Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
28 (2012), 167–176
www.emis.de/journals
ISSN 1786-0091
A NOTE ON THE FOURIER COEFFICIENTS AND PARTIAL
SUMS OF VILENKIN-FOURIER SERIES
GEORGE TEPHNADZE
Abstract. The aim of this paper is to investigate Paley type and HardyLittlewood type inequalities and strong convergence theorem of partial sums
of Vilenkin-Fourier series.
Let N+ denote the set of the positive integers, N := N+ ∪ {0}. Let m :=
(m0 , m1 , . . .) denote a sequence of the positive numbers, not less than 2. Denote
by
Zmk := {0, 1, . . . , mk − 1}
the additive group of integers modulo mk . Define the group Gm as the complete
direct product of the group Zmj with the product of the discrete topologies of
Zmj ’s.
The direct product µ, of the measures
µk ({j}) := 1/mk ,
(j ∈ Zmk )
is the Haar measure on Gm , with µ (Gm ) = 1. If supn mn < ∞, then we call
Gm a bounded Vilenkin group. If the generating sequence m is not bounded
then Gm is said to be an unbounded Vilenkin group. In this paper we discuss
bounded Vilenkin groups only.
The elements of Gm represented by sequences
x := (x0 , x1 , . . . , xj , . . .),
(xk ∈ Zmk ) .
It is easy to give a base for the neighborhood of Gm :
I0 (x) := Gm ,
In (x) := {y ∈ Gm | y0 = x0 , . . . , yn−1 = xn−1 },
(x ∈ Gm , n ∈ N).
−
Denote In := In (0), for n ∈ N and In := Gm \ In .
2010 Mathematics Subject Classification. 42C10.
Key words and phrases. Vilenkin system, Fourier coefficients, partial sums, martingale
Hardy space.
167
168
GEORGE TEPHNADZE
If we define the so-called generalized number system, based on m in the
following way :
M0 := 1, Mk+1 := mk Mk (k ∈ N),
∞
P
then every n ∈ N can be uniquely expressed as n =
nj Mj , where nj ∈ Zmj
j=0
(j ∈ N) and only a finite number of nj ’s differ from zero.
Let |n| := max {j ∈ N : nj 6= 0}. Denote by Nn0 the subset of positive
integers N+ , for which n|n| = n0 = 1. Then every n ∈ Nn0 , Mk < n < Mk+1
can be written as
n = M0 +
k−1
X
nj Mj + Mk = 1 +
j=1
k−1
X
nj Mj + Mk ,
j=1
where nj ∈ {0, mj − 1}, (j ∈ N+ ).
By simple calculation we get
X
Mk−1
(1)
1=
≥ cMk ,
m0
{n:Mk ≤n≤Mk+1 , n∈Nn0 }
where c is absolute constant.
Denote by L1 (Gm ) the usual (one dimensional) Lebesgue space. Next, we
introduce on Gm an orthonormal system, which is called the Vilenkin system.
At first define the complex valued function rk (x) : Gm → C, the generalized
Rademacher functions as
rk (x) := exp (2πιxk /mk ) ,
ι2 = −1, x ∈ Gm , k ∈ N .
Now define the Vilenkin system ψ := (ψn : n ∈ N) on Gm as:
∞
ψn (x) := Π rknk (x) ,
k=0
(n ∈ N) .
Specifically, we call this system the Walsh–Paley one if m ≡ 2. The Vilenkin
system is orthonormal and complete in L2 (Gm ) [1, 14].
Now we introduce analogues of the usual definitions in Fourier-analysis. If
f ∈ L1 (Gm ) we can establish the the Fourier coefficients, the partial sums, the
Dirichlet kernels, with respect to the Vilenkin system in the usual manner:
Z
b
f (k) :=
f ψ k dµ, (k ∈ N) ,
Sn f :=
Gm
n−1
X
fb(k) ψk ,
(n ∈ N+ , S0 f := 0) ,
k=0
Dn :=
n−1
X
k=0
ψk ,
(n ∈ N+ ) .
FOURIER COEFFICIENTS AND PARTIAL SUMS
Recall that
(2)
and
(3)
169
(
Mn , if x ∈ In
DMn (x) =
0,
if x ∈
/ In

