PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 124, Number 3, March 1996
AN IMPROVED MENSHOV-RADEMACHER THEOREM
FERENC MÓRICZ AND KÁROLY TANDORI
(Communicated by J. Marshall Ash)
Abstract. We study the a.e. convergence of orthogonal series defined over
a general measure space. We give sufficient conditions which contain the
Menshov-Rademacher theorem as an endpoint case. These conditions turn
out to be necessary in the particular case where the measure space is the unit
interval [0, 1] and the moduli of the coefficients form a nonincreasing sequence.
We also prove a new version of the Menshov-Rademacher inequality.
1. Introduction
In this note we consider an arbitrary measure space X with measure µ. We
denote by Ω the class of orthonormal systems ϕ := {ϕk (x) : k ≥ 1, x ∈ X} of
functions. That is, we always assume that
Z
Z
ϕ2k (x) dµ(x) = 1 (k, l = 1, 2, . . . ).
ϕk (x)ϕl (x) dµ(x) = 0 (k 6= l),
Here and in the sequel, the integrals are extended over the whole space X.
Given a sequence a = {ak : k ≥ 1} of real numbers we introduce the quantity

1/2
p
!2
Z
X

ka(m, n)k := sup
max ak ϕk (x)
dµ(x)
,

m≤p≤n 
k=m
where m, n are integers, 1 ≤ m ≤ n, and the supremum is taken over all systems
ϕ ∈ Ω. Furthermore, set
kak := lim ka(1, n)k,
n→∞
which may be infinite.
The following theorem proved by the second named author characterizes those
sequences a for which the orthogonal series
∞
X
(1.1)
ak ϕk (x)
k=1
converges a.e. for all ϕ ∈ Ω.
Received by the editors November 1, 1993 and, in revised form, September 26, 1994.
1991 Mathematics Subject Classification. Primary 42C05.
Key words and phrases. Orthonormal system, a.e. convergence, Menshov-Rademacher inequality and theorem.
This research was partially supported by the Hungarian National Foundation for Scientific
Research under Grant #234.
c
1996
American Mathematical Society
877
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878
FERENC MÓRICZ AND KÁROLY TANDORI
Theorem A ([8, Theorem 1]). The orthogonal series (1.1) converges a.e. for all
ϕ ∈ Ω if and only if kak < ∞.
In general, it is a difficult task to check whether the norm kak is finite or not.
Therefore it makes sense to look for necessary or sufficient conditions, which are
easier to apply in concrete situations.
2. Sufficient conditions
The well-known Menshov-Rademacher theorem (see [2], [5], and also [1, p. 80])
states that if
∞
X
(2.1)
a2k (log k)2 < ∞,
k=1
then the orthogonal series (1.1) converges a.e. for all ϕ ∈ Ω. Here and in the sequel,
the logarithms are to the base 2.
We note that condition (2.1) is the best possible in the sense that for any nondecreasing sequence {w(k)} of positive numbers such that
w(k) = o(log k)
(k → ∞)
there exist a sequence {ak } of real numbers and a particular system ϕ ∈ Ω such
that
∞
X
a2k w2 (k) < ∞,
k=1
but the orthogonal series (1.1) diverges a.e. (see, e.g., [1, p. 88]).
The second named author improved the above result by showing (see [9]) that if
(2.2)
∞
X
k=1
a2k (log k) log+
1
< ∞,
a2k
then the orthogonal series (1.1) converges a.e. for all ϕ ∈ Ω. Here we agree to put
(
log(1/a2k ) if 0 < a2k ≤ 1/2,
log+ (1/a2k ) =
1
if a2k > 1/2 or ak = 0.
In [4] we proved that condition (2.2) can be replaced by the somewhat better
condition
!
∞ X
X
2 X 2
2
(2.3)
ak (log k) log
al < ∞,
a2k
r=0
k∈Jr
l∈Jr
where
(2.4)
r
Jr := {nr + 1, nr + 2, . . . , nr+1 } and nr := 22
(r ≥ 0).
