Georgian Mathematical Journal
1(1994), No. 3, 235-242
ON CONVERGENCE SUBSYSTEMS OF ORTHONORMAL
SYSTEMS
G. BARELADZE
Abstract. It is proved that for any sequence {Rk }∞
k=1 of real numbers satisfying Rk ≥ k (k ≥ 1) and Rk = o(k log2 k), k → ∞, there
exists a orthonormal system {ϕn (x)}∞
n=1 , x ∈ (0; 1), such that none
∞ with n ≤ R (k ≥ 1) is a convergence
of its subsystems {ϕnk (x)}k=1
k
k
subsystem.
Let {ϕn (x)} be an orthonormal
P system (ONS) on (0; 1). It is called a
convergence system if the series cn ϕn (x) is convergent
everywhere
P almost
whenever the sequence {cn } of real numbers satisfies
cn2 < ∞.
It is well-known [1] that not every ONS {ϕn (x)} is a convergence system. However [2], [3], each of them contains some convergence subsystem {ϕnk (x)}. A question was formulated later [4] whether there exists a
common estimate of growth rate of numbers nk in the class of all ONS.
B.S.Kashin [5] answered this question in the affirmative: one can determine
a sequence of positive numbers {Rk } such that from any ONS it is possible
to choose a convergence subsystem {ϕnk } with nk ≤ Rk , 1 ≤ k < ∞. In
the same paper [5] the problem of finding {Rk } with a minimal admissible
growth order is formulated and the hypothesis Rk = k 1+ε (ε > 0) is conjectured. G.A.Karagulyan [6] proved that one can take Rk = λk , λ > 1.
However, this upper estimate is rougher than the one expected in [5].
In this paper we shall give the proof of the theorem providing the lower
estimate for {Rk }.
Theorem. For any sequence {Rk }∞
k=1 of real numbers satisfying
Rk ≥ k (k ≥ 1) and Rk = o(k log2 k), k → ∞,
(1)
there exists an ONS {ϕn (x)}∞
n=1 , x ∈ (0; 1), such that none of its subsystems
∞
{ϕnk (x)}k=1
with nk ≤ Rk (k ≥ 1) is a convergence subsystem.
Several lemmas are needed to prove this theorem.
1991 Mathematics Subject Classification. 42C20.
235
236
G. BARELADZE
Lemma 1 (H.Rademacher [7]). For any ONS {ψn (x)}N
n=1 , x ∈ (0; 1),
and any collection of real numbers {cn }N
n=1
Z
1
0

j
N
X
ŒX
Œ‘2
c2n ,
max Œ
cn ψn (x)Œ dx ≤ c log22 (N + 1)
1≤j≤N
n=1
n=1
1 ≤ N < ∞.1
Lemma 2. For any N ≥ 1 there exists an ONS
ˆ
‰N
ψ(N ) := ψnN (x) n=1 ,
x ∈ (0; 1),
satisfying, for any collection of natural numbers 1 ≤ n1 < n2 < · · · < nm ≤
N (1 ≤ m ≤ N ), the inequality
Z
0
1
j
Œ
ŒX
log N
Œ
Œ
ψnNk (x)Œdx ≥ c √2 m.
max Œ
1≤j≤m
N
k=1
(2)
Proof. We shall assure that the requirements of this lemma are satisfied by
the ONS usually used in the proof of the Menshov-Rademacher theorem.
The functions ψnN (x), 1 ≤ n ≤ N , belonging to this ONS (see [8], p.295)
have, in particular, the following properties:
(i)
 √

‘
s−1 2s−1
c N

,
,
x
∈
;

 s−n

 2N 4N ‘
√
N
s
c N
2s−1
ψn (x) := n−s , x ∈ 4N ; 2N ,


‘


n
0,
x ∈ n−1
;
2N 2N ;
1 ≤ s ≤ N,
s 6= n;
(ii) ψnN (x) is constant on each of the intervals
(iii)
€s − 1 s 
,
;
4N 4N
Z
2N + 1 ≤ s ≤ 4N ;
1
0
ψnN (x) dx = 0;
from
(0; 1) onto (−∞; ∞) with period 1.
(iv) ψnN (x) is extended
€

