ON THE L1 NORM OF THE WEIGHTED MAXIMAL
FUNCTION OF FEJÉR KERNELS WITH RESPECT
TO THE WALSH-KACZMARZ SYSTEM
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
KÁROLY NAGY
Institute of Mathematics and Computer Science
College of Nyíregyháza
P.O. Box 166, Nyíregyháza
H-4400 Hungary
EMail: nkaroly@nyf.hu
Received:
24 May, 2007
Accepted:
15 February, 2008
Communicated by:
Zs. Pales
2000 AMS Sub. Class.:
42C10.
Key words:
Walsh-Kaczmarz system, Fejér kernels, Fejér means, Maximal operator.
Abstract:
The main aim of this paper is to investigate the integral of the weighted maximal
function of the Walsh-Kaczmarz-Fejér kernels. We give a necessary and sufficient conditions for that the weighted maximal function of the Walsh-KaczmarzFejér kernels is in L1 . After this we discuss the weighted maximal function of
(C, α) kernels with respect to Walsh-Paley system too.
Title Page
Contents
JJ
II
J
I
Page 1 of 17
Go Back
Full Screen
Close
Contents
1
Introduction and Preliminaries
3
2
The Results
7
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 2 of 17
Go Back
Full Screen
Close
1.
Introduction and Preliminaries
The Walsh-Kaczmarz system was introduced in 1948 by Šneider [9]. He showed that
the behavior of the Dirichlet kernel of the Walsh-Kaczmarz system is worse than of
the kernel of the Walsh-Paley system. Namely, he showed in [9] that the inequality
n (x)|
lim sup |Dlog
≥ C > 0 holds a.e. for the Dirichlet kernel with respect to the Walshn
Kaczmarz system. This allows us to construct examples of divergent Fourier series
[2].
On the other hand, Schipp [6] and Wo-Sang Young [10] proved that the WalshKaczmarz system is a convergence system. Skvorcov [8] verified the everywhere
and uniform convergence of the Fejér means for continous functions. Gát proved [4]
that the Fejér-Lebesgue theorem holds for the Walsh-Kaczmarz system.
It is easy to show that the L1 norm of supn |Dn | with respect to both systems is
infinite. Gát in [3] raised the following problem: "What happens if we apply some
weight function α? That is, on what conditions do we find the inequality
sup Dn < ∞
n α(n) 1
to be valid?" He gave necessary and sufficient conditions for both rearrangements
of the Walsh system. The main aim of this paper to give necessary and sufficient
conditions for the maximal function of Fejér kernels with weight function α for both
rearrangements.
First we give a brief introduction to the theory of dyadic analysis [7, 1].
Denote by Z2 the discrete cyclic group of order 2, that is Z2 = {0, 1}, the group
operation is modulo 2 addition and every subset is open. The normalized Haar
