Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
27 (2011), 257–265
www.emis.de/journals
ISSN 1786-0091
ON THE CONVERGENCE ALMOST EVERYWHERE OF
DOUBLE SERIES WITH RESPECT TO DIAGONAL
BLOCK-ORTHONORMAL SYSTEMS
GIVI NADIBAIDZE
Abstract. The diagonal double block-orthonormal system is introduced.
The two-dimensional generalization of Menshov-Rademacher’s and V.F.
Gaposhkin’s theorems on the almost everywhere convergence of series with
respect to block-orthonormal systems is proved.
Block-orthonormal systems were introduced by Gaposhkin [2]. He proved,
that the Menshov-Rademacher’s theorem [1] and the strong law of large numbers are valid for such systems in certain conditions. In [3] were obtained
some results on convergence and summability of series with respect to blockorthonormal systems. In particular, Menshov-Rademacher’s and Gaposhkin’s
theorems were generalized and the exact Weyl multipliers for the convergence and summability almost everywhere of series with respect to blockorthogonal systems were established in the cases, when Menshov-Rademacher’s
and Gaposhkin’s theorems are not true.
The two-dimensional analog of Menshov-Rademacher’s theorem was obtained in [4]. In [5] was considered the almost everywhere convergence of
multiple orthogonal series.
In the present paper it will be introduced a diagonal block-orthonormal
systems and it will be considered the almost everywhere convergence of double
series with respect to diagonal block-orthonormal systems.
Definition 1. Let {Mk } and{Nk } be the increasing sequences of natural numbers and ∆k = ([1, Mk+1 ] × [1, Nk+1 ])\([1, Mk ] × [1, Nk ]), (k ≥ 1). Let {ϕmn }
be a system of functions from L2 ((0, 1)2 ). The system {ϕmn } will be called a
diagonal ∆k -orthonormal system (∆k -ONS) if:
1. kϕmn k2 = 1, m = 1, 2, . . . , n = 1, 2, . . . ;
2. (ϕij , ϕpq ) = 0, for (i, j) , (p, q) ∈ ∆k , (i, j) 6= (p, q), (k ≥ 1).
2010 Mathematics Subject Classification. 42C20.
Key words and phrases. block-orthonormal systems, diagonal block-orthonormal systems.
The designated research has been fulfilled by financial support of the Georgian National
Science Foundation, Grant GNSF/ST08/3-393.
257
258
GIVI NADIBAIDZE
Let the sequences {Mk }, {Nk } be fixed and {ϕmn } be a diagonal ∆k -ONS.
Let the double series
∞
X
(1)
amn ϕmn (x, y)
m,n=1
P∞
Is given, where m,n=1 a2mn < ∞.
Under the convergence of the series (1) it is understood the convergence in
Pringscame’s sense, that is the existence of the limit
(2)
lim
M,N →∞
M X
N
X
amn ϕmn (x, y),
m=1 n=1
as M and N independently approaches infinity.
Definition 2. Let {ω (m, n)} be a sequence of positive numbers, for which
ω (m, n) ≤ ω (m, n + 1) and ω (m, n) ≤ ω (m + 1, n) (m, n = 1, 2, . . .). The sequence {ω (m, n)} will be called the Weyl multiplier for the convergence almost
