On local property of absolute summability of factored Fourier series

0.17 MB·English

by  Hüseyin Bor

ON LOCAL PROPERTY OF ABSOLUTE SUMMABILITY OF
FACTORED FOURIER SERIES
3
1 HÜSEYİNBOR,DANSHENGYU,ANDPINGZHOU
0
2
Abstract. Weestablishtwo general theorems onthe localproperties of the

n absolutesummabilityoffactoredFourierseriesbyapplyingarecentlydefined
a absolute summability, |A,αn|k summability, and the class S(αn,φn), which
J generalize somewellknownresultsandcanbeappliedtoimprovemanyclas-
9 sicalabsolutesummabilitymethods.
2

]
P 1. Introduction
A
. Let A:=(ank) be a lower triangularmatrix and {sn} the partial sums of an.
h
Let {α } be a nonnegative sequence, then the series a is said to be summable
t n n P
a |A,α | ,k ≥1, if (see [19])
m n k P
∞
[ αn|An−An−1|k <∞,

1 nX=1
v where
n
4
A := a s .
7 n nv v
8 v=1
X
6 In particular, if α = nk−1 , then |A,α | −summability reduces to the |A| -
n n k k
1. summability (see [17]). Let A be the Cesàro matrices C := (cnv) of order α, that
0 is,
3 Aα−1
1 cnv := An−αv, v =0,1,··· ,n,
: n
v where
i Γ(n+α+1)
X Aα := , n=0,1,··· .
n Γ(α+1)Γ(n+1)
r
a Whenα =nδk+k−1,k ≥1,δ ≥0,|A,α | −summabilityisusuallycalled|C,α,δ| -
n n k k
summability. Therefore,aseries a issaidtobesummable|C,α,δ| , k ≥1,α>
n k
−1, if (see [9])
∞ P
nδk+k−1 σα−σα k <∞,
n n−1
n=1
X (cid:12) (cid:12)
where (cid:12) (cid:12)
n Aα−1
σα := n−js .
n Aα j
j=0 n
X
2010 Mathematics Subject Classification. 40D15,40F05,40G05 ,42A24,42B15.
Key words and phrases. Absolute summability, Fourier series, Local property, Cesàro matrices,
Rhalygeneralized Cesàromatrices,p−Cesàromatrices,Hölder’sinequality.
ResearchofthesecondauthorissupportedbyNSFofChina(10901044),andProgramforExcellent
YoungTeachers inHZNU.Researchofthethirdauthor issupportedbyNSERCofCanada.
1
For any positive sequence {p } such that P = p +p +···+p → ∞, the
n n 0 1 n
corresponding Riesz matrix R has the entries
p
v
r := , v =0,1,··· ,n, n=0,1,2,··· .
nv
P
n
δk+k−1
Taking α = Pn and α =nδk+k−1, we get two special absolute summa-
n pn n
bility, N,p ,(cid:16)δ s(cid:17)ummabilityand|R,p ,δ| summability,of|R,α | summability,
n k n k n k
respectively. Inparticular,ifnp ≍P ,then N,p ,δ summabilityand|R,p ,δ|
(cid:12) (cid:12) n n n k n k
(cid:12) (cid:12)
summability are equivalent. See [2] and [3] for more details on N,p ,δ summa-
(cid:12) (cid:12) n k
bility and |R,p ,δ| summability. (cid:12) (cid:12)
n k (cid:12) (cid:12)
One can find more examples of |A,α | −summability for d(cid:12)ifferent(cid:12)weight se-
n k
quences {α } and different summability matrices A discussed in many papers, see
n
[2], [3], [6], [10], and [16] for examples.
Let f be a function with period 2π, integrable (L) over(−π,π). Without loss of
generality we may assume that the constant term in the Fourier series of f(t) is
zero, so that
π
f(t)dt=0
Z−π
and
∞ ∞
f(t)∼ (a cosnt+b sinnt)≡ C (t).
n n n
n=1 n=1
X X
It is well known that (see [18]) the convergence of the Fourier series at t = x is a
localpropertyofthegeneratingfunctionf(t)(i.e.,itdependsonlyonthebehavior
of f in a arbitrarily small neighborhood of x), and hence the summability of the
Fourier series at t = x by any regular linear summability method is also a local
property of the generating function f(t).
