NEGATIVE RESULTS CONCERNING FOURIER SERIES ON THE COMPLETE PRODUCT OF S3

NEGATIVE RESULTS CONCERNING FOURIER SERIES ON THE COMPLETE PRODUCT OF S₃

R. TOLEDO

INSTITUTE OFMATHEMATICS ANDCOMPUTERSCIENCE

COLLEGE OFNYÍREGYHÁZA

P.O. BOX166, NYÍREGYHÁZA, H-4400 HUNGARY

toledo@nyf.hu

Received 26 November, 2007; accepted 13 October, 2008 Communicated by S.S. Dragomir

ABSTRACT. The aim of this paper is to continue the studies about convergence in L^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 ofS3. In this case we proved the existence of an1 < p < 2number for which exists anf ∈ L^p such that its n-th partial sum of Fourier seriesS_ndo not converge to the functionfinL^p-norm. In this paper we extend this ”negative”

result for all1< p <∞andp6= 2numbers.

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 ofS₃, i.e. the symmetric group on 3 elements (see [2]). Section 2 extends the definition of the sequence Ψfor all p ≥ 1. Finally, we use the results of Section 2 to 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 ofGhave the same systemϕat 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 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].

1. REPRESENTATIVE PRODUCTSYSTEMS

Letm := (m_k, k ∈ N)be a sequence of positive integers such thatm_k ≥ 2andG_k a finite group with orderm_k, (k ∈N). Suppose that each group has discrete topology and normalized Haar measure µ_k. LetG be the compact group formed by the complete direct product of G_k with the product of the topologies, operations and measures(µ). Thus eachx ∈ Gconsists of sequencesx := (x₀, x₁, . . .), wherex_k ∈ G_k,(k ∈ N). We call this sequence the expansion of

348-08

x. The compact totally disconnected group Gis called a bounded group if the sequence mis bounded.

If M₀ := 1and M_k+1 := m_kM_k, k ∈ N, then every n ∈ N can be uniquely expressed as n =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 by Σ_k the dual object of the finite group G_k (k ∈ N). Thus each σ ∈ Σ_k is a set of continuous irreducible unitary representations ofG_k which are equivalent to some fixed representationU^(σ). Letd_σbe the dimension of its representation space and let{ζ₁, ζ₂, . . . , ζ_dσ} be a fixed but arbitrary orthonormal basis in the representation space. The functions

u^(σ)_i,j(x) :=hU_x^(σ)ζi, ζji (i, j ∈ {1, . . . , dσ}, x∈Gk)

are called the coordinate functions forU^(σ) and the basis {ζ₁, ζ₂, . . . , ζ_dσ}. In this manner for eachσ ∈Σ_kwe obtaind²_σ number of coordinate functions, in totalm_knumber 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 groupG_kand suppose thatϕ⁰_k≡1. Thus for every0≤s < m_kthere exists aσ∈Σ_k,i, j ∈ {1, . . . , d_σ}such that

ϕ^s_k(x) = p

d_σu^(σ)_i,j (x) (x∈G_k).

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

∞

Y

k=0

ϕⁿ_k^k(x_k) (x∈G), where n is of the form n = P∞

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

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

n<Mk

kψnk1kψnk∞ (k ∈N).

It seems that the boundedness of the sequence Ψplays an important role in the norm conver- gence of Fourier series.

For an integrable complex function f defined in G we define the Fourier coefficients 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 for f ∈L^p(G)if and only if there exists aCp >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 systemψ 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 ifm_k = 2andG_k:=Z₂, the cyclic group of order 2 for allk ∈N. Moreover, we obtain the Vilenkin systems if the sequence mis an arbitrary sequence of integers greater than 1 andG_k :=Z_mk, the cyclic group of order m_k for allk ∈N.

Letm_k = 6for allk ∈ NandS₃ be the symmetric group on 3 elements. Let G_k := S₃ for allk ∈N. S₃has 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 indexkis 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 2

√ 6 3

√ 6 2

ϕ⁵ 0 0 −

√ 6 2

√ 6

2 −

√ 6 2

√ 6 2

√ 6 3

√ 6 2

Notice that the functionsϕ^s_kcan take the value 0, and the product system ofϕis not uniformly bounded. These facts encumber the study of these systems. On the other hand, max

0≤s<6kϕ^sk₁kϕ^sk∞

= ⁴₃, thusΨ_k = ⁴₃k

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

2. THESEQUENCE OF FUNCTIONSΨ_k(p) We extend the definition of the sequenceΨfor allp≥1as follows:

Ψ_k(p) := max

n<M_kkψ_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 functionsΨ_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 product kfk_pkfk_q 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. LetG be a finite group with discrete topology and normalized Haar measureµ, and letf be a normalized complex valued function onG(kfk₂ = 1). Thus,

(1) ifkfk₁kfk∞= 1, thenkfk_pkfk_q = 1for allp≥1and ¹_p +¹_q = 1.