everywhere (a. e.) of series
respect to diagonal ∆k -ONS {ϕmn } if the
P(1) with
2
convergence of the series ∞
a
m,n=1 mn ω(m, n) < ∞ guarantees the existence of
2
the limit (2) a. e. on (0, 1) .
In this paper, the logarithms are to the base 2.
Theorem. Let the sequences {Mk }, {Nk } be fixed and {ω1 (m)}, {ω2 (n)} be the
nondecreasing sequences of positive numbers. In order that a double sequence
{ω1 (m)ω2 (n)} be the Weyl multiplier for the convergence a. e. of series (1)
with respect to all diagonal ∆k -ONS {ϕmn }, it is necessary and sufficient that
the following two conditions be fulfilled:
∞
X
(3)
1
< ∞,
ω (Mp )ω2 (Nq )
p,q=1 1
(4)
log2 m = O(ωi (m)), i = 1, 2, (m → ∞).
Below we shall use the following lemma, which is the two-dimensional analog
of well-known lemma: (see [1, Lemma 2.3.1], [4, lemma 1]).
Lemma 1. Let {ϕmn } be a orthonormal system from L2 ((0, 1)2 ). Then for all
numbers {amn }0≤m≤M ,0≤n≤N are fulfilled:
Z
1
Z
(5)
0
0
1
2
m X
n
X
aij ϕij (x, y) dx dy
max 0≤m≤M 0≤n≤N
i=0 j=0
≤ c log (M + 2) log (N + 2)
2
2
M X
N
X
i=0 j=0
a2ij
ON THE CONVERGENCE ALMOST EVERYWHERE OF DOUBLE SERIES
Z
1
Z
1
(6)
0
0
Z
1
m N
2
N
M X
X X
X
2
max aij ϕij (x, y) dx dy ≤ c log (M + 2)
a2ij ,
0≤m≤M i=0 j=0
Z
1
(7)
0
259
0
i=0 j=0
2
M X
N
M X
n
X
X
2
a2ij .
aij ϕij (x, y) dx dy ≤ c log (N + 2)
max 0≤n≤N i=0 j=0
i=0 j=0
For (5), (6) and (7) we have generalizations of Kantorovich ([1, p. 89]):
2
m n
Z 1Z 1
X X
aij ϕij (x, y) dx dy
(8)
max 0≤m≤M,
0
0
0≤n≤N
i=0 j=0
≤c
M X
N
X
a2ij log2 (i + 2) log2 (j + 2),
i=0 j=0
Z
1
0
2
m X
N
M X
N
X
X
max aij ϕij (x, y) dx dy ≤ c
a2ij log2 (i + 2),
0≤m≤M Z
1
Z
1
(9)
0
Z
1
i=0 j=0
(10)
0
0
i=0 j=0
2
M X
n
M X
N
X
X
max aij ϕij (x, y) dx dy ≤ c
a2ij log2 (i + 2).
0≤n≤N i=0 j=0
i=0 j=0
Proof of Theorem. Sufficiency. Let for sequence {ω1 (m)ω2 (n)} the conditions
(3), (4) are fulfilled and let for sequence {amn } have:
∞
X
a2mn ω1 (m)ω2 (n) < ∞.
m,n=1
Let {ϕmn } be arbitrary diagonal ∆k -ONS. In first we shall prove that the
limit
(11)
lim SMp ,Nq (x, y) =
p,q→∞
M p Nq
X
X
amn ϕmn (x, y)
m=1 n=1
exists almost everywhere on (0, 1)2 .
Without loss of generality it can be assumed that M0 = N0 = 0 and ω1 (0) =
ω2 (0) = 1. Then
SMp+s ,Nq+r (x, y) − SMp ,Nq (x, y)
Mp+s Nq+r
Nq+r
Mp
X X
X X
= amn ϕmn (x, y) +
amn ϕmn (x, y)
m=Mp +1 n=1
m=1 n=Nq +1
260
GIVI NADIBAIDZE
Nj+1
∞ X
∞ M
i+1
X
X
X
≤
a
ϕ
(x,
y)
mn
mn
i=p j=0 m=Mi +1 n=Nj +1
Nj+1
∞ X
∞ M
i+1
X
X
X
= Ip (x, y) + Jq (x, y).
+
a
ϕ
(x,
y)
mn
mn
i=0 j=q m=Mi +1 n=Nj +1
We shall prove that the double series
Nj+1
∞ M
i+1
X
X
X
(12)
amn ϕmn (x, y)
i,j=0 m=Mi +1 n=Nj +1
converges a. e. on (0, 1)2 . Indeed, we have
Nj+1
∞ Z 1Z 1 M
i+1
X
X
X
amn ϕmn (x, y) dx dy
0 m=M +1 n=N +1
i,j=0 0
i
j
2