∞
X
Dn = ψn 
DMj
j=0
mj −1
X

rju  .
u=mj −nj
The norm (or quasinorm) of the space Lp (Gm ) is defined by
Z
1/p
p
kf kp :=
|f | dµ
(0 < p < ∞) .
Gm
The space Lp,∞ (Gm ) consists of all measurable functions f for which
kf kLp,∞ := supλµ (f > λ)1/p < +∞.
λ>0
The σ-algebra, generated by the intervals {In (x) : x ∈ Gm } will be denoted by
zn (n ∈ N). The conditional expectation operators relative to zn (n ∈ N) are
denoted by En . Then
Z
M
n −1
X
1
En f (x) = SMn f (x) =
fb(k) wk =
f (x)dµ(x),
|I
n (x)| In (x)
k=0
where |In (x)| = Mn−1 denotes the
length of In (x).
A sequence f = f (n) , n ∈ N of functions fn ∈ L1 (G) is said to be a dyadic
martingale if
(i) f (n) is zn measurable, for all n ∈ N,
(ii) En f (m) = f (n) , for all n ≤ m
(for details see e.g. [15]).
The maximal function of a martingale f is denoted by
f ∗ = sup f (n) .
n∈N
In case f ∈ L1 , the maximal functions are also be given by
Z
1 ∗
.
f
(u)
µ
(u)
f (x) = sup
n∈N |In (x)|
In (x)
For 0 < p < ∞, the Hardy martingale spaces Hp (Gm ) consist of all martingales, for which
kf kHp := kf ∗ kp < ∞.
If f ∈ L1 , then it is easy to show that the sequence (SMn f : n ∈ N) is a martingale. If f = f (n) , n ∈ N is martingale, then the Vilenkin-Fourier coefficients
170
GEORGE TEPHNADZE
must be defined in a slightly different manner:
Z
(4)
fb(i) := lim
f (k) (x) Ψi (x) dµ (x) .
k→∞
Gm
The Vilenkin-Fourier coefficients of f ∈ L1 (Gm ) are the same as those of
the martingale (SMn f : n ∈ N) obtained from f .
A bounded measurable function a is p-atom, if there exist a dyadic interval
I, such that
R
(i) I adµ = 0
(ii) kak∞ ≤ µ (I)−1/p
(iii) supp (a) ⊂ I.
The Hardy martingale spaces Hp (Gm ), for 0 < p ≤ 1 have an atomic characterization. Namely, the following theorem is true.
Theorem W (Weisz, [17]). A martingale f = f (n) , n ∈ N is in Hp (0 < p ≤ 1)
if and only if there exist a sequence (ak , k ∈ N) of p-atoms and a sequence
(µk , k ∈ N) of a real numbers, such that for every n ∈ N:
∞
X
µk SMn ak = f (n) ,
(5)
k=0
∞
X
|µk |p < ∞.
k=0
p 1/p
,
k=0 |µk | )
P∞
Moreover, kf kHp v inf (
decomposition of f of the form (5).
where the infimum is taken over all
When 0 < p ≤ 1, the Hardy martingale space Hp is proper subspace of
Lebesgue space Lp . It is well known that for 1 < p < ∞ the space Hp is
nothing but Lp .
The classical inequality of Hardy type is well known in the trigonometric as
well as in the Vilenkin-Fourier analysis. Namely,
b ∞
X f (k)
≤ c kf kH1 ,
k
k=1
where the function f belongs to the Hardy space H1 and c is an absolute constant. This was proved in the trigonometric case by Hardy and Littlewood [6]
(see also Coifman and Weiss [2]) and for Walsh system it can be found in [8].
FOURIER COEFFICIENTS AND PARTIAL SUMS
171