measure on Z2 is given in the way that the measure of a singleton is 1/2, that is,
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 3 of 17
Go Back
Full Screen
Close
µ({0}) = µ({1}) = 1/2. Let
∞
G := × Z2 ,
k=0
G is called the Walsh group. The elements of G can be represented by a sequence
x = (x0 , x1 , . . . , xk , . . . ), where xk ∈ {0, 1} (k ∈ N) (N := {0, 1, . . . }, P :=
N\{0}).
The group operation on G is coordinate-wise addition (denoted by +), the measure (denoted by µ) and the topology are the product measure and topology. Consequently, G is a compact Abelian group. Dyadic intervals are defined by
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
I0 (x) := G,
In (x) := {y ∈ G : y = (x0 , . . . , xn−1 , yn , yn+1 . . . )}
for x ∈ G, n ∈ P. They form a base for the neighborhoods of G. Let 0 = (0 : i ∈
N) ∈ G and In := In (0) for n ∈ N.
Furthermore, let Lp (G) denote the usual Lebesgue spaces on G (with the corresponding norm k · k). The Rademacher functions are defined as
rk (x) := (−1)xk
(x ∈ G, k ∈ N).
P∞
i
Each natural number n can be uniquely expressed as n =
i=0 ni 2 , ni ∈
{0, 1} (i ∈ N), where only a finite number of ni ’s are different from zero. Let the
order of n > 0 be denoted by |n| := max{j ∈ N : nj 6= 0}. That is, |n| is the
integer part of the binary logarithm of n.
Define the Walsh-Paley functions by
ωn (x) :=
∞
Y
k=0
P|n|
(rk (x))nk = (−1)
k=0
nk xk
.
Title Page
Contents
JJ
II
J
I
Page 4 of 17
Go Back
Full Screen
Close
Let the Walsh-Kaczmarz functions be defined by κ0 = 1 and for n ≥ 1
|n|−1
κn (x) := r|n| (x)
Y
P|n|−1
(r|n|−1−k (x))nk = r|n| (x)(−1)
k=0
nk x|n|−1−k
.
k=0
The Walsh-Paley system is ω := (ωn : n ∈ N) and the Walsh-Kaczmarz system
is κ := (κn : n ∈ N). It is well known that
L1 Norm of the Weighted
Maximal
{κn : 2k ≤ n < 2k+1 } = {ωn : 2k ≤ n < 2k+1 }
for all k ∈ N and κ0 = ω0 .
A relation between Walsh-Kaczmarz functions and Walsh-Paley functions was
given by Skvorcov in the following way [8]. Let the transformation τA : G → G be
defined by
τA (x) := (xA−1 , xA−2 , . . . , x1 , x0 , xA , xA+1 , . . . )
for A ∈ N. We have that
κn (x) = r|n| (x)ωn−2|n| (τ|n| (x))
(n ∈ N, x ∈ G).
Dnφ :=
n
φk ,
k=0
where φn = ωn or κn (n ∈ P).
It is known [7] that
D0φ , K0φ
(
2n ,
D2n (x) =
0,
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 5 of 17
Define the Dirichlet and Fejér kernels by
n−1
X
Károly Nagy
1X φ
Knφ :=
D ,
n k=1 k
:= 0.
x ∈ In ,
otherwise (n ∈ N).
Go Back
Full Screen
Close
Let α, β : [0, ∞) → [1, ∞) be monotone increasing functions and define the