 21
Mi+1 Nj+1
Z
Z
∞
1
1
X
X
X
dx dy 

≤
a
ϕ
(x,
y)
mn
mn
0
0
m=Mi +1 n=Nj +1
i,j=0
1
2

! 21
Nj+1
Mi+1
∞
∞
X
X
X
X

a2mn  ≤ c
<∞
=
a2mn ω1 (m)ω2 (n)
i,j=0
m=Mi +1 n=Nj +1
i,j=1
Hence Levi’s theorem implies, that series (12) converges a. e. on (0, 1)2 . Then
almost everywhere on (0, 1)2 we have
lim Ip (x, y) = 0 and lim Jq (x, y) = 0.
p→∞
q→∞
Therefore the limit (11) exists almost everywhere on (0, 1)2 .
Let k, l be the natural numbers, for which
Mp < k ≤ Mp+1 ,
We have
max
Mp <k≤Mp+1
Nq <l≤Nq+1
Nq < l ≤ Nq+1 .
Sk,l (x, y) − SMp ,Nq (x, y)
X
X
Nq
Mp
k
l
X
X
amn ϕmn (x, y)
max amn ϕmn (x, y) +
≤ max Nq <l≤Nq+1 Mp <k≤Mp +1 m=Mp +1 n=1
m=1 n=Nq +1
k
l
X
X
amn ϕmn (x, y)
+ max Mp <k≤Mp+1 m=Mp +1 n=Nq +1
N <l≤N
q
q+1
ON THE CONVERGENCE ALMOST EVERYWHERE OF DOUBLE SERIES
M
l2
i+1
X
X
≤
sup
a
ϕ
(x,
y)
mn mn
N ≤l <l <∞
i=0 q 1 2
m=Mi +1 n=l1 +1
k2
Nj+1
∞
X
X
X
+
sup
a
ϕ
(x,
y)
mn
mn
M
≤k
<k
<∞
m=k1 +1 n=Nj +1
j=0 p 1 2
k
l
X
X
amn ϕmn (x, y)
+ max Mp <k≤Mp+1 m=Mp +1 n=Nq +1
N <l≤N
∞
X
q
∞
X
=
q+1
αiq (x, y)
+
∞
X
i=0
βjp (x, y) + δp,q (x, y).
j=0
It’s clear, that the sequences
αq (x, y) =
∞
X
αiq (x, y) and βp (x, y) =
i=0
∞
X
βjp (x, y)
j=0
are increasing sequences. Show that a. e. on (0, 1)2
lim αq (x, y) = 0 and lim βp (x, y) = 0.
q→∞
p→∞
Indeed, using lemma we have:
∞ Z
X
i=0
≤
1
Z
1
αiq (x, y)dx dy
0
0
∞
X
Z
(
1
Z
0
i=0
X
Mi+1
−
≤
∞ Z
X
i=0
X
Mi+1
1
sup
|
0
1
Z
[αiq (x, y)]2
dx dy
0
l2
X
0 Nq ≤l1 <l2 <∞ m=M +1 n=N +1
q
i
l1
X
12
1
amn ϕmn (x, y)
1
amn ϕmn (x, y)|2 dx dy) 2
m=Mi +1 n=Nq +1
2
 21
M
l
i+1
1
X
X
amn ϕmn (x, y) dx dy 
sup ≤c
0 Nq <l m=M +1 n=N +1
0
i=0
q
i
2
 12


Z 1Z 1 X
Nj+1
∞ M
∞
i+1
X
X
X


amn ϕmn (x, y) dx dy 
≤c
0
0
j=q m=Mi +1 n=Nj +1
i=0
2
 21

Mi+1
Z
Z
l
∞
∞
1
1
X
X
X
X

amn ϕmn (x, y) dx dy 
max +c
0 j=q Nj <l≤Nj+1 m=M +1 n=N +1
0
i=0
i
j
∞
X

Z

1Z
261
262
GIVI NADIBAIDZE
2
 21
Mi+1 Nj+1
X
X
≤c
amn ϕmn (x, y) dx dy 
0 m=M +1 n=N +1
i=0
j=q 0
i
j
2
 12 