It is easy to see that condition (2.3) follows from (2.2), and condition (2.2) follows
from (2.1), but not conversely.
Before stating the main result of this paper, we introduce the following notations:
X
(2.5)
Iν := {2ν + 1, 2ν + 2, . . . , 2ν+1 }, A2ν :=
a2k
(ν ≥ 0),
k∈Iν
and we agree to put a2k log(2A2ν /a2k ) = 0 in case ak = 0.
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AN IMPROVED MENSHOV-RADEMACHER THEOREM
879
Theorem 1. If for some 0 < ε ≤ 2 we have
2−ε
∞ X
X
2A2
(2.6)
a2k (log k)ε log 2ν
< ∞,
ak
ν=0
k∈Iν
then the orthogonal series (1.1) converges a.e. for all ϕ ∈ Ω.
It turns out that even more is true. Namely, under (2.6), the rearranged series
∞
X
(2.7)
akj ϕkj (x)
j=1
converges a.e. for all ϕ ∈ Ω and for all (so-called weak) permutations {k1 , k2 , . . . }
of the positive integers {1, 2, . . . } such that {kj : j ∈ Iν } is a permutation of Iν in
the case of ν large enough.
It is instructive to consider the special case of (2.6) corresponding to ε = 1:
(2.8)
∞ X
X
a2k (log k) log
ν=0 k∈Iν
2A2ν
< ∞.
a2k
We claim that even this particular condition is better than (2.3). Indeed, a simple
estimation gives
∞ X
X
a2k (log k) log
ν=1 k∈Iν
2A2ν
a2k
∞ 2X
−1 X
X
r+1
=
a2k (log k) log
r=0 ν=2r k∈Iν
≤
∞ X
X
a2k (log k) log
r=0 k∈Jr
=
∞
X
X
2A2ν
a2k
2 X 2
al
a2k
l∈Jr
a2k (log k) log
r=0 k∈Jr
2 X 2
al
a2k
!
!
,
l∈Jr
which justifies our claim.
In order to prove Theorem 1, first we prove a new version of the famous MenshovRademacher inequality (see, e.g., [1, p. 79] and [3, Theorem 3]), which says
!2
Z
N
n
X
X
(2.9)
ak ϕk (x)
dµ(x) ≤ (log 2N )2
a2k
(N ≥ 1).
max 1≤n≤N k=1
k=1
Lemma 1. For every ε, 0 < ε ≤ 2, there exists a constant C depending only on ε
such that for all sequences a, all ϕ ∈ Ω, and all integers N ≥ 1, we have
n
!2
Z
X
max ak ϕk (x)
dµ(x)
1≤n≤N k=1
(2.10)
2−ε
N
N
X
X
2A2
≤ C(log 2N )ε
a2k log 2
,
A2 :=
a2k .
ak
k=1
k=1
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880
FERENC MÓRICZ AND KÁROLY TANDORI
Proof of Lemma 1. Inequality (2.10) is obvious in case N = 1 and 2, provided
C ≥ 2. In case N ≥ 3, we may assume that
r
for some r ≥ 1.
N = nr := 22
(2.11)
Otherwise, we may supplement the given coefficients a1 , a2 , . . . , aN by an appropriate number of zeros, which does not affect the left-hand side in (2.10), while the
right-hand side increases at most by 2ε .
We rearrange the |ak | in a descending order of magnitude:
|ad(1) | ≥ |ad(2) | ≥ · · · ≥ |ad(nr ) |.
(2.12)
For each p = 0, 1, . . . , r − 1, we consider the integers of the block {d(k) : k ∈ Jp }
in their original (increasing) order, where Jp is defined in (2.4). More precisely, let
{e(k) : k ∈ Jp } be the permutation of Jp such that
d(e(np + 1)) < d(e(np + 2)) < · · · < d(e(np+1 )).
Our key estimate is the following:
n
X
ak ϕk (x)
M (x) := max 1≤n≤N k=1
n
X
≤|ad(1) ϕd(1) (x)| + |ad(2) ϕd(2) (x)| +
max ad(e(k)) ϕd(e(k)) (x) .
n∈Jp p=0
k=np +1
r−1
X
Applying Minkowski’s inequality and (2.9) gives
Z
(2.13)

1/2
1/2
r−1
X

X
M 2 (x) dµ(x)
≤ |ad(1) | + |ad(2) | +
(log 2np+1 )
a2d(k)
.


p=0
k∈Jp
By the Cauchy-Schwarz inequality, we obtain

1/2
r−1  X

X
a2d(k) (log 2k)2


p=0
k∈Jp
≤ 4ε/2

r−1 
X
2pε
p=0
(2.14)
≤ 4ε/2
X

(r−1
X
k∈Jp
2pε
)1/2 r−1
X X

p=0
4ε/2
≤ ε
(2 − 1)1/2
a2d(k) (log 2k)2−ε
(
2
rε
1/2


a2d(k) (log 2k)2−ε
p=0 k∈Jp
nr
X
(2.15)
k=3
a2d(k) (log 2k)2−ε

)1/2
a2d(k) (log 2k)2−ε
.
k=3
By (2.12), we have ka2d(k) ≤ A2 for all k. Thus,
nr
X
1/2

