Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
30 (2014), 91–101
www.emis.de/journals
ISSN 1786-0091
ON ALMOST EVERYWHERE CONVERGENCE OF SOME
SUB-SEQUENCES OF FEJÉR MEANS FOR INTEGRABLE
FUNCTIONS ON UNBOUNDED VILENKIN GROUPS
NACIMA MEMIĆ
Abstract. By means of Gát’s methods in [2] our aim is to prove the
almost everywhere convergence of some sub-sequences of (σn f )n to f , for
every integrable function f on unbounded Vilenkin groups. These are in fact
sub-sequences of the form (σan Mn f )n , where the numbers an are bounded.
This result can be considered as a generalization of Gát, s result concerning
the almost everywhere convergence of the sequence (σMn f )n on unbounded
Vilenkin groups for every integrable function f .
1. Introduction
Many results concerning the a.e. convergence of the Fejér means (σn f )n have
been obtained for Vilenkin groups. On bounded groups, mean convergence
holds almost everywhere for integrable functions [4]. However, using different
methods on unbounded groups, G. Gát [1] proved this result for Lp functions
when p > 1, and obtained in [2] that σMn f → f , a.e. for every integrable
function f . The same author [3] established the mean convergence almost
everywhere of the full sequence for integrable functions on rarely unbounded
groups. In the present paper we establish the almost everywhere convergence
of sub-sequences of the form (σan Mn f )n , where the numbers an are bounded,
to the integrable function f .
Let (m0 , m1 , . . . , mn , . . . ) be an unbounded sequence of integers not less
than 2. We
Q∞denote by P the set of positive integers and let N = P ∪ {0}.
Let G := n=0 Zmn , where Zmn denotes the discrete group of order mn , with
addition mod mn . Each element from G can be represented as a sequence
(xn )n , where xn ∈ {0, 1, . . . , mn − 1}, for every integer n ≥ 0. Addition in G
is obtained coordinatewise.
2010 Mathematics Subject Classification. 42C10.
Key words and phrases. Unbounded Vilenkin groups, Fejér means, almost everywhere
convergence.
91
92
NACIMA MEMIĆ
The topology on G is generated by the subgroups In := {x = (xi )i ∈ G, xi =
0 for i < n}, and their translations In (y) := {x = (xi )i ∈ G, xi = yi for i < n}.
Sometimes we write In (y) in the form In (y) = In (y0 , . . . , yn−1 ).
Define the sequence (Mn )n as follows: M0 = 1 and Mn+1 = mn Mn .
If µ(In ) denotes the normalized product measure of In then it can be easily
seen that µ(In ) = Mn−1 .
The generalized Rademacher functions are defined by
rn (x) := e
2πixn
mn
, n ∈ N, x ∈ G,
For every non-negative integer n, there exists a unique sequence (ni )i so that
∞
P
n=
ni Mi .
i=0
and the system of Vilenkin functions by
ψn (x) :=
∞
Y
n ∈ N, x ∈ G.
rini (x),
i=0
The Fourier coefficients, the partial sums of the Fourier series, the mean
values, the Dirichlet kernels, the Fejér means, and the Fejér kernels with respect
to the Vilenkin system are respectively defined as follows
Z
fˆ(n) =
f (x)ψ̄n (x)dx,
n−1
X
Sn f =
fˆ(k)ψk ,
En (f ) = SMn f,
k=0
Dn =
n−1
X
1X
Sk f,
n k=1
n
ψk ,
σn f =
k=0
for every f ∈ L (G).
It can be easily seen that
Z
Sn f (y) = Dn (y − x)f (x)dx,
1X
Dk ,
n k=1
n
Kn =
1
and
DMn (x) = Mn 1In (x),
Z
En f (y) = Mn
f (x)dx.
In (y)
Let A, s, j be fixed positive integers such that j ≤ A and s < mA , then
following G.Gát in the definition of the operators Hj,A and Hj in [2], we define
the operators
Z
s
H̃j,A f (y) = MA−j |
S
xA−j 6=yA−j
s
f (x)r̄A
(x)
IA (y0 ,...,yA−j−1 ,xA−j ,yA−j+1 ,...,yA−1 )
1
1 − rA−j (y − x)
dx|,
ON ALMOST EVERYWHERE CONVERGENCE OF SUB-SEQUENCES. . .
