈ G. For this reason we obtain an equality in the Hölder inequality for all1 < p < ∞,

1

p + ¹_q = 1and the equality 1 =

Z

G

|f|² dµ=kfk_pkfk_q holds.

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2. Suppose there is a1< p <2such that kfk_pkfk_q = 1 =

Z

G

|f|² dµ.

Then the equality in the Hölder inequality holds. For this reason there are nonnegative numbersAandB not both 0 such that

A|f(x)|^p =B|f(x)|^q (x∈G).

Thus, there is ac > 0such that|f| = cor|f| = 0for allx ∈G(c =kfk∞).

Then |f| · kfk_∞ = |f|². Integrating boths part of the last equation we have kfk₁kfk_∞ = 1. We obtain a contradiction.

However, the following lemma states much more.

Lemma 2.2. Let Gbe a finite group with discrete topology and normalized Haar measureµ, and letf be a complex valued function onG. Thus, the functionΨ(p) :=

kfk_pkfk_q(¹_p + ¹_q = 1)is a monotone decreasing function on the interval[1,2].

Proof. Letf₀ := _kfk^f

∞. Then Ψ(p) = kfk²_∞kf₀k_pkf₀k_q. Letm be the order of the groupG. We take the elements ofGin the order,G ={g₁, g₂, . . . , g_m}, to obtain the numbers

a_i :=|f₀(g_i)| ≤1 (i= 1, . . . , m), with which we write

Ψ(p) = kfk²_∞ m

m

X

i=1

a^p_i

!¹_p _m X

i=1

a^q_i

!¹_q .

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Sinceq = _p−1^p ,we have

∂q

∂p =− 1

(p−1)² =−q² p². Therefore,

∂Ψ

∂p = Ψ(p)

"

− 1 p² log

m

X

i=1

a^p_i

! + 1

p Pm

i=1a^p_i loga_i Pm

i=1a^p_i

#

+ Ψ(p)

"

−1 q² log

m

X

i=1

a^q_i

! +1

q Pm

i=1a^q_i loga_i Pm

i=1a^q_i

#

−q² p²

.

The condition1< p <2ensures that

−1 q · q²

p² =− 1

p(p−1) <−1 p, from which we have

1 Ψ(p)

∂Ψ

∂p ≤ 1 p²

"

log

m

X

i=1

a^q_i

!

−log

m

X

i=1

a^p_i

!#

+ 1 p

Pm

i=1a^p_i loga_i Pm

i=1a^p_i − Pm

i=1a^q_i loga_i Pm

i=1a^q_i

. Both addends in the sum above are not positive. Indeed, the facts a_i ≤ 1 for all 1≤ i ≤ mandp < qimply thata^q_i ≤ a^p_i for all1≤ i ≤ m, from which it is clear that

(2.3) log

m

X

i=1

a^q_i

!

−log

m

X

i=1

a^p_i

!

≤0.

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Secondly,

h(x) :=

Pm

i=1a^x_i logai

Pm i=1a^x_i is a monotone increasing function. Indeed,

h⁰(x) = Pm

i=1a^x_i log²a_i Pm

i=1a^x_i −(Pm

i=1a^x_i loga_i)² (Pm

i=1a^x_i)²

= Pm

i,j=1a^x_ia^x_j(loga_i−loga_j)² (Pm

i=1a^x_i)² ≥0.

Consequently, we have (2.4)

Pm

i=1a^p_i loga_i Pm

i=1a^p_i − Pm

i=1a^q_i loga_i Pm

i=1a^q_i ≤0.

By (2.3) and (2.4) we obtain ^∂Ψ_∂p ≤ 0for all1 < p < 2, which completes the proof of the lemma.

We can apply Lemma2.1 and Lemma2.2 to obtain similar properties forΥk(p) andΨ_k(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 G_k be a coordinate group of G such that kϕ^s_kk₁ = 1 for all s < m_k. Then Υ_k(p) ≡ 1. Otherwise, the function Υ_k(p) is a strictly monotone decreasing function on the interval[1,2].

The functionΨ_k(p) ≡ 1ifkϕ^s_ik₁ = 1for alls < m_i andi ≤ k. Otherwise, the functionΨ_k(p)is a strictly monotone decreasing function on the interval[1,2].