weighted maximal function of the Dirichlet kernels Dαφ,∗ and of the Fejér kernels
Kαφ,∗ :
Dαφ,∗ (x)
|Dnφ (x)|
:= sup
,
n∈N α([log n])
Kαφ,∗ (x)
|Knφ (x)|
:= sup
n∈N α([log n])
(x ∈ G),
where φ is either the Walsh-Paley, or the Walsh-Kaczmarz system. For the the
weighted maximal function of the Dirichlet kernels with respect
P to 1the Walsh-Paley
system Dαω,∗ Gát [3] proved that Dαω,∗ ∈ L1 if and only if ∞
A=0 α(A) < ∞. Moreover, he proved that
∞
∞
X
1
1X 1
≤ kDαω,∗ k1 ≤ 2
.
2 A=0 α(A)
α(A)
A=0
For the Walsh-Kaczmarz P
system, he showed that the situation is changed, namely
A
Dακ,∗ ∈ L1 if and only if ∞
A=1 α(A) < ∞. Moreover, he proved that there exists a
positive constant C such that
∞
1 X A
− C.
kDακ,∗ k1 ≥
25 A=1 α(A)
The two conditions are quite different for the two rearrangements of the Walsh system.
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 6 of 17
Go Back
Full Screen
Close
2.
The Results
For kKαω,∗ k1 , we immediately obtain from Gát’s result the following lemma:
P
1
Lemma 2.1. Kαω,∗ ∈ L1 if and only if ∞
A=0 α(A) < ∞. Moreover,
∞
∞
X
1X 1
1
ω,∗
≤ kKα k1 ≤ 2
.
4 A=0 α(A)
α(A)
A=0
L1 Norm of the Weighted
Maximal
Károly Nagy
Proof. The upper estimation follows trivially from
n
n
1 X |Djω (x)|
1 X ω,∗
|Knω (x)|
≤
≤
D (x) ≤ Dαω,∗ (x),
α(|n|)
n j=1 α(|j|)
n j=1 α
vol. 9, iss. 1, art. 16, 2008
Title Page
that is
Kαω,∗ (x) ≤ Dαω,∗ (x)
Contents
(x ∈ G).
The lower estimation for φ = ω or κ comes from the following. On the set
IA \IA+1 we have
2A
2A + 1
1 X
φ
K2A (x) = A
k=
.
2 k=1
2
Thus, we have
JJ
II
J
I
Page 7 of 17
Go Back
Full Screen
kKαφ,∗ k1 =
=
∞ Z
X
A=0
∞
X
Kαφ,∗ (x)dµ(x) ≥
IA \IA+1
1
α(A)
A=0
Z
IA \IA+1
∞ Z
X
A=0
A
K2φA (x)
IA \IA+1
∞
α(A)
2 +1
1X 1
dµ(x) ≥
.
2
4 A=0 α(A)
dµ(x)
Close
We will show that we can obtain as good an estimation for kKακ,∗ k1 as for kKαω,∗ k1 .
This means that the behavior of the Walsh-Kaczmarz-Fejér kernels is better than the
behavior of the Walsh-Kaczmarz-Dirichlet kernels. This is the main reason, why
we have so many convergence theorems for Walsh-Kaczmarz-Fejér means [4, 8].
Namely,
Theorem 2.2. There is positive absolute constant C such that
∞
∞
X
1
1X 1
≤ kKακ,∗ k1 ≤ C
.
4 A=0 α(A)
α(A)
A=0
Corollary 2.3. Kακ,∗ ∈ L1 if and only if
P∞
1
A=0 α(A)
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
< ∞.
Title Page
Skvorcov in [8] proved that for n ∈ P, x ∈ G
|n|−1
nKnκ (x)
=1+
X
Contents
|n|−1
i
2 D2i (x) +
i=0
X
2i ri (x)K2ωi (τi (x))
i=0
|n|
+ (n − 2 )(D2|n| (x) +
ω
r|n| (x)Kn−2
|n| (τ|n| (x))).
To prove Theorem 2.2, we will use two lemmas by Gát [4].
Lemma 2.4. Let A, t ∈ N, A > t. Suppose that x ∈ It \It+1 . Then