93
and for fixed s, j
s
H̃js f (y) = sup H̃j,A
f (y).
A:s<mA
The notation C will be used for an absolute positive constant that may vary
in different contexts.
2. Main results
Lemma 2.1. For every fixed s the operator H̃1s is bounded on L2 .
Proof. Let f ∈ L2 . Using the proof of [2, Lemma 2.3.] we can write
s
s
H̃1,A
f = H1,A (f r̄A
),
from which we get
s
s
s 2
kH̃1,A
f k22 = kH1,A (f r̄A
)k22 ≤ Ckf r̄A
k2 ≤ Ckf k22 .
Since
s
s
H̃1,A
f = H̃1,A
(EA+1 f ),
and
s
H̃1,A
(EA f ) = 0,
it follows
s
s
(EA+1 f − EA f )k22
f k22 = k sup H̃1,A
k sup H̃1,A
A:s<mA
A:s<mA
≤
X
s
kH̃1,A
(EA+1 f − EA f )k22
A:s<mA
≤C
X
k(EA+1 f − EA f )k22 ≤ Ckf k22 .
A
Lemma 2.2. For every fixed s the operator H̃1s is of weak type (L1 , L1 ).
Proof. We proceed as in the proof of [2, Lemma 2.4.]. Namely, let f ∈ L1 be
such that
β
[
supp(f ) ⊂
Ik (z, j) = I,
j=α
where Ik (z, j) = Ik+1 (z0 , z1 , . . . , zk−1 , j), for some fixed z ∈ G and j ∈ {α, α +
1, . . . , β} ⊂ {0, 1, . . . , mk − 1}.
R
R
s
R If ss < min(mk , mk+1 ), then suppose that I f dµ = 0, I f r̄k dµ = 0 and
f r̄k+1 dµ = 0. We construct the set 6I as done in [2, Lemma 2.4.].
I
Take any y ∈ Ik (z) \ 6I. Clearly,
s
s
f )(y).
f (y) = H1,k+1 (r̄k+1
H̃1s f (y) = H̃1,k+1
94
NACIMA MEMIĆ
From the proof of [2, Lemma 2.4.] it follows that
Z
Z
s
s
H̃1 f (y)dy =
H̃1,k+1
f (y)dy
Ik (z)\6I
Ik (z)\6I
Z
s
=
H1,k+1 (r̄k+1
f )(y)dy ≤ Ckf k1 .
Ik (z)\6I
Now if y ∈ Ik−1 (z) \ Ik (z), we get
s
H̃1s f (y) = H̃1,k
f (y) = H1,k (f r̄ks )(y) = 0,
R
because I f r̄ks dµ = 0.
For y ∈ Il−1 (z) \ Il (z) for any l ≤ k − 1, we get
R
because I f dµ = 0.
It follows that
s
f (y) = 0,
H̃1s f (y) = H̃1,l
Z
H̃1s f (y)dy ≤ Ckf k1 .
G\6I
R
If
m
≤
s
<
m
,
then
we
only
suppose
that
f
satisfies
f dµ = 0 and
k+1
k
I
R
s
f
r̄
dµ
=
0.
k
I
Then it is easily seen that H̃1s fR (y) = 0 for every
R y ∈s G \ 6I.
If mk ≤ s < mk+1 , then for I f dµ = 0 and I f r̄k+1 dµ = 0, we get in a
similar way that
Z
H̃1s f (y)dy ≤ Ckf k1 ,
Ik (z)\6I
moreover, H̃1s f (y) = 0 for every y ∈ G \ Ik (z).
R
Finally, if s ≥ max(mk , mk+1 ), then we only suppose that I f dµ = 0. In
this case we also obtain that H̃1s f (y) = 0 for every y ∈ G \ 6I
We follow the steps in the proof of [2, Lemma 2.4.] and introduce a decomposition lemma but this latter will be the same as the decomposition made in
[5, Lemma 2]. For an arbitrary function f ∈ L1 , if λ > 0 is such that kf k1 ≤ λ
and (αk )k is a sequence of integers defined by αk = −s if s < mk and αk = 0
Sβj
otherwise, there exist mutually disjoint intervals Jj = l=α
I (z j , l), j ∈ P,
j kj
and integrable functions b and g such that
(1) f = b + g,
(2) kgk∞ ≤ Cλ,
(3) kgk1 ≤ CkfSk1 ,
Jj ,
(4) supp(b) ⊂ ∞
R
R j=1αkj
(5) Jj bdµ = Jj brkj dµ = 0, for every j ∈ P,
R
R
(6) Jj |b|dµ ≤ C Jj |f |dµ, for every j ∈ P,
P
kf k1
(7) ∞
j=1 µ(Jj ) ≤ λ .
