Journal of Inequalities in Pure and
Applied Mathematics
ON RELATIONS OF COEFFICIENT CONDITIONS
LÁSZLÓ LEINDLER
Bolyai Institute
Jozsef Attila University
Aradi vertanuk tere 1
H-6720 Szeged
Hungary.
volume 5, issue 4, article 92,
2004.
Received 27 April, 2004;
accepted 20 October, 2004.
Communicated by: Alexander G.
Babenko
EMail: leindler@math.u-szeged.hu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
We analyze the relations of three coefficient conditions of different type implying
one by one the absolute convergence of the Haar series. Furthermore we
give a sharp condition which guaranties the equivalence of these coefficient
conditions.
2000 Mathematics Subject Classification: 26D15, 40A30, 40G05.
Key words: Haar series, Absolute convergence, Equivalence of coefficient conditions.
On Relations of Coefficient
Conditions
László Leindler
Partially supported by the Hungarian NFSR Grand # T042462.
Title Page
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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3
5
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1.
Introduction
A known result of P.L. Ul’janov [4] asserts that the condition
(1.1)
σ1 :=
∞
X
a
√n < ∞
n
n=3
(an ≥ 0)
implies the absolute convergence of the Haar series, i.e.
m
∞ X
2
∞
X
(k) (k) X
bm χm (x) ≡
|an χn (x)| < ∞
m=0 k=1
On Relations of Coefficient
Conditions
n=0
László Leindler
almost everywhere in (0, 1). He also verified, among others, that if the sequence
{an } is monotone then the condition (1.1) is not only sufficient, but also necessary to the absolute convergence of the Haar series.
In [1] we verified that if the condition
(1.2)
σ2 :=
∞
X
(
m+1
2X
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) 21
a2n
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<∞
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holds then the Haar series is absolute (C, α)-summable for any α ≥ 0, consequently the condition (1.2) also guarantees the absolute convergence of the Haar
series.
Recently, in [3], we showed that if the sequence {an } is only locally quasi
decreasing, i.e. if
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m=1
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J. Ineq. Pure and Appl. Math. 5(4) Art. 92, 2004
an ≤ K am
for
m ≤ n ≤ 2m
and for all
m,
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and the Haar series is absolute (C, α ≥ 0)-summable almost everywhere, then
(1.2) holds.
Here and in the sequel, K and Ki will denote positive constants, not necessarily the same at each occurrence. Furthermore we shall say that a sequence
{an } is quasi decreasing if
(0 ≤)an ≤ K am
holds for any n ≥ m. This will be denoted by {an } ∈ QDS, and if the sequence
{an } is a locally quasi decreasing, then we use the short notion {an } ∈ LQDS.
P.L. Ul’janov [5], implicitly, gave a further condition in the form
(1.3)
σ3 :=
∞
X
(
1
m=3 m(log m)
1
2
∞
X
On Relations of Coefficient
Conditions
László Leindler
) 12
a2n
<∞
n=m
which also implies the absolute convergence of the Haar series.
These results propose the question: What is the relation among these conditions?
We shall show that the condition (1.3) claims more than (1.2), and (1.2)
demands more than (1.1); and in general, they cannot be reversed. In order to
get an opposite implication, a certain monotonicity condition on the sequence
{an } is required.
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2.
Results
We establish the following theorem.
Theorem 2.1. Suppose that a := {an } is a sequence of nonnegative numbers.
Then the following assertions hold:
σ1 ≤ K σ2 ,
(2.1)
and if a ∈ LQDS then
On Relations of Coefficient
Conditions
σ2 ≤ K σ1 .
(2.2)
Similarly
László Leindler
σ2 ≤ K σ3 ,
(2.3)
and if the sequence {Am } defined by
(
Am :=
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m+1
2X
) 12
a2k
k=2m +1
belongs to QDS then
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σ1 ≤ K σ3 ,
(2.5)
and if the sequence
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σ3 ≤ K σ2 .
(2.4)
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{n a2n }
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∈ QDS then
J. Ineq. Pure and Appl. Math. 5(4) Art. 92, 2004
(2.6)
σ3 ≤ K σ1 .
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Corollary 2.2. If the sequence {n a2n } ∈ QDS then the conditions (1.1), (1.2)
and (1.3) are equivalent.
Next we show that the assumption {n a2n } ∈ QDS in a certain sense is sharp.
Namely if we claim only that the sequence {nα a2n } ∈ QDS with α < 1, then
already the implication (1.1) ⇒ (1.3), in general, does not hold.