≤
nr
X
k=3
a2d(k)
2A2
log 2
ad(k)
!2−ε
.
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AN IMPROVED MENSHOV-RADEMACHER THEOREM
881
Taking into account (2.11), (2.13)–(2.15), and the fact that log(2A2 /a2d(k) ) ≥ 1,
we conclude
Z
1/2
M 2 (x) dµ(x)
≤ 2


ε/2
4
+2
(log N )ε

(2ε − 1)1/2
nr
X
a2d(k)
k=1
!2−ε 1/2

2A
log 2
.

ad(k)
2
This proves (2.10), since the right-hand side is symmetric in the terms a1, a2, . . . ,
anr .
Denote by
sn (x) :=
n
X
ak ϕk (x)
k=1
the nth partial sum of series (1.1).
Lemma 2 (see, e.g., [1, p. 83]). If
∞
X
(2.16)
a2k (log log k)2 < ∞,
k=2
then the subsequence {s (x) : ν ≥ 0} of the partial sums converges a.e. for all
ϕ ∈ Ω.
2ν
Proof of Theorem 1. From (2.6) it follows that
∞
X
a2k (log k)ε < ∞,
k=2
which in turn implies (2.16). By Lemma 2, s2ν (x) converges a.e. as m → ∞.
It remains to estimate the maximal fluctuation
n
X
Mν (x) := max ak ϕk (x)
(ν ≥ 1).
n∈Iν ν
k=2 +1
By Lemma 1,
Z
Mν2 (x) dµ(x) ≤ C(ν + 1)ε
X
k∈Iν
2−ε
2A2
a2k log 2ν
.
ak
By (2.6),
∞ Z
X
Mν2 (x) dµ(x) < ∞,
ν=1
whence, by the dominated convergence theorem,
n
X
lim max ak ϕk (x) = 0 a.e.
ν→∞ n∈Iν ν
k=2 +1
This completes the proof of Theorem 1.
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882
FERENC MÓRICZ AND KÁROLY TANDORI
3. Necessary conditions
From now on, let the measure space X be the unit interval [0, 1] with Borel
measurable subsets and Lebesgue measure µ. Clearly, the condition
∞
X
a2k < ∞
k=1
is necessary for the orthogonal series (1.1) to converge a.e. for all ϕ ∈ Ω. To see
this, it is enough to take the Rademacher system in the capacity of ϕ (see, e.g., [1,
p. 54]).
The second named author gave the following nontrivial necessary condition.
Theorem B ([6, Theorem 1]). If the orthogonal series (1.1) converges a.e. for all
ϕ ∈ Ω, then
2
∞
X
1
(3.1)
a2k log+ 2
< ∞.
ak
k=1
It is plain that (3.1) implies
∞ X
X
(3.2)
ν=0 k∈Iν
2
2A2
a2k log 2ν
< ∞.
ak
This latter condition coincides with (2.6) for ε = 0.
We note that condition (3.1) is not sufficient in general for the orthogonal series
(1.1) to converge a.e. for all ϕ ∈ Ω. This is shown by the next
Example 1. The second named author proved in [7] that if |a1 | ≥ |a2 | ≥ · · · , then
all rearrangements (2.7) of the orthogonal series (1.1) converge a.e. for all ϕ ∈ Ω if
and only if
(
)1/2
∞
X
X
2
2
(3.3)
ak (log k)
<∞
r=0
k∈Jr
where the blocks Jr are defined by (2.4).
Now, for the sequence
ak := r−1 2−r (nr+1 − nr )−1/2
if k ∈ Jr (r ≥ 0)
condition (3.1) is satisfied, but (3.3) is not. Consequently, there exist a system
{ϕk } ∈ Ω on the unit interval [0, 1] and a permutation {kj } of the positive integers
such that the rearranged series (2.7) diverges a.e. (see [7]). It remains to observe
that condition (3.1) is invariant under a permutation of its terms.
On the other hand, condition (2.6) for some ε > 0 is not necessary for the
orthogonal series (1.1) to converge a.e. for all ϕ ∈ Ω. To see this, we present the
following
Example 2. Let
(
r−2
ak :=
0
r
if k = nr := 22 (r ≥ 1),
otherwise.
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AN IMPROVED MENSHOV-RADEMACHER THEOREM
883
Then condition (2.6) is not satisfied for any ε > 0. Nevertheless, series (1.1)
converges a.e. for all ϕ ∈ Ω, due to
∞
X
|ak | =
k=1
∞
X
r−2 < ∞.
r=1
4. Analysis of condition (2.6)
Given a sequence a := {ak : k ≥ 1} of real
P numbers, consider the series occurring