In1939,BosanquetandKestleman([8])showedthateventhesummability|C,1|
of the Fourier series at a point is not a local property of f. Mohanty ([11]) subse-
quently observed that the summability |R,logn,1| of the factored series
C (t)/log(n+1),
n
atanypointis alocalpropertyXoff, whereasthe summability |C,1| ofthis seriesis
not. SeveralgeneralizationsofMohanty’s resulthave been made by many authors,
for examples, see, Bhatt ([1]), Bor ([3]-[5]), Borwein ([7]), Sarigöl ([14], [15]), etc.
For any lower triangular matrix A, associated it with two lower triangular ma-
trices A and A defined by
n
ba = a , v =0,1,2,··· ,n and n=0,1,2,··· ,
nv nr
r=v
X
and
anv =anv−an−1,v, v =0,1,··· ,n−1;n=1,2,3,··· . ann =ann =ann.
Sarigöl ([15]) proved the following theorem:
b b
TheoremA.Let Abealowertriangularmatrixwithnonnegativeentriessatisfying
(i) an−1,v ≥anv for n≥v+1,
(ii) a =1, n=0,1,··· ,
n0
2
(iii) n−1a a =O(a ),
v=1 vv n,v+1 nn
(iv) ∆X =O 1 , X =(na )−1,n=1,2,··· ,X =0.
P n n n nn 0
If {θn} holds forbthe following conditions,
(v) ∞ (θ a(cid:0))k(cid:1)−1Xk−11λk <∞,
v=1 v vv v v v
(vi) ∞ (θ a )k−1Xk∆λ <∞,
P v=1 v vv v v
(vii) ∞ (θ a )k−1|∆a |=O (θ a )k−1a
P n=v+1 n nn n,v+1 v vv vv
and (cid:16) (cid:17)
(viii)P∞ (θ a )k−1a b =O (θ a )k−1 ,
n=v+1 n nn n,v+1 v vv
then the summability of A,θk−1 ,k ≥1(cid:16),ofthe seri(cid:17)es λ X C (t) at any point
P n k n n n
is a local property of f,wherbe {λ } is a convex sequence such that n−1λ is
(cid:12) (cid:12) n P n
convergent. (cid:12) (cid:12)
P
TheoremAgeneralizedsomewellknownresultsonthelocalpropertyofsumma-
bility of factored Fourier series. Although, there are some matrices satisfying the
conditions in Theorem A, a Cesàro’s matrix may not satisfy all the conditions (i)-
(iii). Infact,(ii)and(iii)donotholdforanyα>1orα<1. Furthermore,Rhaly’s
generalized Cesàro matrices and the p−Cesàro matrices do not satisfy the condi-
tions of TheoremA neither (see Section 3 for the definitions of Rhaly’s generalized
Cesàro matrices and the p−Cesàro matrices).
In the present paper, we establish a new factor theorem which generalizes The-
orem A, and can be applied to many well known matrices, including the ones
mentioned above. We need the following class of matrices, S(α ,φ ), which is
n n
recently introduced by Yu and Zhou ([20]):
Definition 1. Let {α }, {φ } be sequences of positive numbers. We say that
n n
a lower triangular matrix A := (a ) ∈ S(α ,φ ), if it satisfies the following
nk n n
conditions
n−1
|∆ a |=O(φ ); (T1)
i ni n
i=0
X
|ani|=O(φbn), i=0,1,··· ,n; (T2)
∞
b αnφnk−1|∆iani|=O αiφki ; (T3)
n=i+1
X (cid:0) (cid:1)
∞
b
α φk−1|a |=O α φk−1 . (T4)
n n n,i+1 i i
n=i+1
X (cid:0) (cid:1)
Our main results are the following:b
Theorem1. Let {α },and{φ }besequencesofpositivenumbers.Let{λ }∈BV
n n n
1
be a sequenceof complex numbers such that λ =O(|λ |) for n=1,2,··· , and
n+1 n
∞
(A) α φkXk λk <∞,
n=0 n n n n
(B) ∞ α φk−1Xk|∆λ |<∞.
Pn=0 n n (cid:12)n (cid:12) n
If A∈S(α ,φ ) satisfi(cid:12)es(cid:12)
n n
P
n
|a a |=O(φ ), (1.1)
vv n,v+1 n
v=0
X
b
1 ∞
Wesayasequenceofcomplexnumbers {λn}∈BV,ifPn=1|∆λn|<∞.
3
φ
n
∆X =O(φ ), X = , (1.2)
n n n
a
nn
thenthesummabilityof|A,α | for k≥1,of theseries C (t)λ X atanypoint
n k n n n
is a local property of f.
P
Remark 1. The restrictions of {λ } in Theorem A are relaxed in Theorem 1 to