ON ALMOST EVERYWHERE CONVERGENCE OF SUB-SEQUENCES. . .
95
In [5, Lemma 2] it was proved that for every j ∈ P there exist constants akj
and bkj such that
αk
b(x) = f (x) − akj − bkj r̄kj j (x),
We introduce the functions
−1
∀x ∈ Jj .
Z
hj (x) = [b(x) − (µ(Jj )) (
Jj
f r̄ksj +1 dµ)rksj +1 (x)]1Jj (x),
j ∈ P,
if s < mkj +1 and hj (x) = b(x)1Jj (x), otherwise.
Notice that for s < mkj +1
Z
Z
Z
Z
−1
s
hj dµ =
bdµ − (µ(Jj ))
f r̄kj +1 dµ
rksj +1 dµ = 0,
because
Z
Jj
Jj
R
Jj
Jj
Jj
R
rksj +1 dµ = 0. But also , since Jj rksj +1 r̄ksj dµ = 0, we have
Z
Z
Z
−1
s
s
s
f r̄kj +1 dµ
rksj +1 r̄ksj dµ = 0.
br̄kj dµ − (µ(Jj ))
hj r̄kj dµ =
Jj
Jj
Jj
Besides
Z
Z
Jj
hj r̄ksj +1 dµ
Jj
Z
br̄ksj +1 dµ
−
f r̄ksj +1 dµ
J
Jj
Zj
Z
s
=
f r̄kj +1 dµ − akj
r̄ksj +1 dµ
Jj
Jj
Z
Z
αkj s
− b kj
r̄kj r̄kj +1 dµ −
f r̄ksj +1 dµ = 0.
=
Jj
Jj
For s ≥ mkj +1 , we obviously have
Z
Z
αk
hj dµ =
hj rkj j dµ = 0.
Jj
Jj
In this way we have proved that
Z
Z
s
H̃1 hj (y)dy ≤ Ckhj k1 ≤ C
|f |dµ,
G\6Jj
∀j ∈ P.
Jj
Following the steps used in [2, Lemma 2.4.], we obtain that
∞
∞
X
CX
C
s
µ(H̃1
hj > λ) ≤
khj k1 ≤ kf k1 .
λ j=1
λ
j=1
From
f =b+g =
∞
X
hj +
j=1
=:
∞
X
j=1
hj + G.
∞
X
j=1
Z
(µ(Jj )) ( f r̄ksj +1 dµ)rksj +1 (x)1Jj + g
−1
Jj
96
NACIMA MEMIĆ
The mutually disjoint intervals (Jj )j∈P were constructed such that
Z
Z
−1
s
−1
(µ(Jj )) |
f r̄kj +1 dµ| ≤ (µ(Jj ))
|f |dµ ≤ 3λ.
Jj
Jj
Therefore, the function G remains bounded. Moreover,
kGk1 ≤ kgk1 +
∞
X
Z
−1
Z
(µ(Jj )) |
Jj
j=1
f r̄ksj +1 dµ|
dµ ≤ kgk1 + kf k1 ≤ Ckf k1 .
Jj
Proceeding as in [2, Lemma 2.4.], we get
µ(H̃1s G
kH̃1s Gk22
kGk22
C
C
> λ) ≤ C
≤ C 2 ≤ kGk1 ≤ kf k1 .
2
λ
λ
λ
λ
Finally,
µ(H̃1s f > 2λ) ≤ µ(H̃1s
∞
X
hj > λ) + µ(H̃1s G > λ) ≤
j=1
C
kf k1 .
λ
Lemma 2.3. There exists an absolute constant C > 0 such that for all j ∈ P,
f ∈ L1 and λ > 0, we have
j 2C
µ(H̃js f > 2λ) ≤ j kf k1 .