Mi+1
Z
Z
∞
l
1
1
X
X
X

+
max amn ϕmn (x, y) dx dy  
0 Nj <l≤Nj+1 m=M +1 n=N +1
j=q 0
i
j

 21 
 12 

Nj+1
Nj+1
Mi+1
Mi+1
∞
∞
∞
X
X X
X X
X X

2
2 

αmn
+
αmn
log2 (n + 2) 
≤c

∞
X
i=0

∞ Z
X

1
Z
1
m=Mi+1 n=Nj +1
i=q


Mi+1
∞ X
∞
X
X

≤ c
i=0 i=q
≤ c

X
Nj+1
 21
2 
αmn
ω1 (Mi )ω2 (Nj ) ·
X
 21
Nj+1
2
αmn
log2 (n + 2)ω1 (Mi ) ·
i=0 i=q m=Mi +1 n=Nj +1
∞
∞
X
X
∞ X
∞
X
i=0 i=q
m=Mi +1 n=Nj +1

Mi+1
∞ X
∞
X
X
≤ c

i=q m=Mi +1 n=Nj +1
∞
X
i=0
1
ω1 (Mi )
 12
2
αmn
ω1 (m)ω2 (n) ,
m=1 n=Nq +1
hence
Z
1
Z
1
lim
q→∞
αq (x, y)dx dy = 0.
0
0
Then by Fatou’s theorem
lim αq (x, y) = 0 a. e. on (0, 1)2 .
(13)
q→∞
Similarly we obtain
lim βp (x, y) = 0 a. e. on (0, 1)2 .
(14)
p→∞
Now we prove that
lim δp,q (x, y) = 0 a. e. on (0, 1)2 .
(15)
p,q→∞
Indeed, using inequality (8) we get
∞ X
∞ Z 1Z 1
X
2
(x, y)dx dy
δp,q
p=0 q=0
≤c
0
0
∞ X
∞
X
X
Mp+1
X
Nq+1
p=0 q=0 m=Mp +1 n=Nq +1
1
ω1 (Mi )ω2 (Nj )
a2mn log2 (m + 2) log2 (n + 2)
! 21
! 21
ON THE CONVERGENCE ALMOST EVERYWHERE OF DOUBLE SERIES
∞
X
≤c
P∞
263
a2mn ω1 (m)ω2 (n) < ∞,
m,n=1
2
hence p,q=0 δp,q
(x, y) < ∞ almost everywhere on (0, 1)2 . Then we obtain
(15). Therefore taking into account (13), (14) and (15) we get
Sk,l (x, y) − SMp ,Nq (x, y) = 0
max
Mp <k≤Mp+1 , Nq <l≤Nq+1
almost everywhere on (0, 1)2 . Finally taking into account (11) we finished
proof of sufficiency.
Necessity. a) Let
∞
X
1
= ∞.
ω (Mp )ω2 (Nq )
p,q=1 1
Without loss of generality it can be assumed that
∞
X
1
= ∞.
ω
(M
)
1
p
p=1
Then there exist numbers cp > 0 such that
∞
X
c2p ω1 (Mp ) < ∞ and
p,q=1
∞
X
cp = ∞.
p=1
Take aMp ,N1 = cp , (p = 1, 2, . . .), amn = 0, ((m, n) 6= (Mp , N1 ), m ∈ N, n ∈
N, p ∈ N). Let ϕMp N1 (x, y) = 1, (p = 1, 2, . . .), (x, y) ∈ (0, 1)2 and choose
as other functions an arbitrary ONS orthogonal to 1. The system {ϕmn } is
diagonal ∆k -ONS , for which
∞
∞
X
X
amn ϕmn (x, y) =
cp = ∞ (x, y) ∈ (0, 1)2
m,n=1
Though
∞
X
p=1
a2mn ω1 (m)ω2 (n)
=
m,n=1
∞
X
c2p ω1 (Mp )ω2 (N1 ) < ∞.
p=1
b) Let condition (4) is not fulfilled. Without loss of generality it can be
assumed that the condition log2 m = O(ω1 (m)), (m → ∞) is not fulfilled.
Then there exist (see [3, Theorem 1.]), numbers bm and (Mp , Mp+1 ]-ONS {ϕm }
such that
∞
X
(16)
b2m ω1 (m) < ∞,
m=1
though
(17)
∞
X
m=1
bm ϕm (x)
264
GIVI NADIBAIDZE
diverges a. e. on (0, 1).
Take am,1 = bm , (m = 1, 2, . . .), amn = 0, (m ∈ N, n ≥ 2). Let {ψn } be an