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It is important to remark that the functions Υ_k(p) and Ψ_k(p) are monotone increasing if p > 2. It follows from the property Υ_k(p) = Υ_k

p p−1

. In order to illustrate these properties we plot the values ofΥ(p)for the groupS3.

1 1.05 1.1 1.15 1.2 1.25 1.3

2 4 6 8 10

p

Figure 1: Values ofΥ(p)for the groupS3

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3. Negative Results

Theorem 3.1. Letpbe a fixed number on the interval(1,2)and ¹_p +¹_q = 1. IfGis a group with unbounded sequenceΨ_k(p), then the operatorS_n is not of type (p, p) or(q, q).

Proof. To prove this theorem, choosei_k < m_k the index for which the normalized coordinate functionϕⁱ_k^k of the finite groupG_k satisfies

ϕⁱ_k^k p

ϕⁱ_k^k

q = max

s<mk

kϕ^s_kk_pkϕ^s_kk_q. Define

f_k(x) :=ϕⁱ_k^k(x)

ϕⁱ_k^k(x)

q−2 (x∈G_k).

(3.1)

Z

G_k

f_kϕⁱ_k^kdµ_k

ϕⁱ_k^k

p =kf_kk_p ϕⁱ_k^k

q

ϕⁱ_k^k p. Ifkis an arbitrary positive integer andn:=Pk−1

j=0i_jM_j, then defineF_k ∈L^p(G)by F_k(x) :=

k−1

Y

j=0

f_j(x_j) (x= (x₀, x₁, . . .)∈G).

SincekFkkp =Qk−1

j=0kfjkp,it follows from (3.1) that kSn+1Fk−SnFkkp =

Z

G

Fkψ_ndµ

kψnkp

(3.2)

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=

k−1

Y

j=0

Z

G

f_jϕ^s_jdµ_j

kϕ^s_jk_p ≥Ψ_k(p)kF_kk_p.

On the other hand, ifS_nis of type(p, p), then there exists aC_p >0such that kSn+1Fk−SnFkkp ≤ kSn+1Fkkp+kSnFkkp ≤2CpkFkkp

for eachk >0, which contradicts (3.2) because the sequenceΨ_k(p)is not bounded.

For this reason, the operatorsS_nare not uniformly of type(p, p). By a duality argu- ment (see [6]) the operators S_n cannot be uniformly of type (q, q). This completes the proof of the theorem.

By Theorem3.1we obtain:

Theorem 3.2. LetGbe a bounded group and suppose that all the same finite groups appearing in the product ofGhave the same systemϕat all of their occurrences. If the sequenceΨis unbounded, then the operatorS_nis not of type(p, p)for allp6= 2.

Proof. If the sequenceΨ_k = Ψ_k(1)is not bounded, there exists a finite groupF with system{ϕ^s : 0≤s <|F|}(|F|is the order of the groupF) which appears infinitely many times in the product ofGand

Υ(1) := max

s<|F|kϕ^sk₁kϕ^sk_∞>1.

Hence by Theorem2.3we have

Υ(p) := max

s<|F|kϕ^skpkϕ^skq >1

for allp6= 2. Denote byl(k)the number of times the groupF appears in the firstk

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coordinates ofG. Thusl(k)→ ∞ifk → ∞and Ψ_k(p)≥

l(k)

Y

i=1

Υ(p)→ ∞ ifk→ ∞,

for allp6= 2. Consequently, the groupGsatisfies the conditions of Theorem3.1for all1< p <2. This completes the proof of the theorem.

Corollary 3.3. If G is the complete product ofS₃ with the system ϕ appearing in Section2, then the operatorS_nis not of type(p, p)for allp6= 2.

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References

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[2] G. GÁT AND R. TOLEDO,L^p-norm convergence of series in compact totally disconected groups, Anal. Math., 22 (1996), 13–24.

[3] E. HEWITTANDK. ROSS Abstract Harmonic Analysis I, Springer-Verlag, Hei- delberg, 1963.

[4] F. SCHIPP, On Walsh function with respect to weights, Math. Balkanica, 16 (2002), 169–173.

[5] P. SIMON, On the divergence of Fourier series with respect to weighted Walsh systems, East Journal on Approximations, 9(1) (2003), 21–30.

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Paed. Nyíregyháza, 19(1) (2003), 43–50.