in condition (2.6) and denote its sum by ε for some ε ≥ 0. That is, let
2−ε
∞ X
X
P
2A2ν
2
ε
ak (log k) log 2
,
ε :=
ak
ν=0
k∈Iν
which may be infinite. We note that condition (2.6) reduces to (2.1) if ε = 2, to
(2.8) if ε = 1, and to (3.2) if ε = 0.
We shall prove the following
Theorem 2. Let 0 ≤ δ < ε ≤ 2. Then there exists an absolute constant C1 such
that for every sequence a = {ak }
P
P
(4.1)
δ ≤ C1
ε.
If the sequence {|ak |} is nonincreasing, then the converse inequality
P
P
(4.2)
ε ≤ C2
δ
also holds, with another absolute constant C2 .
Combining Theorems 1 and 2 with Theorem B provides the following known
characterization (see, e.g., [8] and [9]).
Corollary. If a sequence {|ak |} is nonincreasing, then conditions (2.1), (2.8), and
(3.2) are pairwise equivalent, and each of them is necessary and sufficient for the
orthogonal series (1.1) to converge a.e. for all ϕ ∈ Ω.
Before proving Theorem 2, we prove a simplified version for finite sequences.
Lemma 3. There exists an absolute constant C3 such that for all sequences a =
{ak } and all N ≥ 1 we have
2
N
N
N
X
X
X
2A2
(4.3)
a2k log 2
≤ C3
a2k (log 2k)2 ,
A2 :=
a2k .
ak
k=1
k=1
k=1
If the sequence {|ak |} is nonincreasing, then
2
N
N
X
X
2A2
(4.4)
a2k (log 2k)2 ≤
a2k log 2
.
ak
k=1
k=1
Proof of Lemma 3. First, we assume that the sequence {|ak |} is nonincreasing.
Without loss of generality, we may assume that A = 1. By monotonicity,
ka2k ≤ 1,
whence 2k ≤ 2/a2k (1 ≤ k ≤ N ).
This proves (4.4).
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884
FERENC MÓRICZ AND KÁROLY TANDORI
From now on, we do not use monotonicity of {|ak |}. We shall distinguish between
two cases accordingly as
(i) k 4 a2k ≥ 1, whence log a22 ≤ 4 log 2k;
k
4
(ii) k 4 a2k < 1, whence a2k log a22 ≤ C
k2 ,
k
where
2
2
C4 := max t log 2
.
0<t≤1
t
(4.5)
To sum up,
N
X
a2k
k=1
2
N
N
X
X
2
1
≤ 16
a2k (log 2k)2 + C4
.
log 2
ak
k2
k=1
Hence (4.3) follows with C3 := 16 +
k=1
2
C4 π6 .
Proof of Theorem 2. It is plain that if
P
ε
∞
X
A :=
< ∞ for some ε > 0, then
a2k < ∞.
k=1
Without loss of generality, we may assume that A = 1.
Again, we distinguish between two cases accordingly as
(i) k 4 a2k ≥ A2ν , whence log
2A2ν
a2k
(ii) k 4 a2k > A2ν , whence a2k (log
≤ 4 log 2k;
2A2ν ε−δ
)
a2k
where C4 is defined in (4.5). Hence
P
δ
≤4ε−δ
∞ X
X
ν
≤ C4 A
k2 ,
2−ε
2A2
a2k (log 2k)ε log 2ν
ak
ν=0 k∈Iν
∞ X
X
+ C4
ν=0 k∈Iν
(log 2k)δ
Aν .
k2
Since Aν ≤ A = 1, we get
P
δ
Due to the fact that
P
ε
≤ 16
P
ε
+ C4
∞
X
(log 2k)2
k2
k=2
.
≥ A2 = 1, this gives (4.1) with
C1 := 16 + C4
∞
X
(log 2k)2
k=2
k2
.
Relying on (4.1) just proved, it is enough to check (4.2) in the special case when
δ = 0 and ε = 2. By Lemma 3, for ν ≥ 6 we have
1
2
(4.6)
ν+1
2X
a2k (log 2k)2
≤
k=2ν +2ν−1
ν+1
2X
a2k (log 2(k − 2ν ))2
k=2ν +2ν−1
≤
X
k∈Iν
a2k
2
2A2ν
log 2
.
ak
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AN IMPROVED MENSHOV-RADEMACHER THEOREM
885
By monotonicity, for ν ≥ 6 we have
(4.7)
1 X 2
ak (log 2k)2 ≤
8
k∈Iν+1
ν+1
2X
a2k (log 2k)2 .
k=2ν +2ν−1
Combining (4.6) and (4.7) yields (4.2).
Acknowledgment
The authors are grateful to the referee for the valuable suggestions which improved the presentation of this paper.
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Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary
E-mail address: [email protected]
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