Proposition 2.3. If (0 ≤) α < 1 then there exists a sequence {an } such that the
sequence {nα a2n } ∈ QDS, furthermore
σ1 < ∞ but
σ3 = ∞.
On Relations of Coefficient
Conditions
László Leindler
Finally we verify the following.
Proposition 2.4. The requirements
(2.7)
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{n a2n } ∈ QDS
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and the following two assumptions jointly
(2.8)
{Am } ∈ QDS
and
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{an } ∈ LQDS
are equivalent.
Acknowledgement 1. I would like to sincerest thanks to the referee for his
worthy suggestions, exceptionally for the remark that the inequality (2.6) also
follows from (2.2), (2.4) and Proposition 2.4.
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3.
Lemma
We require the following lemma being a special case of a theorem proved in [2,
Satz] appended with the inequality (3.2) which was also verified, in the same
paper, in the proof of the "Hilfssatz" (see p. 217).
Lemma 3.1. The inequality (1.3) holds if and only if there exists a nondecreasing sequence {µn } of positive numbers with the properties
∞
X
1
< ∞ and
n
µ
n
n=1
(3.1)
∞
X
a2n µn < ∞.
n=1
On Relations of Coefficient
Conditions
László Leindler
Furthermore
(3.2)
(∞
X
∞
X
1
n=3
n(log n) 2
also holds.
1
k=n
) 12
a2k
≤K
) 12
) 12 ( ∞
X 1
a2n µn
n µn
n=3
n=1
(∞
X
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4.
Proofs
Proof of Theorem 2.1. The inequality (2.1) can be verified by then Hölder inequality. Namely
m+1
∞
2X
X
∞
X
a
√n ≤
σ1 =
n m=1
m=1 n=2m +1
(
) 12 (
m+1
2X
a2n
n=2m +1
m+1
2X
1
n
n=2m +1
) 12
≤ σ2 .
To prove the inequality (2.2) we utilize the monotonicity assumption and thus
we get that
On Relations of Coefficient
Conditions
László Leindler
σ2 ≤ K
∞
X
2m/2 a2m +1 ≤ K1
m=1
m+1
2X
∞
X
1
√ an = K1 σ1 .
n
m=1 n=2m +1
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The inequality (2.3) also comes via the Hölder inequality. Let Rm :=
∞
P
n=m
Then
ν+1
σ2 =
∞ 2X
−1
X
(
ν=0 m=2ν
≤
∞
X
ν=0
2ν/2
) 21
m+1
2X
n=22ν +1

a2n
 12


.
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a2n
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n=2m +1
 ν+1
2
 2X
a2n
Contents
12
≤
∞
X
ν=0
2ν/2

∞
 X
n=22ν +1

a2n
1
2
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
J. Ineq. Pure and Appl. Math. 5(4) Art. 92, 2004
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≤ R3 + K
2ν
∞
X
2
X
ν=1 n=22ν−1 +1
1
n(log n)
1
2
R
22ν +1
≤ K1
∞
X
1
n=3
n(log n) 2
1
Rn = K1 σ3 .
In order to prove (2.4) first we define a nondecreasing sequence {µn } as follows.
Let
µn := max A−1
for 2m < n ≤ 2m+1 ,
m = 1, 2, . . . ,
k
1≤k≤m
furthermore let µ1 = µ2 = µ3 . It is clear by {Am } ∈ QDS that
(4.1)
−1
A−1
m ≤ µ2m+1 ≤ K Am
(m ≥ 1),
holds. Hence we obtain by (1.2) and (4.1) that
On Relations of Coefficient
Conditions
László Leindler
2m+1
(4.2)
∞
X
X
a2n µn ≤ K σ2 < ∞
m=1 n=2m +1
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and
∞
∞
X
X
1
1
≤K
n µn
n µn
n=1
n=3
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m+1
=K
(4.3)
∞
2X
X
1
n µn
m=1 n=2m +1
≤ K1
≤ K1
∞
X
m=1
∞
X
m=1
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µ2m+1
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Am = K1 σ2 < ∞.
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Finally, using the inequality (3.2), the estimations (4.2) and (4.3) clearly imply
the statement (2.4).
The assertion (2.5) is an immediate consequence of (2.1) and (2.3).
The proof of the declaration (2.6) is analogous to that of (2.4). The assumption {n a2n } ∈ QDS enables us to define again a nondecreasing sequence {µn }
satisfying the inequalities in (3.1). We can clearly assume that all ak > 0,
otherwise (2.6) is trivial if {n a2n } ∈ QDS. Let for n ≥ 3
1
√ ,
1≤k≤n a
k k
µn := max
and µ1 = µ2 = µ3 .