0
if x − xt et 6∈ IA ,
ω
K2A (x) =
2t−1 if x − x e ∈ I .
t t
A
If x ∈ IA , then K2ωA (x) =
2A +1
.
2
JJ
II
J
I
Page 8 of 17
Go Back
Full Screen
Close
Set
ω
Ka,b
:=
a+b−1
X
Djω
(a, b ∈ N),
j=a
and n
(s)
:=
P∞
i=s
i
ni 2 (n, s ∈ N). Using simple calculations, we have
nKnω
=
|n|
X
ns Knω(s+1) ,2s + Dnω
(n ∈ P).
L1 Norm of the Weighted
Maximal
s=0
Lemma 2.5. Let s, t, n ∈ N, and x ∈ It \It+1 . If s ≤ t ≤ |n|, then |Knω(s+1) ,2s (x)| ≤
c2s+t . If t < s ≤ |n|, then we have

0
if x − xt et 6∈ Is ,
ω
Kn(s+1) ,2s (x) =
ω
s+t−1
if x − xt et ∈ Is .
n(s+1) (x)2
Throughout the remainder of the paper C will denote a positive absolute constant,
though not always the same at different occurences.
Proof of the Theorem 2.2. We will use Skvorcov’s result and
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 9 of 17
|n|−1
X
1
1
1
+
2i D2i (x) +
(n − 2|n| )D2|n| (x)
nα(|n|) nα(|n|) i=0
nα(|n|)
Go Back
Full Screen
|n|−1
≤
1
1 X i D2i (x)
1
+
2
+ Dαω,∗ (x) ≤
+ CDαω,∗ (x).
α(1) n i=0
α(i)
α(1)
Now, we discuss
|n|−1
X
1
2i ri (x)K2ωi (τi (x)).
nα(|n|) i=0
Close
Let Jti := {x ∈ G : xi−1 = · · · = xi−t = 0, xi−t−1 = 1} and J0i := {x ∈
G : xi−1 = 1}. For every 1 ≤ i ∈ N we can decompose G as the disjoint union:
S
i
G := Ii ∪ i−1
t=0 Jt .
By Gát’s Lemma 2.4, if x ∈ Jti , then K2ωi (τi (x)) 6= 0 only in the case when
xi−t−2 = · · · = x0 = 0, and in this case K2ωi (τi (x)) = 2t−1 .
Z
Z
Z
ω
ω
K2ωi (τi (x))dµ(x)
|ri (x)K2i (τi (x))|dµ(x) =
K2i (τi (x))dµ(x) +
G
Ii
Ii
2i + 1 1
≤
· i+
2
2
i−1 Z
X
≤1+
≤1+
t=0
i−1
X
t=0
i−1 Z
X
t=0
Jti
L1 Norm of the Weighted
Maximal
Károly Nagy
K2ωi (τi (x))dµ(x)
2t−1 dµ(x)
vol. 9, iss. 1, art. 16, 2008
Title Page
{x∈G:xi−t−1 =1,xj =0 if j<i and j6=i−t−1}
2t−1
≤ 2.
2i
Thus, we have
|n|−1
X
1
i
ω
sup 2
r
(x)K
(τ
(x))
i
i
i
2
n nα(|n|)
i=0
1
q−1
∞ Z
X
X
1
≤
sup q
2i |ri (x)K2ωi (τi (x))|dµ(x)
2
α(q)
q=0 G |n|=q
i=0
Z
q−1
∞
X 1 X
2i
|ri (x)K2ωi (τi (x))|dµ(x)
≤
q
2 α(q) i=0
G
q=0
Contents
JJ
II
J
I
Page 10 of 17
Go Back
Full Screen
Close
≤
∞
X
q=0
q−1
∞
X 1
1 X i+1
2
≤
C
.
2q α(q) i=0
α(q)
q=0
We have to discuss
n − 2|n|
ω
sup r|n| (x)Kn−2|n| (τ|n| (x)) .
nα(|n|)
n
n − 2|n|
ω
sup r|n| (x)Kn−2|n| (τ|n| (x)) dµ(x)
nα(|n|)
G n
Z
∞
X
1
n − 2|n| ω
≤
sup
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) G |n|=l
n
l=1
Z
∞
X
1
n − 2|n| ω
=
sup
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) Il |n|=l
n
l=1
Z
∞
X
1
n − 2|n| ω
+
sup
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) Il |n|=l
n
l=1
L1 Norm of the Weighted
Maximal
Károly Nagy
Z
=: S 1 + S 2 .
ω
If x ∈ I|n| , then τ|n| (x) ∈ I|n| and Kn−2|n| (τ|n| (x)) ≤ C(n − 2|n| ) and
Z
∞
X
(n − 2|n| )2
1
1
