ON EQUIVALENCE OF COEFFICIENT CONDITIONS
L. LEINDLER
Bolyai Institute, University of Szeged,
Aradi vértanúk tere 1,
H-6720 Szeged, Hungary
EMail: leindler@math.u-szeged.hu
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
Title Page
Received:
24 January, 2007
Accepted:
12 March, 2007
Communicated by:
J.M. Rassias
2000 AMS Sub. Class.:
26D15, 40D15.
Key words:
Numerical series, Equiconvergence, Weyl multiplier.
Abstract:
Two equivalence theorems and two corollaries are proved pertaining to
the equiconvergence of numerical series.
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1
Introduction
3
2
Results
6
3
Proofs
8
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
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1.
Introduction
Some papers, see [1], [2] and [5], have dealt with the equivalence of coefficient
conditions, e.g. in [3] it was proved that two conditions which guarantee that a factorsequence should be a Weyl multiplier for a certain property of a given orthogonal
series is equivalent to one assumption. An example of these results is the following
general theorem proved in [2].
Theorem 1.1. Let 0 < P
p < q, {λn } and {cn } be sequences of nonnegative numbers,
furthermore let Λn := nk=1 λk . The inequality
! pq
∞
∞
X
X
(1.1)
S11 :=
λn
cqk
<∞
n=1
S12 :=
∞
X
cqn µn < ∞
n=1
(1.3)
S13 :=
n=1
λn
Λn
µn
µn := Λn Cn−1 ,
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p
q−p
In the special case p = 1, q = 2, with an and
author in [3] showed that if
(1.4)
vol. 8, iss. 1, art. 8, 2007
Page 3 of 12
and
∞
X
L. Leindler
k=n
holds if and only if there exists a nondecreasing sequence {µn } of positive numbers
satisfying the following conditions
(1.2)
Equivalence of Coefficient
Conditions
< ∞.
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λ−1
n
where Cn :=
in the place of cn and λn , the
∞
X
k=n
! 21
c2k
,
then (1.1) implies (1.2) and (1.3).
In this special case it is easy to see that if εn → 0, then with µ∗n := εn µn in place
of µn , the condition (1.2) is also satisfied, but it can be that
∞
X
Λn
(1.5)
λn ∗ = ∞
µn
n=1
will occur.
This raises the question: Do (1.2) and (1.5) with the sequence {µ∗n } also imply
(1.1) for arbitrary {cn }? In [3] we showed that the answer is negative. In other
words, this verified the necessity of condition (1.3).
This shows that condition (1.2), in general, does not imply the inequality (1.3).
Thus, we can ask, promptly in connection with the general case considered in
Theorem 1.1: What is the "optimal sequence {µn }" when (1.2) implies (1.3) and
conversely?
We shall show that the optimal sequence is
! 1q
∞
X
(1.6)
µn := Λn Cnp−q , where Cn :=
cqk
,
k=n
and with this {µn } (1.2) holds if and only if (1.3) also holds, that is, the assumptions
(1.2) and (1.3) are equivalent.
Since the following symmetrical analogue of Theorem 1.1 was also verified in
[2], therefore we shall set the same question pertaining to the series appearing in it.
Theorem
1.2. If p, q, {λn } and {cn } are as in Theorem 1.1, furthermore Λ̃n :=
P∞
λ
,
then
k=n k
! pq
∞
m
X
X
cqk
(1.7)
S17 :=
λm
<∞
m=1
k=1
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
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holds if and only if there exists a nondecreasing sequence {µn } of positive numbers
satisfying conditions (1.2) and
p
! q−p
∞
X
Λ̃n
(1.8)
S18 :=
< ∞.
λn
µ
n
n=1
In order to verify our assertions made above, first we shall prove two theorems
regarding the equiconvergence of two special series.
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
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2.
Results
We prove the following assertions.
Theorem 2.1. Let 0 < P
α < 1, {an } and P
{λn } be sequences of nonnegative numbers,
n
α−1
furthermore let Λn := k=1 λk , An := ∞
k=n ak and µn := Λn An . Then the sum
(2.1)
S21 :=
∞
X
Equivalence of Coefficient
Conditions
an µ n < ∞
n=1
L. Leindler
vol. 8, iss. 1, art. 8, 2007
if and only if
(2.2)
S22 :=
∞
X
λn Aαn < ∞.
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n=1
Theorem
2.2. If P
α, {an } and {λn } are as in Theorem 2.1, furthermore Λ̃n :=
P∞
n
α−1
k=n λk , Ãn :=
k=1 ak , Ã0 := 0, and µ̃n := Λ̃n Ãn , then
(2.3)
S23 :=
∞
X
an µ̃n < ∞
n=1
if and only if
(2.4)
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S24 :=
∞
X
λn Ãαn < ∞.
n=1
Corollary 2.3. If p, q, {λn }, {cn } and Λn are as in Theorem 1.1, and µn is defined
in (1.6), then the sums in (1.1), (1.2) and (1.3) are equiconvergent.
Corollary 2.4. If p, q, {λn }, {cn } and Λ̃n are as in Theorem 1.2, furthermore
µn := Λ̃n C̃np−q ,
where
C̃n :=
n
X
! 1q
cqk
,
k=1
then the sums in (1.2), (1.7) and (1.8) are equiconvergent.
Remark 1. Corollary
P 2.3 shows that if (1.1) implies a certain property of a fixed
orthogonal series ∞
n=1 cn ϕn (x), then there is no exact universal Weyl multiplier
concerning this property, namely the multiplier sequence {µn } depends on {cn }.