2λ
Proof. Since
H̃js f
=
sup
j≤A,s<mA
≤
j−1
X
≤
s
H̃j,A
f
sup
j≤A,s<m
k=0 A≡k modA2j
j−1
X
sup
j≤A,s<m
k=0 A≡k modAj
s
H̃j,A
f
+
s
H̃j,A
f
j−1
X
k=0 A≡k
sup
j≤A,s<mA
mod j,A6=k mod 2j
s
H̃j,A
f.
N
using the properties of the operators Hj,k
introduced in [2, Lemma 2.5.], it is
easily seen that we only need to prove that for every k ∈ {0, 1, . . . , j − 1} the
operators
s
2j
sup
H̃j,A
f,
j≤A≤N j+k
A≡k mod 2j
and
2j
sup
A≡k
j≤A≤N j+k
mod j,A6=k mod 2j
s
f
H̃j,A
are of weak type (L1 , L1 ) uniformly on N .
We use a similar function as the permutation α introduced in [2, Lemma
2.5.]. Put
• α0 (n) = n, if n ≥ N j + k or n 6= k + 2lj, k − 1 + (2l + 1)j, for any l ∈ N,
• α0 (k + 2jl) = k + (2l + 1)j − 1, if 2l < N ,
ON ALMOST EVERYWHERE CONVERGENCE OF SUB-SEQUENCES. . .
97
• α0 (k + (2l + 1)j − 1) = k + 2lj, if 2l < N .
Let G0 be the Vilenkin group generated by the sequence (mα0 (i) )i . Then, for
A ≤ N j + k, A ≡ k mod j with A 6= k mod 2j, we have α0 (A − j) = A − 1,
α0 (A − 1) = A − j, but α0 (A) = A.
Z
1
s
s
f (x)r̄A (x)
H̃j,A f (y) = MA−j dx
S
1 − rA−j (y − x) xA−j 6=yA−j IA (y0 ,...,yA−j−1 ,xA−j ,yA−j+1 ,...,yA−1 )
Z
1
0 0
0
0 s
0
= MA−j f
(x
)(r̄
(x
))
dx
A
0
S
0
0
0
0
0 1 − r
(y 0 − x0 ) x0 6=y0 IA (y0 ,...,yA−j−1 ,yA−j ,yA−j+1 ,...,xA−1 ) A−1
A−1
=
A−1
MA−j s 0 0
s
f 0 (y 0 ),
H̃1,A f (y ) ≤ 21−j H̃1,A
MA−1
where (x0i )i = (xα0 (i) )i ∈ G0 , for every x ∈ G, (rn0 )n is the convenient set of
Rademacher functions for G0 and f 0 is defined on G0 by f 0 (x0 ) = f (x).
Following the steps of [2, Lemma 2.5.] we get that
2j
sup
j≤A≤N j+k
A≡k mod j
A6=k mod 2j
s
H̃j,A
f
is of weak type (L1 , L1 ) uniformly on N .
In a similar way if we introduce the permutation α00 :
• α00 (n) = n, if n ≥ N j + k or n 6= k + (2l + 1)j, k − 1 + 2lj, for any l ∈ N,
• α00 (k + (2l + 1)j) = k + (2l + 2)j − 1, if 2l + 1 < N ,
• α00 (k + (2l + 2)j − 1) = k + (2l + 1)j, if 2l + 1 < N ,
let G00 be the Vilenkin group generated by the sequence (mα00 (i) )i .
If A = k + 2lj, l ∈ P, we have α00 (A − j) = A − 1, α00 (A − 1) = A − j, but
00
α (A) = A, then we have
Z
1
s
s
H̃j,A
f (y) = MA−j f
(x)r̄
(x)
dx
A
S
IA (y0 ,...,yA−j−1 ,xA−j ,yA−j+1 ,...,yA−1 ) 1 − rA−j (y − x)
xA−j 6=yA−j
Z
1
00 00
00
00 s
00 f
(x
)(r̄
(x
))
= MA−j dx
A
00
00 − x00 )
S
00
00
00
1
−
r
(y
IA (y000 ,...,yA−j−1
,yA−j
,yA−j+1
,...,x00
)
A−1
A−1
x00 6=y00
A−1
=
A−1
MA−j s 00 00
s
H̃1,A f (y ) ≤ 21−j H̃1,A
f 00 (y 00 ),
MA−1
where (x00i )i = (xα00 (i) )i ∈ G00 , for every x ∈ G, (rn00 )n is the convenient set of
Rademacher functions for G00 and f 00 is defined on G00 by f 00 (x00 ) = f (x).