The definition of µn and the assumption {n a2n } ∈ QDS certainly imply that
1
K
√ ≤ µn ≤ √
an n
an n
(4.4)
is valid. The definition of σ1 given in (1.1) and (4.4) convey the estimations
∞
X
n=3
and
a2n
∞
X
a
√n ≤ K σ1 < ∞
µn ≤ K
n
n=3
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Conditions
László Leindler
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∞
X
n=1
∞
X
∞
X
1
a
1
√n = K σ1 < ∞.
≤K
=K
n µn
n µn
n
n=3
n=3
These estimations and (3.2) verify (2.6).
Herewith the whole theorem is proved.
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Proof of Corollary 2.2. The inequalities (2.1), (2.3) and (2.6) proved in the theorem obviously deliver the assertion of the corollary. The proof is ready.
Proof of Proposition 2.3. Setting
m
νm := 22 ,
α−1
2
εm := 2−m/2 νm+1
and
a2n := ε2m n−α
νm < n ≤ νm+1 ,
if
m = 0, 1, . . .
Then
On Relations of Coefficient
Conditions
νm+1
∞
∞
X
X
X
1+α
an
√ =
εm
n− 2
n m=0 n=ν +1
n=3
László Leindler
m
≤
∞
X
1−α
2
εm νm+1 =
m=0
however, with Rn :=
∞
P
a2k
∞
X
2−m/2 < ∞,
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m=0
1
2
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,
k=n
σ3 =
=
≥
∞
X
n=3
∞
X
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1
n(log n)
νm+1
X
m=0 n=νm +1
∞
X
1
2
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n(log n)
1
Rν 2m/2 ,
4 m=0 m+1
1
2
Rn
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J. Ineq. Pure and Appl. Math. 5(4) Art. 92, 2004
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furthermore
Rν2m
≥
≥
νk+1
∞
X
X
a2n
=
k=m n=νk +1
∞
1 X 2 1−α
ε ν
K k=m k k+1
∞
X
k=m
=
1
K
νk+1
ε2k
X
k −α
n=νk +1
∞
X
−k
2
k=m
≥
1 −m
2 .
K
From the last two estimations we clearly get that σ3 = ∞, as stated.
The proof is complete.
Proof of Proposition 2.4. First we prove that the assumption (2.7) implies both
properties claimed in (2.8). Namely by {n a2n } ∈ QDS we get that if µ > m
then
A2m
=
m+1
2X
a2n n
1
1
≥ m+1 2m a22m+1 2m+1
n
2
K
n=2m +1
µ+1
2X
1 2 µ
1
≥
a µ2 ≥
a2
2K 2 2
2K 3 n=2µ +1 n
=
1 2
A ,
2K 3 µ
i.e. {n a2n } ∈ QDS ⇒ {An } ∈ QDS holds.
The implications {n a2n } ∈ QDS ⇒ {an } ∈ QDS ⇒ {an } ∈ LQDS are
trivial.
On Relations of Coefficient
Conditions
László Leindler
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To prove the implication (2.8) ⇒ (2.7) we first prove by {an } ∈ LQDS that
if µ > m then
m+1
2X
a2k ≤ K 2m a22m
k=2m +1
and
µ
2
X
a2k ≥ 2µ−1
k=2µ−1 +1
1 2
a µ,
K 2
thus by {An } ∈ QDS we obtain that
2µ a22µ ≤ K1 2m a22m
holds, whence {n a2n } ∈ QDS plainly follows.
The proof is ended.
On Relations of Coefficient
Conditions
László Leindler
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References
[1] L. LEINDLER, Über die absolute Summierbarkeit der Orthogonalreihen,
Acta Sci. Math. (Szeged), 22 (1961), 243–268.
[2] L. LEINDLER, Über einen Äquivalenzsatz, Publ. Math. Debrecen, 12
(1965), 213–218.
[3] L. LEINDLER, Refinement of some necessary conditions, Commentationes
Mathematicae Prace Matematyczne, (in press).
[4] P.L. UL’JANOV, Divergent Fourier series, Uspehi Mat. Nauk (in Russian),
16 (1961), 61–142.
[5] P.L. UL’JANOV, Some properties of series with respect to the Haar system,
Mat. Zametki (in Russian), 1 (1967), 17–24.
On Relations of Coefficient
Conditions
László Leindler
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