Weisz [15, 18] generalized this result for Vilenkin system and proved:
Theorem A (Weisz). Let 0 < p ≤ 2. Then there is an absolute constant cp ,
depend only p, such that
b p
∞
X f (k)
(6)
≤ cp kf kHp ,
k 2−p
k=1
for all f ∈ Hp .
Paley [7] proved that the Walsh–Fourier coefficients of a function f ∈
Lp (1 < p < 2) satisfy the condition
∞ X
b k 2
f 2 < ∞.
k=1
This results fails to hold p = 1. However, it can be verified for functions
f ∈ L1 , such that f ∗ belongs L1 , i.e. f ∈ H1 (see e.g. Coifman and Weiss [2]).
For the Vilenkin system we have the following theorem.
Theorem B (Weisz [11]). Let 0 < p ≤ 1. Then there is an absolute constant
cp , depend only p, such that
!
m
∞
k −1 2 1/2
X
X
b
2−2/p
Mk
≤ cp kf kHp ,
(7)
f (jMk )
k=1
j=1
for all f ∈ Hp .
It is well-known that Vilenkin system forms not basis in the space L1 . Moreover, there is a function in the dyadic Hardy space H1 , such that the partial
sums of f are not bounded in L1 -norm. However, in Simon [9] the following
strong convergence result was obtained for all f ∈ H1 :
n
1 X kSk f − f k1
lim
= 0,
n→∞ log n
k
k=1
where Sk f denotes the k-th partial sum of the Walsh–Fourier series of f . (For
the trigonometric analogue see Smith [12], for the Vilenkin system by Gát [3]).
For the Vilenkin system Simon proved:
Theorem C (Simon [10]). Let 0 < p < 1. Then there is an absolute constant
cp , depends only p, such that
(8)
∞
X
kSk f kpp
k=1
for all f ∈ Hp .
k 2−p
≤ cp kf kpHp ,
172
GEORGE TEPHNADZE
Strong convergence theorems of two-dimensional partial sums was investigate by Weisz [16], Goginava [4], Gogoladze [5], Tephnadze [13].
The main aim of this paper is to prove the following theorem:
Theorem 1. Let {Φn }∞
n=1 is any nondecreasing sequence, satisfying the condition lim Φn = +∞. Then there exists a martingale f ∈ Hp , such that
n→∞
b p
∞
f
(k)
Φk
X
(9)
= ∞, for 0 < p ≤ 2,
2−p
k
k=1
∞
X
Φ Mk
(10)
m
k −1 X
2/p−2
j=1
k=1 Mk
2
b
f (jMk ) = ∞, for 0 < p ≤ 1
and
∞
X
kSk f kpLp,∞ Φk
(11)
k 2−p
k=1
= ∞, for 0 < p < 1.
Proof. Let 0 < p ≤ 2 and {Φn }∞
n=1 is any nondecreasing, nonnegative sequence,
satisfying condition lim Φn = ∞.
n→∞
For this function Φ (n), there exists an increasing sequence {αk ≥ 2 : k ∈ N+ }
of the positive integers such that:
∞
X
1
< ∞.
(12)
p/4
k=1 ΦMα
k
Let
X
f (A) (x) :=
λk ak (x) ,
{k; αk <A}
where
λk =
1/p−1
1
1/4
ΦMα
, ak (x) =
Mαk
M
DMαk +1 (x) − DMαk (x) ,
k
and M = supn∈N mn .
It is easy to show that the martingale f = f (1) , f (2) , . . . , f (A) , . . . ∈ Hp .
Indeed,
(
ak (x) αk < A
(13)
SMA (ak (x)) =
0,
αk ≥ A,
Z
supp(ak ) = Iαk ,
ak dµ = 0,
Iαk
and
1/p−1
kak k∞ ≤
M αk
M
Mαk +1 ≤ Mα1/p
= µ(supp ak )−1/p .
k
FOURIER COEFFICIENTS AND PARTIAL SUMS
173
If we apply Theorem W and (12) we conclude that f ∈ Hp .
It is easy to show that