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

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)|= 0for all x∈G, therefore|f(x)|=kfk∞or|f(x)|= 0for allx∈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.

(2) Suppose there is a1< p <2such that kfkpkfkq = 1 =

Z

G

|f|² dµ.

Then the equality in the Hölder inequality holds. For this reason there are nonnegative numbersAandBnot both 0 such that

A|f(x)|^p =B|f(x)|^q (x∈G).

Thus, there is a c > 0 such that|f| = cor |f| = 0 for all x ∈ G (c = kfk∞). Then

|f| · kfk∞ = |f|². Integrating boths part of the last equation we havekfk₁kfk∞ = 1.

We obtain a contradiction.

However, the following lemma states much more.

Lemma 2.2. LetG be a finite group with discrete topology and normalized Haar measureµ, and letfbe 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 := _kf^f_k

∞. ThenΨ(p) = kfk²_∞kf₀k_pkf₀k_q. Letmbe 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 . 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_iloga_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 factsa_i ≤1for all1≤i≤ mand p < 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.

Secondly,

h(x) :=

Pm

i=1a^x_i loga_i 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 ≤ 0 for all 1 < p < 2, which completes the proof of the

lemma.

We can apply Lemma 2.1 and Lemma 2.2 to obtain similar properties forΥ_k(p)and Ψ_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 ofG such thatkϕ^s_kk₁ = 1for alls < 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 function Ψ_k(p) ≡ 1if kϕ^s_ik₁ = 1for all s < m_i andi ≤ k. Otherwise, the function Ψ_k(p)is a strictly monotone decreasing function on the interval[1,2].

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 propertyΥ_k(p) = Υ_k

p p−1

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

3. NEGATIVERESULTS

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_nis not of type(p, p)or(q, q).

1 1.05 1.1 1.15 1.2 1.25 1.3

2 4 6 8 10

p

Figure 2.1: Values ofΥ(p)for the groupS3

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

ϕⁱ_k^k p

ϕⁱ_k^k

q = max

s<m_kkϕ^s_kk_pkϕ^s_kk_q. Define

f_k(x) :=ϕⁱ_k^k(x)

ϕⁱ_k^k(x)

q−2 (x∈G_k).

Thus, |fk(x)|^p = |ϕⁱ_k^k(x)|^q and fk(x)ϕⁱ_k^k(x) = |ϕⁱ_k^k(x)|^q ∈ R⁺ if ϕⁱ_k^k(x) 6= 0. Hence both equalities hold in Hölder’s inequality. For this reason

(3.1)

Z

Gk

f_kϕⁱ_k^kdµ_k

ϕⁱ_k^k

p =kf_kk_p ϕⁱ_k^k

q

ϕⁱ_k^k p. Ifk is 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).

SincekF_kk_p =Qk−1

j=0kf_jk_p,it follows from (3.1) that kS_n+1F_k−S_nF_kk_p =

Z

G

F_kψ_ndµ

kψ_nk_p (3.2)

=

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, ifSnis of type(p, p), then there exists aCp >0such that

kS_n+1F_k−S_nF_kk_p ≤ kS_n+1F_kk_p+kS_nF_kk_p ≤2C_pkF_kk_p

for each k > 0, which contradicts (3.2) because the sequence Ψ_k(p)is not bounded. For this reason, the operators S_n are not uniformly of type (p, p). By a duality argument (see [6]) the operatorsS_ncannot be uniformly of type(q, q). This completes the proof of the theorem.

By Theorem 3.1 we obtain:

Theorem 3.2. LetGbe a bounded group and suppose that all the same finite groups appearing in the product of G have the same systemϕ at all of their occurrences. If the sequence Ψis unbounded, then the operatorS_nis not of type(p, p)for allp6= 2.

Proof. If the sequence Ψk = Ψk(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 Theorem 2.3 we 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 firstkcoordinates 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 Theorem 3.1 for all1< 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 operatorS_nis not of type(p, p)for allp6= 2.

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