2s−1
Denote δs := s−1
;
≤ s ≤ N . When x ∈ δs , because of (i) we
,
1
2N
4N
have ψpN (x) ≥ 0 for 1 ≤ p ≤ s and ψpN (x) ≤ 0 for s ≤ p ≤ N (1 ≤ s ≤ N ).
1 Here
and in what follows c denotes positive absolute constants which, in general,
may differ from one equality (inequality) to another.
ON CONVERGENCE SUBSYSTEMS
237
Therefore for fixed numbers 1 ≤ n1 < n2 < · · · < nm ≤ N (1 ≤ m ≤ N )
and for each x ∈ δs (1 ≤ s ≤ N ) we obtain
m
Œ
Œ
X
X
X
ΠN
Œ
Œ
ΠX N
Œψn (x)Œ =
+
=Œ
ψnk (x)Œ +
k
k=1
k:nk ≤s
k:nk >s
k:nk ≤s
j
Œ
ŒX
ΠX
Œ
Œ
Œ
Œ
Œ
N
ψnNk (x)Œ.
+Œ
ψnk (x)Œ ≤ 3 max Œ
1≤j≤m
k:nk >s
Hence
k=1
Z X
j
m
ŒX
Œ
Œ
Œ
1
Œ
Œ
ŒψnN (x)Œdx =
max Œ
ψnNk (x)Œdx ≥
k
3 δs
δs 1≤j≤m k=1
k=1
X
1
c
=√
, 1 ≤ s ≤ N;
|nk − s|
N
Z
Z
k:1≤k≤m,
nk 6=s
1
N Z
Œ
Πj
X
Œ
ŒX N
ψnk (x)Œdx ≥
max Œ
0 1≤j≤m k=1
s=1
j
Œ
ŒX
Œ
Œ
ψnNk (x)Œdx ≥
max Œ
δs 1≤j≤m k=1
N
1
c X X
=
≥√
N s=1 k:1≤k≤m, |nk − s|
nk 6=s
m
c X X
1
log N
=√
≥ c √2 m. „
N k=1 s:1≤s≤N, |nk − s|
N
s6=nk
N
is an ONS on (0; 1)
Remark 1. For any positive integer Q, {ψnN (Qx)}n=1
also satisfying the inequality (2).
Remark 2. Let N0 , N1 , Q0 , Q1 , p be positive integers. If Q1 = 4pN0 Q0 ,
N0
then functions belonging to different collections {ψnN0 (Q0 x)}n=1
and
N1
N1
{ψn (Q1 x)}n=1 are mutually orthogonal and pairwise stochastically independent on (0; 1).
Both conclusions follow readily from (i)–(iv).
Lemma 3. Suppose a function f ∈ L1(0;1) is not equivalent to zero and
Then
ˆ
‰
1
A := x ∈ (0; 1) : |f (x)| > kf kL1
.
(0;1)
2
mes A ≥ kf k2L1
(0;1)
Ž
4kf k2L2
(0;1)
.
(3)
238
G. BARELADZE
Proof. Indeed, we obtain (using Hölder’s inequality)
Z
Z
|f (x)|dx +
kf kL1 =
|f (x)|dx ≤
(0;1)
A
(0;1)\A
≤(mes A)1/2 · kf kL2
(0;1)
1
+ kf kL1 ,
(0;1)
2
which immediately implies (3).
Lemma 4. Let N, Q, m (1 ≤ m ≤ N ) be positive integers. Then for any
collection of natural numbers 1 ≤ n1 < n2 < · · · < nm ≤ N
j
ΠJo
ŒX
n
m
Œ
Œ
ψnNk (Qx)Œ >
mes x ∈ (0; 1) : max Œ
≥c ,
1≤j≤m
2
N
k=1
where ψnN (x) ∈ ψ(N ), 1 ≤ n ≤ N (see Lemma 2), and
J :=
Z
1
j
ŒX
Œ
Œ
Œ
max Œ
ψnNk (Qx)Œ dx.
0 1≤j≤m k=1
Proof. By Lemmas 1 and 2 (see also Remark 1) we have
Z
1
0