1/p−1

1 Mαk

, if j ∈ {Mαk , . . . , Mαk +1 − 1} , k = 1, 2 . . . ,
 M Φ1/4
Mα
b
k
(14)
f (j) =
∞
S


if j ∈
/
{Mαk , . . . , Mαk +1 − 1} .
0,
k=1
First we prove equality (9). Using (14) we can
p
p
Mαk +1 −1 b
M
−1
b
k
α
+1
n
f (l) Φl
X f (l) Φl X X
=
2−p
l
l2−p
n=1 l=Mαn
l=1
p
p
Mαk +1 −1 b
Mαk +1 −1 b
X f (l) Φl
X f (l)
≥
≥ cΦMαk
2−p
l
l2−p
l=M
l=M
αk
≥ cΦMαk
1/2
αk
≥
1
1/2
≥ cΦMα Mα1−p
k
k
Mαk +1 −1
Mαk +1 −1
X
Mα1−p
1
k
p/4
2−p
ΦMα l=Mα l
k
k
Mα1−p
k +1
X
1
l=Mαk
Mα2−p
k +1
1/2
cΦMα Mα1−p
k
k
≥ cΦMα → ∞, when k → ∞.
k
Next we prove equality (10). Let 0 < p ≤ 1. Using (14) we get
k
X
Mα2−2/p
ΦMαl
l
mαl −1 mαk −1 2
2
X X 2−2/p
b
f (jMαl ) ≥ Mαk ΦMαk
fb(jMαk )
j=1
l=1
j=1
mαk −1
≥ cMα2−2/p
ΦMαk
k
X Mα2/p−2
k
1/2
j=1
≥
1/2
cΦMα
k
ΦMα
k
→ ∞, when k → ∞.
Finally we prove equality (11). Let 0 < p < 1 and Mαk ≤ j < Mαk +1 . From
(14) we have
Mαk−1 +1 −1
Sj f (x) =
X
fb(l)ψl (x) +
l=0
=
η +1 −1
k−1 MαX
X
η=0 v=Mαη
j−1
X
fb(l)ψl (x)
l=Mαk
fb(v)ψv (x) +
j−1
X
fb(v)ψv (x)
v=Mαk
j−1
η +1 −1
k−1 MαX
1/p−1
1/p−1
X
X
1 M αk
1 Mαη
ψv (x) +
ψv (x)
=
M Φ1/4
M Φ1/4
η=0 v=Mαη
Mαη
Mα
v=Mα
k
k
174
GEORGE TEPHNADZE
k−1
1/p−1 X
1 Mαη
=
D
(x)
−
D
(x)
Mαη +1
Mαη
M Φ1/4
η=0
Mαη
1/p−1 1 M αk
+
Dj (x) − DMαk (x)
1/4
M ΦM
α
k
= I + II.
Let j ∈ Nn0 and x ∈ Gm \I1 . Since j − Mαk ∈ Nn0 and
Dj+Mαk (x) = DMαk (x) + ψMαk (x) Dj (x) ,
when j < Mαk . Combining (2) and (3) we can write
ψMαk Dj−Mα (x)
1/p−1
(15)
1 Mαk
|II| =
M Φ1/4
Mα
k
k
ψMαk (x) ψj−Mαk (x) r0m0 −1 (x) D1 (x)
1/p−1
1 Mαk
=
M Φ1/4
Mα
k
=
1
M
1/p−1
Mαk
.
1/4
ΦMα
k
Applying (2) and condition αn ≥ 2 (n ∈ N) for I we have
I = 0, for x ∈ Gm \I1 .
(16)
It follows that
1/p−1
|Sj f (x)| = |II| =
1 Mαk
, for x ∈ Gm \I1 .
M Φ1/4
Mα
k
Hence

1
2M
1/p−1
M αk
µ x
1/4
Φ Mα
k
∈ Gm : |Sj (f (x))| >
≥
1
2M
1/p−1
M αk
µ x
1/4
Φ Mα
k
∈ Gm \I1 : |Sj (f (x))| >
=
1 M αk
1/4
2M ΦM
α
≥
1/p−1
cMαk
.
1/4
Φ Mα
k
kSj (f (x))kLp,∞ ≥
1
2M

(17)
1/p−1
k
|Gm \I1 |
1/p
1/p−1
M αk 
1/4
Φ Mα
k
1
2M
1/p
1/p−1
Mαk 
1/4
ΦMα
k
FOURIER COEFFICIENTS AND PARTIAL SUMS
175
Combining (1) and (17) we have
Mαk +1 −1
X kSj (f (x))kpLp,∞ Φj
j=1
j 2−p
Mαk +1 −1
X kSj (f (x))kpLp,∞ Φj
≥
j 2−p
j=Mαk
X
≥ ΦMαk
{j:Mk ≤j≤Mk+1 ,
p/4
ΦMα
k
3/4
≥ cΦMα Mα1−p
k
k
X
{j:Mk ≤j≤Mk+1 ,
X
{j:Mk ≤j≤Mk+1 ,
1
j∈Nn0 }
k
{j:Mk ≤j≤Mk+1 ,
j 2−p
1
j∈Nn0 }
X
3/4
ΦMα
Mαk +1
j 2−p
j∈Nn0 }
Mα1−p
k
≥ cΦMαk
≥c
kSj (f (x))kpLp,∞
Mα2−p
k +1
1
j∈Nn0 }
3/4
≥ cΦMα → ∞, when k → ∞. k
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Received August 28, 2012.
Department of Mathematics,
Faculty of Exact and Natural Sciences,
Tbilisi State University,
Chavchavadze str. 1, Tbilisi 0128, Georgia
E-mail address: giorgitephnadze@gmail.com