98
NACIMA MEMIĆ
Consequently,
2j
sup
j≤A≤N j+k
A≡k mod 2j
s
H̃j,A
f
is of weak type (L1 , L1 ) uniformly on N , and the lemma is proved.
Lemma 2.4. Let s ∈ P be fixed. Then the operator
s
sup |rN
DMN ∗ f |
N :s<mN
is of weak type (L1 , L1 ).
Proof. We first prove that the mentioned operator is bounded on L2 . For
g ∈ L2 , we have
s
krN
DM N
∗
gk22
2
Z Z
s
dy
=
r̄
(x)g(x)dx
N
IN (y)
Z
Z Z
2
s
2
2
|g(x)| dx dy
|r̄N (x)| dx
≤ MN
IN (y)
IN (y)
Z Z
2
= MN
|g(x)| dx dy = kgk22 .
MN2
IN (y)
s
s
s
Since rN
DMN ∗ (EN f ) = 0 and rN
DMN ∗ (f ) = rN
DMN ∗ (EN +1 f ), the same
s
argument used in Lemma 2.1 gives that sup |rN DMN ∗ f | is bounded on L2 .
N :s<mN
Now we use the decomposition mentioned in Lemma 2.2 with the same
notations for some fixed function f ∈ L1 and λ > kf k1 . Put bj = b · 1Jj for
∞
P
every j ∈ P. We can write f =
bj + g.
Let y ∈ G \ (
∞
S
j=1
Jj ), then for every j ∈ P, N ∈ N with s < mN ,
j=1
Z
s
(x)bj (x)dx = 0.
r̄N
IN (y)
From which we get
Z
s
(x)b(x)dx = 0,
r̄N
IN (y)
consequently,
s
DMN ∗ b)(y)| = 0.
sup |(rN
N :s<mN
ON ALMOST EVERYWHERE CONVERGENCE OF SUB-SEQUENCES. . .
99
Using the boundedness of the operator on L2 and the argument used in
Lemma 2.2, we obtain
∞
[
s
µ({ sup |rN DMN ∗ f | > 2λ} ∩ (G \
Jj ))
N :s<mN
j=1
≤ µ({ sup
N :s<mN
s
|rN
DMN
∗ b| > λ} ∩ (G \
∞
[
Jj ))
j=1
+ µ( sup
N :s<mN
s
|rN
DMN
∗ g| > λ)
s
DMN ∗ g| > λ) ≤ C
= µ( sup |rN
N :s<mN
kf k1
kgk1
≤C
.
λ
λ
Theorem 2.5. Let f ∈ L1 , l ∈ P fixed. Then SaN MN f → f almost everywhere
uniformly on aN ∈ {1, 2, . . . , min(l, mN − 1)}.
Proof. Let aN ≤ min(l, mN − 1). We write DaN MN in the following form
Da N M N =
MX
N −1
ψi +
i=0
Since
2M
N −1
X
ψi + . . . +
aNX
MN −1
ψi = DMN +
i=(aN −1)MN
i=MN
aX
N −1
s
rN
DMN .
s=1
s
s
sup |rN
DMN ∗ f | is of weak type (L1 , L1 ), and from rN
DMN ∗ g → 0,
N :s<mN
for every polynomial g, repeating the method of [2, Theorem 2.1.], we get that
s
rN
DMN ∗ f → 0, almost everywhere. The result follows from the fact that
DMN ∗ f → f , almost everywhere.
Theorem 2.6. Let f ∈ L1 , l ∈ P fixed. Then σaN MN f → f almost everywhere
uniformly on aN ∈ {1, 2, . . . , min(l, mN − 1)}.
Proof. Let aN ≤ min(l, mN − 1).