j
ŒX
Œ‘ 2
ψnNk (Qx)Œ dx ≤ cm log22 (m + 1),
max Œ
1≤j≤m
k=1
m2
log22 N ;
J2 ≥ c
N
1 ≤ m ≤ N < ∞.
Thus, applying Lemma 3 , we obtain
j
ŒX
ΠJo
n
m
log2 N
m
Œ
Œ
mes x ∈ (0; 1) : max Œ
≥c · 2 2
ψnNk (Qx)Œ >
≥c . „
1≤j≤m
2
N log2 (m + 1)
N
k=1
Proof of the theorem. If the conditions (1) are fulfilled, then Rk = k log2 k/ε(k),
2 ≤ k < ∞, where ε(k) tends to infinity. Moreover, without loss of generality, it can be assumed that
a) R1 = 1;
b) ε(k) is a nondecreasing sequence of positive integers;
c) the sets ∆m := {k : ε(k) = m} have the form (νm−1 ; νm ] ∩ N
(m = 1, 2, . . . ), where ν0 = 0, log2 log2 log2 νm = pm (m ≥ 1) and
{pm }∞
m=1 is some increasing sequence of positive integers.
In particular, for k ∈ ∆m (m ≥ 2) we have
pm−1
m−1
log2 k > log2 νm−1 = 22
≥ 22
> m = ε(k),
ε(k log2 k) ≤ ε(νm log2 νm ) < ε(νm+1 ) = m + 1 ≤ 2ε(k).
(4)
ON CONVERGENCE SUBSYSTEMS
239
Denote
T1 := 0,
Tk := 2[k log2 k log2 log2 k] , 2 ≤ k < ∞,
(5)
and Em := {k : Rk ≤ Tm }, 2 ≤ m < ∞ (here [x] is the integer part of the
number x).
Since the function ϕ(x) = xε(x)/ log2 x increases on (e; ∞), taking into
account (4) and (5), we have, for m > ν1 ,
‰
ˆ
‰
ˆ
k log2 k
≤ Tm =
Em = {1; 2} ∪ k ≥ 3 : Rk ≤ Tm = {1; 2} ∪ k ≥ 3 :
ε(k)
 k log k ‘
ˆ
‰
2
= {1; 2} ∪ k ≥ 3 : ϕ
≤ ϕ(Tm ) ⊃
ε(k)
‰
‰ ˆ
ˆ
1
⊃ {1; 2} ∪ k ≥ 3 : 2k ≤ ϕ(Tm ) = k ≥ 1 : k ≤ ϕ(Tm ) .
2
Therefore for a large m (m ≥ m1 > ν1 )
1 Tm ε(Tm )
1
,
ϕ(Tm ) − 1 ≥
2
3 log2 Tm
1
1 Tm+1 ε(Tm+1 )
Tm+1
· Tm ≥ 2Tm ;
|Em+1 | ≥
≥
3 log2 Tm+1
3 Tm log2 Tm+1
|Em | >
(6)
(7)
Because of
X
X
1
1
1
1
=
> (pm − pm−1 ) ≥
(m ≥ 2)
log2 Tk
[k log2 k log2 log2 k]
8
8
k∈∆m
k∈∆m
we can select a subsequence Tqk ≡ Tek (1 ≤ k < ∞; Te1 = 0, Te2 = ν1 ) such
that
X
X
1
1
1
1
(8)
≡
<
≤ 2 , m ≥ 2,
e
T
2m2
log
m
2 qk
k:qk ∈∆m
k:qk ∈∆m log2 Tk
and hence
≥
X
X
k:qk ∈∆m
k:qk ∈∆m
ε(Tek )
=
log2 Tek
ε(qk )
=m
log2 Tqk
X
k:qk ∈∆m
X
k:qk ∈∆m
ε(Tqk )
≥
log2 Tqk
1
1
>
,
log2 Tqk
2m
m ≥ 2.
(9)
Let
Q2 := 1,
Nm := Tem − Tem−1 ,
Qm+1 := 4Nm Qm , 2 ≤ m < ∞.
(10)
240
G. BARELADZE
Consider the orthonormal collections
ˆ Nm
‰Nm
ψn (Qm x) n=1 , 2 ≤ m < ∞,
and construct with their aid the desired ONS {ϕn (x)}∞
n=1 as
ϕn (x) := ψkNm (Qm x),
(11)
where n ∈ (Tem−1 ; Tem ], k = n − Tem−1 , 2 ≤ m < ∞, x ∈ (0; 1) (the orthonormality follows from (10) and Remark 2).
Let {nk }∞
k=1 be a sequence of positive integers with k ≤ nk ≤ Rk , 1 ≤
k < ∞. We set
ˆ
‰
Gm := k : Tem−1 < nk ≤ Tem ,
Mm := |Gm |,
−1/2
€
for n ∈ (Tem−1 ; Tem ]; m ≥ 2.
an := (1 + Mm ) log2 Tem
On account of (8)
∞
X
∞
X
an2 k =
k=1
<
m=2 k:T
e
∞
X
k=2
X
an2 k =
e
∞
X
m−1 <nk ≤Tm
1
=
log2 Tqk
m=2
∞
X
m=2
X
k:qk ∈∆m
Mm
(1 + Mm ) log2 Tem
<
1
< ∞.
log2 Tqk
It is thus sufficient to show that the series
∞
X
ank ϕnk (x)
(12)
k=1
diverges on some set of positive measure.
Note that
Gm = {k : nk ≤ Tem }\{k : nk ≤ Tem−1 } ⊃
⊃ {k : Rk ≤ Tem }\{k : k ≤ Tem−1 }, m ≥ 2.
Consequently in view of (6) and (7)
Mm ≥ |{k : Rk ≤ Tem }| − |{k : k ≤ Tem−1 }| = |{k : Rk ≤ Tqm }| − Tqm−1 >
1
1 Tem ε(Tem )
1
,
> |Eqm | − |Eqm−1 +1 | ≥ |Eqm | ≥
2
2
6 log2 Tem
m ≥ m1 .
(13)
ON CONVERGENCE SUBSYSTEMS
Hence by (11) and Lemma 2
Z 1
Œ
Œ
e
Jm : =
max
Œ
em−1 <j≤Tem
0 T
≥c
and therefore
em−1 <nk ≤j
k:T
Œ
Œ
ank ϕnk (x)Œdx ≥
log Nm
· √2
· Mm ≥
Nm
(1 + Mm ) log2 Tem
≥c· q
s
X
241
1
q
Mm log2 Tem
≥ c ε(Tem ), m ≥ m1 ,
Tem
lim Jem = ∞.
(14)
m→∞
If Am denotes the set
Œ
n
Œ
x ∈ (0; 1) :
max
Œ
e
e
<j≤
Tm
Tm−1
X
em−1 <nk ≤j
k:T
then by (11), Lemma 4, (13) and (9)
Π1 o
Œ
ank ϕnk (x)Œ > Jem , m ≥ 2,
2
1 ε(Tem )
Mm
Mm
>
, m ≥ m1 ;
>
e
Nm
6 log2 Tem
Tm
∞
1 X
ε(Tqk )
1 X ε(Tek )
>
=
≥
e
6
6
log2 Tqk
k=m1 log2 Tk
k:qk ≥qm
mes Am ≥
∞
X
mes Am
m=2
>
1
6
1
X
k:qk >νm1
X
1
ε(Tqk )
=
log2 Tqk
6 m=1+m
1
X
k:qk ∈∆m
ε(Tqk )
= ∞.
log2 Tqk
(15)
It is easy to verify (see (10),(11), Remark 2) that {Am }∞
m=2 is a sequence
of stochastically independent sets. Therefore by (15) and the Borel-Cantelli
lemma
€