Remark 2. The interested reader can check that the proofs of the implications (1.1)⇒(1.2)
and (1.7)⇒(1.2) given by our corollaries are shorter than those in [2].
Remark 3. As far as we know, Y. Okuyama and T. Tsuchikara [4]were the first to
study conditions of the type (1.7).
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
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3.
Proofs
Proof of Theorem 2.1. First we show that (2.1) implies (2.2). Since An → 0, then
(3.1)
∞
X
λn Aαn =
n=1
∞
X
n=1
=
∞
X
λn
∞
X
(Aαm − Aαm+1 )
m=n
(Aαm
−
Aαm+1 )
m=1
m
X
λn .
Equivalence of Coefficient
Conditions
L. Leindler
n=1
vol. 8, iss. 1, art. 8, 2007
An easy consideration yields that if 0 ≤ a < b, 0 < α < 1 and
b α − aα
= α ξ α−1 ,
b−a
(3.2)
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then
ξ ≥ α1/(1−α) b =: ξ 0 ,
(3.3)
namely if a = 0 then ξ = ξ 0 .
Using the relations (3.2) and (3.3) we obtain that
This and (3.1) yield that
∞
X
n=1
λn Aαn ≤
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Aαm − Aαm+1 = am αξ α−1 ≤ am Aα−1
m .
(3.4)
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∞
X
am Aα−1
m Λm =
m=1
Herewith the implication (2.1)⇒(2.2) is proved.
∞
X
m=1
am µ m .
The proof of (2.2)⇒(2.1) is similar. We use the first part of (3.4), α < 1 and
ξ < Am . Thus
∞
X
am µ m =
m=1
=
∞
X
m=1
∞
X
am Aα−1
m Λm
(Am − Am+1 )Aα−1
m Λm
Equivalence of Coefficient
Conditions
m=1
≤α
−1
∞
X
(Aαm
Aαm+1 )
−
m=1
= α−1
∞
X
= α−1
L. Leindler
λn
λn
∞
X
(Aαm − Aαm+1 )
λn Aαn ,
that is, (2.2)⇒(2.1) is verified.
The proof of Theorem 2.1 is complete.
Proof of Theorem 2.2. The proof is almost the same as that of Theorem 2.1. By (3.2)
and (3.3) we get that
α−1
.
Ãαm − Ãαm−1 = am αξ α−1 ≤ am Ãm
Utilizing this at the final step we have
∞
X
n=1
λn Ãαn
=
∞
X
n=1
λn
n
X
k=1
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m=n
n=1
(3.5)
vol. 8, iss. 1, art. 8, 2007
n=1
n=1
∞
X
m
X
(Ãαk − Ãαk−1 )
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=
∞
X
(Ãαk
−
Ãαk−1 )
k=1
∞
X
λn ≤
n=k
∞
X
ak Ãα−1
Λ̃k
k
≡
∞
X
ak µ̃k ;
k=1
k=1
this proves the implication (2.3)⇒(2.4).
To verify (2.4)⇒(2.3) we use
α−1 (Ãαm − Ãαm−1 ) ≥ am Ãα−1
m ,
Equivalence of Coefficient
Conditions
which follows from the first part of (3.5) by α < 1 and ξ < Ãm .
Thus we get that
∞
X
n=1
an µ̃n =
∞
X
L. Leindler
vol. 8, iss. 1, art. 8, 2007
an Ãα−1
n Λ̃n
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n=1
≤α
−1
∞
X
(Ãαn
−
Ãαn−1 )
n=1
=α
−1
= α−1
∞
X
k=1
∞
X
∞
X
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λk
k=n
λk
k
X
(Ãαn − Ãαn−1 )
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Summing up, the proof of Theorem 2.2 is complete.
as well as S21 ≡ S12 ,
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λk Ãαk .
k=1
S22 ≡ S11 ≡ S13
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n=1
Proof of Corollary 2.3. We shall use the results of Theorem 2.1 with α =
p−q
P
q q
an = cqn . Then µn = Λn ( ∞
, and
k=n ck )
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p
q
and
moreover, by Theorem 2.1, S21 and S22 are equiconvergent, herewith Corollary 2.3
is proved.
Proof of Corollary 2.4. Now we utilize Theorem 2.2 with α =
p−q
P
µ̃n = Λ̃n ( nk=1 cqk ) q , furthermore
S24 ≡ S17 ≡ S18
p
q
and an = cqn . Then
and S23 ≡ S12 ,
hold. Since, by Theorem 2.2, S23 and S24 are equiconvergent, thus Corollary 2.4 is
verified.
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
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References
[1] L. LEINDLER, On relations of coefficient-conditions, Acta Math. Hungar., 39
(1982), 409–420.
[2] L. LEINDLER, On equivalence of coefficient conditions and application, Math.
Inequal. Appl., 1 (1998), 41–51.
[3] L. LEINDLER, An addendum of an equivalence theorem, Math. Inequal. Appl.,
10(1) (2007), 49–55.
Equivalence of Coefficient
Conditions
L. Leindler
vol. 8, iss. 1, art. 8, 2007
[4] Y. OKUYAMA AND T. TSUCHIKURA, On the absolute Riesz summability of
orthogonal series, Analysis Math., 7 (1981), 199–208.
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[5] V. TOTIK AND I. VINCZE, On relations of coefficient conditions, Acta Sci.
Math. (Szeged), 50 (1986), 93–98.
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