K a N MN
aN
MN
X
1
Dk
=
aN MN k=1

2M
MN
XN
1 X
Dk + . . . +
Dk +
=
aN MN k=1
k=M +1
N
1
=
aN MN
MN
X
Dk +
k=1
MN
X
DMN +k + . . . +
k=1
aX
MN
N −1 X
1
DsMN +k
=
aN MN s=0 k=1
1
=
aN MN
MN
X
k=1
Dk +
aX
MN
N −1 X
s=1 k=1
aN
MN
X

Dk 
k=(aN −1)MN +1
MN
X
D(aN −1)MN +k
k=1
!
s
Dk )
(DsMN + rN
!
100
NACIMA MEMIĆ
aN −1
MN
aX
MN
N −1
X
1 X
1 X
1
s
=
DsMN +
Dk +
r
Dk
aN s=1
aN MN k=1
aN MN s=1 N k=1
aN −1
aN −1
1 X
1
1 X
s
=
DsMN +
K MN +
rN
KMN .
aN s=1
aN
aN s=1
Using [2, Theorem 2.1.] and Theorem 2.5, in order to prove that σaN MN f =
KaN MN ∗ f → f almost everywhere uniformly on aN ∈ {1, 2, . . . , min(l, mN −
s
1)}, it suffices to prove that rN
KMN ∗ f → 0, almost everywhere uniformly on
s ∈ {1, 2, . . . , min(l, mN − 1)}.
We use the method of [2, Theorem 2.1.]. Namely, we prove that the operator
s
s
sup |rA
KMA ∗ f | is of weak type (L1 , L1 ), then noticing that rA
K MA ∗ g
A:s<mA
vanishes whenever the polynomial g is constant on IN -cosets, the result will
follow.
In fact, since KMA (z) = 1−rMtt(z) if z − zt et ∈ IA , KMA (z) = MA2+1 if z ∈ IA ,
and KMA (z) = 0 otherwise, it follows
Z
s
|(rA
KM A
∗ f )(y)| = |
s
KMA (y − x)r̄A
(x)f (x)dx|
Z
s
≤ KMA (y − x)r̄A (x)f (x)dx
IA (y)
A−1
X Z
s
+
KMA (y − x)r̄A (x)f (x)dx
t=0
It (y)\It+1 (y)
≤ SMA |f |(y) +
A−1
X
t=0
≤ SMA |f |(y) +
A−1
X
Z
Mt S
x 6=y
t
t
1
s
dx
f (x)r̄A
(x)
1
−
r
(y
−
x)
t
IA (y0 ,...,yt−1 ,xt ,yt+1 ,...,yA−1 )
s
H̃A−t,A
f (y)
t=0
= SMA |f |(y) +
A
X
s
f (y).
H̃j,A
j=1
Hence,
sup
A:s<mA
s
K MA
|rA
∗ f |(y) ≤ sup SMA |f |(y) +
A:s<mA
∞
X
H̃js f (y).
j=1
Following the steps in the proof of [2, Theorem 2.1.] and replacing Hj by
the result follows by applying Lemma 2.3.
H̃js ,
ON ALMOST EVERYWHERE CONVERGENCE OF SUB-SEQUENCES. . .
101
3. Acknowledgement
I would like to thank Professor György Gát for his kind assistance, suggestions and comments during the preparation of this manuscript.
References
[1] G. Gát. Pointwise convergence of the Fejér means of functions on unbounded Vilenkin
groups. J. Approx. Theory, 101(1):1–36, 1999.
[2] G. Gát. Cesàro means of integrable functions with respect to unbounded Vilenkin systems. J. Approx. Theory, 124(1):25–43, 2003.
[3] G. Gát. Almost everywhere convergence of Fejér means of L1 functions on rarely unbounded Vilenkin groups. Acta Math. Sin. (Engl. Ser.), 23(12):2269–2294, 2007.
[4] J. Pál and P. Simon. On a generalization of the concept of derivative. Acta Math. Acad.
Sci. Hungar., 29(1-2):155–164, 1977.
[5] W. S. Young. Mean convergence of generalized Walsh-Fourier series. Trans. Amer. Math.
Soc., 218:311–320, 1976.
Received June 1, 2014.
Department of Mathematics,
University of Sarajevo,
Zmaja od Bosne 33-35,
Sarajevo,
Bosnia and Herzegovina
E-mail address: nacima.o@gmail.com