sup
dµ(x)
S ≤C
α(l)
n
|n|=l
I
l
l=1
Z
∞
X
1
≤C
sup (n − 2|n| )dµ(x)
α(l)
|n|=l
I
l
l=1
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 11 of 17
Go Back
Full Screen
Close
Z
∞
∞
X
X
1
1
l
≤C
2 dµ(x) ≤ C
.
α(l)
α(l)
I
l
l=1
l=1
Now, we investigate S 2 .
∞
l−1 Z
X
1 X
n − 2|n| ω
S ≤
sup sup
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) t=0 Jtl |n|=l |n−2|n| |=q
n
l=1
2
q<l
≤
∞
X
l=1
1
α(l)
q
1 X ω
sup sup
ns Kn(s+1) ,2s (τ|n| (x)) dµ(x)
Jtl |n|=l |n−2|n| |=q n s=0
l−1 Z
X
t=0
q<l
X
K
+
X
.
D
ω
l
Let x ∈ Jt . By Lemma 2.5 of Gát, if s ≤ t, then Kn(s+1) ,2s (τ|n| (x)) ≤ 2s+t , if
q ≥ s > t, then Knω(s+1) ,2s (τ|n| (x)) 6= 0 if and only if xl−t−2 = · · · = xl−s = 0, and
in this case Knω(s+1) ,2s (τ|n| (x)) = 2s+t .
X
K
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
q<l
∞
l−1 Z
X
1 X
1 ω
+
sup sup
Dn−2|n| (τ|n| (x)) dµ(x)
α(l) t=0 Jtl |n|=l |n−2|n| |=q n
l=1
=:
L1 Norm of the Weighted
Maximal
q
l−1 Z
l−1
∞
X
X
1 X ω
1 X
sup
≤C
Kn(s+1) ,2s (τ|n| (x)) dµ(x)
l
q
α(l) t=0 Jtl |n|=l q=0 2 + 2 s=0
l=1
q Z
l−1 t
∞
X
1 XX 1 X
≤C
2s+t dµ(x)
l
q
l
α(l) t=0 q=0 2 + 2 s=0 Jt
l=1
Title Page
Contents
JJ
II
J
I
Page 12 of 17
Go Back
Full Screen
Close
t Z
∞
l−1 l−1
X
1 X X
1 X
2s+t dµ(x)
+C
l
q
α(l) t=0 q=t+1 2 + 2 s=0 Jtl
l=1
Z
q
∞
l−1 l−1
X
X
1 X X
1
+C
2s+t dµ(x)
l + 2q
l
α(l)
2
t=0 q=t+1
s=t+1 {x∈Jt :xl−t−2 =···=xl−s =0}
l=1
q
∞
l−1 t
∞
l−1
X
X
1 XX 1 X s
1 X 2t (l − t)
≤C
2 +C
α(l) t=0 q=0 2l + 2q s=0
α(l) t=0
2l
l=1
l=1
l−1
∞
X
1 X 2t (l − t)2
+C
α(l) t=0
2l
l=1
∞
X
1
≤C
.
α(l)
l=1
D
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
ω
The inequality Dn−2|n| (τ|n| (x)) ≤ n − 2|n| gives
X
L1 Norm of the Weighted
Maximal
∞
l−1 Z
X
1 X
n − 2|n|
≤
sup sup
dµ(x)
l |n|=l
α(l)
n
|n| |=q
J
|n−2
t
t=0
l=1
JJ
II
J
I
Page 13 of 17
Go Back
q<l
∞
l−1
∞
X
X
1 X −t
1
2 ≤C
.
≤C
α(l) t=0
α(l)
l=1
l=1
The lower estimation comes from Lemma 2.1.
This completes the proof of Theorem 2.2.
Let α ∈ R, and define the nth (C, α) Fejér kernel Knφ,α and the weighted maximal
Full Screen
Close
function of the (C, α) Fejér kernels Kβφ,α,∗ by
Knφ,α
n
1 X α−1 φ
|Knφ,α |
:= α
An−k Dk , Kβφ,α,∗ := sup
,
An k=0
n∈N β([log n])
where φ = ω or κ and Aαn := (1+α)...(n+α)
for any n ∈ N, α ∈ R(α 6= −1, −2, . . . ).
n!
α
α
It is known that An ∼ n .
To investigate Kβω,α,∗ , we have to use the following lemma of Gát and Goginava
[5]:
(A)
p=1
j=2
Theorem 2.7. Let 0 < α ≤ 1, then there are positive absolute constants c, C (c, C
depend only on α) such that
c
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
A
Lemma 2.6 (G. Gát, U. Goginava). Let α ∈ (0, 1) and n := n = nA 2 + · · · +
n0 20 , then