mes lim sup Am = 1.
m→∞
Hence we conclude because of (14) and the definition of sets Am that the
series (12) diverges almost everywhere on (0;1). „
References
1. D.E. Menshov, Sur les séries des fonctions orthogonales I. Fund. Math.
4(1923), 82-105.
242
G. BARELADZE
2. J. Marcinkiewicz, Sur la convergence des séries orthogonales. Studia
Math. 6(1936), 39-45.
3. D.E. Menshov, Sur la convergence et la sommation des séries des
fonctions orthogonales. Bull. Soc. Math. de France 64(1936), 147-170.
4. G. Bennet, Lectures on matrix transformation of lp spaces. Notes in
Banach spaces, pp. 39-80, Univ. Texas Press, Austin, Tex., 1980.
5. B.S. Kashin, The choice of convergence subsystem from the given
orthonormal system. (Russian) Uspehi Mat. Nauk 40(1985), No.2(242),
181-182. English translation: Russ. Math. Surv. 40(1985), 215-216.
6. G.A. Karagulyan, The choice of convergence subsystem with logarithmic density from an arbitrary orthonormal system. (Russian) Mat. Sb.
(N.S.) 136(178)(1988), No.1, 41-45.
7. H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen. Math. Ann. 87(1922), 112-138.
8. B.S. Kashin, A.A. Saakyan, Orthogonal series. (Russian) ”Nauka”,
Moscow, 1984. English translation: Translations of Mathematical Monographs, 75, American Math. Society, Providence, RI, 1989.
(Received 25.02.1993)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., 380043 Tbilisi
Republic of Georgia