ONS from L2 (0, 1) such that ψ1 (y) = 1, y ∈ (0, 1). The system ϕmn (x, y) =
ϕm (x)ψn (y) is a diagonal ∆k -orthonormal system. Then taking into account
(16), (17) we have
∞
∞
X
X
b2m ω1 (m) < ∞ ,
a2mn ω1 (m)ω2 (n) = ω2 (1)
m=1
m,n=1
though the series
∞
X
amn ϕmn (x, y) =
m,n=1
∞
X
bm ϕm (x)ψ1 (y)
m=1
diverges a. e. on (0, 1)2 .
Corollary. If we take ω1 (m) = ω2 (m) = log2 m then we obtain the following
theorem:
a) If
∞
X
1
< ∞,
(18)
2
2
log
(M
)
log
(N
)
p
q
p,q=1
then for every diagonal ∆k -ONS {ϕmn } the condition
∞
X
a2mn log2 m log2 n < ∞
m,n=1
guarantees the convergence a. e. on (0, 1)2 of the series (1).
b) If however
∞
X
1
= ∞,
2
log Mp log2 Nq
p,q=1
then there exist numbers bmn and diagonal ∆k -ONS {ψmn } such that the
series
∞
X
bmn ψmn (x, y)
m,n=1
2
diverges a. e. on (0, 1) though
∞
X
b2mn log2 m log2 n < ∞ .
m,n=1
α
α
Remark 1. For example if we take Mp = 2p , Nq = 2q , α > 21 , then the
condition (18) is fulfilled. Therefore the two-dimensional analog of MenshovRademacher’s
(see
any ∆k-ONS {ϕmn }.
pTheorem
qα[4],
theorem 1)1 is fulfilled
for
α
2
If Mp = 2 , Nq = 2 , 0 < α ≤ 2 , then log m log2 n will be the
Weyl multiplier for the convergence a. e. not for each ∆k -ONS. From proved
ON THE CONVERGENCE ALMOST EVERYWHERE OF DOUBLE SERIES
265
o
n 1
1
Theorem it follows that in that case log α +ε m log α +ε n (ε > 0) is the Weyl
multiplier.
References
[1] G. Alexits. Convergence problems of orthogonal series. Translated from the German by
I. Földer. International Series of Monographs in Pure and Applied Mathematics, Vol. 20.
Pergamon Press, New York, 1961.
[2] V. F. Gaposhkin. Series of block-orthogonal and block-independent systems. Izv. Vyssh.
Uchebn. Zaved. Mat., (5):12–18, 1990.
[3] G. Nadibaidze. On the convergence and summability of series with respect to blockorthonormal systems. Georgian Math. J., 2(5):517–534, 1995.
[4] S. G. Pandgakidze. On the menshov-rademacher’s theorem for the double orthogonal
series. Bull. Acad. Sci. Georg. SSR., 39(2):277–282, 1965.
[5] S. G. Pandgakidze. On the convergence of multiple orthogonal series. Bull. Acad. Sci.
Georg. SSR., 89(3):553–555, 1978.
Received April 03, 2011.
Department of Mathematics,
Tbilisi State University,
Chavchavadze av. 1, 0128, Tbilisi, Georgia
E-mail address: g.nadibaidze@gmail.com