p −1
A X
i
2X

X
c(α)
|Knω,α | ≤ α
2p(α−1)
|Kjω | + 2iα |K2ωi −1 | + 2iα D2i .


n
p−1
i=0
L1 Norm of the Weighted
Maximal
∞
X
∞
X
1
1
≤ kKβω,α,∗ k1 ≤ C
.
β(A)
β(A)
A=0
A=0
This means that the behavior of the weighted maximal function of the (C, α)
kernels is the same as the behavior of the weighted maximal function of the (C, 1)
kernels with respect to this issue.
P
1
Corollary 2.8. Kβω,α,∗ ∈ L1 if and only if ∞
A=0 β(A) < ∞.
Title Page
Contents
JJ
II
J
I
Page 14 of 17
Go Back
Full Screen
Close
Proof. α = 1 is given by Lemma 2.1.
Let |n| = A. Then by Lemma 2.6 of Gát and Goginava we have


p −1
i
A X
2X

ω,α
X
|Kn |
C(α)
p(α−1)
ω
iα
ω
iα
≤ Aα
2
|Kj | + 2 |K2i −1 | + 2 D2i

β(A)
2 β(A) i=0  p=1
j=2p−1


p −1
A
i
2

ω
ω
|K i |
|Kj |
C(α) X X p(α−1) X
Di
≤ Aα
2
+ 2iα 2 −1 + 2iα 2

2
β(p − 1)
β(i − 1)
β(i) 
p−1
p=1
i=0
L1 Norm of the Weighted
Maximal
Károly Nagy
j=2
vol. 9, iss. 1, art. 16, 2008
≤ C(α)(Kβω,∗ + Dβω,∗ ).
This, Lemma 2.1 and [3] of Gát gives that the upper estimation holds for Kβω,α,∗ .
To make the lower estimation we need to investigate K2φ,α
A , where φ = ω or κ.
On the set IA \IA+1 we have
A
2
X
A
Aα−1
Dφ (x) =
2A −j j
j=0
2
X
A
Aα−1
j=
2A −j
j=0
2
X
Aα−1
(2A − l).
l
l=0
A
Alα−1 (2A
− l) =
l=0
A −2
2X
(Aα−1
l
≥
l
X
A
(2 − j) +
j=1
l=0
Aα−1
2A −1
−
Aα−1
l+1 )
A −1
2X
l=1
(2A − l) = Aα−1
2A −1
Contents
JJ
II
J
I
Page 15 of 17
α−1 α+l
Therefore by an Abel transformation and Aα−1
< Aα−1
it follows that
l
l+1 = Al
l+1
2
X
Title Page
Aα−1
2A −1
A −1
2X
l=1
2A (2A − 1)
>0
2
Go Back
Full Screen
A
(2 − l)
Close
and
A
K2φ,α
A (x)
2
1 X α−1 φ
2A (2A − 1)
1
= α
.
A2A −j Dj (x) ≥ α Aα−1
A
A2A j=0
A2A 2 −1
2
Thus,
kKβφ,α,∗ k1
=
∞ Z
X
A=0
≥
IA \IA+1
φ,α K2A (x)
∞ Z
X
A=0
∞
X
IA \IA+1
1
≥
β(A)
A=0
≥c
Kβφ,α,∗ (x)dµ(x)
β(A)
Z
IA \IA+1
∞
X
1
.
β(A)
A=0
This completes the proof of Theorem 2.7.
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
dµ(x)
1 α−1 2A (2A − 1)
AA
dµ(x)
Aα2A 2 −1
2
Title Page
Contents
JJ
II
J
I
Page 16 of 17
Go Back
Full Screen
Close
References
[1] G.H. AGAEV, N.Ja. VILENKIN, G.M. DZHAFARLI AND A.I. RUBINSTEIN,
Multiplicative systems of functions and harmonic analysis on 0-dimensional
groups, Izd. ("ELM"), Baku, (1981), (Russian).
[2] L.A. BALAŠOV, Series with respect to the Walsh system with monotone coefficients, Sibirsk Math. Ž., 12 (1971), 25–39.
[3] G. GÁT, On the L1 norm of the weighted maximal function of the WalshKaczmarz-Dirichlet kernels, Acta Acad. Paed. Agriensis Sectio Matematicae,
30 (2003), 55–66.
[4] G. GÁT, On (C, 1) summability of integrable functions with respect to the
Walsh-Kaczmarz system, Studia Math., 130(2) (1998), 135–148.
[5] G. GÁT AND U. GOGINAVA, Almost everywhere convergence of (C, α)
quadratical partial sums of double Vilenkin-Fourier series, Georgian Math.
Journal, 13(3) (2006), 447–462
[6] F. SCHIPP, Certain rearrangements of series in the Walsh series, Mat. Zametki,
18 (1975), 193–201.
[7] F. SCHIPP, W.R. WADE, P. SIMON AND J. PÁL, Walsh Series. An Introduction
to Dyadic Harmonic Analysis, Adam Hilger (Bristol-New York 1990).
[8] V.A. SKVORCOV, On Fourier series with respect to the Walsh-Kaczmarz system, Analysis Math., 7 (1981), 141–150.
[9] A.A. ŠNEIDER, On series with respect to the Walsh functions with monotone
coefficients, Izv. Akad. Nauk SSSR Ser. Math., 12 (1948), 179–192.
[10] W.S. YOUNG, On the a.e converence of Walsh-Kaczmarz-Fourier series, Proc.
Amer. Math. Soc., 44 (1974), 353–358.
L1 Norm of the Weighted
Maximal
Károly Nagy
vol. 9, iss. 1, art. 16, 2008
Title Page
Contents
JJ
II
J
I
Page 17 of 17
Go Back
Full Screen
Close