n
the simple conditions that {λ }∈BV and λ =O(|λ |), which obviously hold
n n+1 n
when {λ } is a convex sequence such that n−1λ is convergent.
n n
Theorem 2. The result of Theorem 1 alsoPholds when (1.1) and (1.2) are replaced
by
n
|a φ |=O(φ ), (1.3)
n,v+1 v n
v=0
X
b
1
∆X =O n−1 , X = ,n=1,2,··· ,X =0, (1.4)
n n 0
nφ
n
(cid:0) (cid:1)
respectively.
Remark 2. If the matrix A satisfies the condition a = 1, n = 0,1,··· , then
n0
the indexes of the summations in (A), (B), (1.1) and (1.3) only need to run from 1
instead of 0, which can be observed in the proofs of the theorems.
Remark 3. Let φ := a ,α = θk−1. If the matrix A satisfies the conditions in
n nn n n
TheoremA, then we can easily have that A∈S(α ,φ ). That is, Theorem A can
n n
be regardedas a corollary of Theorem 2.
We prove the theorems in Section 2. In Section 3, we show that some well
known matrices such as Cesàro’s matrices, Rhaly’s generalized Cesàro matrices,
the p−Cesàro matrices, and Riesz’s matrices are in S(α ,φ ) for some certain
n n
sequences{α }and{φ },andthenderivesomenewtheoremsonthelocalproperty
n n
of some factored Fourier series, as applications of the above theorems.
2. Proofs of the Main Results
We prove Theorem 1 in this section. The proof of Theorem 2 is similar.
The behavior of the Fourier series, as far as convergence is concerned, at a
particular value of x, depends on the behavior of the function in the immediate
neighborhood of this point only. Therefore, in order to prove the theorem, it is
sufficient to prove that if {s } is bounded, then under the conditions of Theorem
n
1, a λ X is summable |A,α | , k ≥ 1. Let T be the n−th term of the
n n n n k n
A−transform of n λ a X . Then
P i=0 i i i
Pn v n n n
T = a a λ X = a λ X a = a a λ X .
n nv i i i i i i nv ni i i i
v=0 i=0 i=0 v=i i=0
X X X X X
4
Thus,
n n−1
Tn−Tn−1 = aniaiλiXi− an−1,iaiλiXi
i=0 i=0
X X
n n
= aniaiλiXi = aniλiXi(si−si−1)
i=0 i=0
X X
n−1b b
= (a λ X −a λ X )s +a λ s X
ni i i n,i+1 i+1 i+1 i nn n n n
i=0
X
n−1 b b n−1 n−1
= a ∆λ X s + a λ ∆X s + (∆ a )λ X s
n,i+1 i i i n,i+1 i+1 i i i ni i i i
i=0 i=0 i=0
X X X
+a λ X s
bnn n n n b b
=:T +T +T +T .
n1 n2 n3 n4
Therefore, it is sufficient to prove that
∞
α |T |k <∞, for i=1,2,3,4. (2.1)
n ni
n=1
X
Applying Hölder’s inequality, we have
m+1 m+1 n−1 k
α |T |k =O(1) α |a ||X ||∆λ |
n n1 n n,i+1 i i
!
n=1 n=1 i=0
X X X
m+1 n−1 b n−1
=O(1) α |a | Xk |∆λ | |a ||∆λ | .
n n,i+1 i i n,i+1 i
! !
n=1 i=0 i=0
X X (cid:12) (cid:12) X
b (cid:12) (cid:12) b
Since {λ }∈BV, we have
n
n−1
|a ||∆λ |=O(φ ),
n,i+1 i n
i=0
X
b
by (T2). Hence
m+1 m+1 n−1
α |T |k =O(1) α φk−1 |a | Xk |∆λ |
n n1 n n n,i+1 i i
n=1 n=1 i=0
X X X (cid:12) (cid:12)
m mb+1 (cid:12) (cid:12)
=O(1) Xk |∆λ | α φk−1|a |
i i n n n,i+1
i=0 n=i+1
X(cid:12) (cid:12) X
m (cid:12) (cid:12) b
=O(1) α φk−1 Xk |∆λ |=O(1), (2.2)
i i i i
i=0
X (cid:12) (cid:12)
(cid:12) (cid:12)
by (T3), and (B) of Theorem 1.
5
It follows from (1.2) that ∆X = O(a X ). Then by Hölder’s inequality, (1.1)
i ii i
and condition (A) of the Theorem 1, we have
m+1 m+1 n−1 k
α |T |k =O(1) α |a λ ∆X |
n n2 n n,i+1 i+1 i
!
n=1 n=1 i=0
X X X
m+1 n−1 b k
=O(1) α |a λ a X |
n n,i+1 i ii i
!
n=1 i=0
X X
m+1 n−1 b n−1 k−1
=O(1) α |a a | λk Xk |a a |
n n,i+1 ii i i n,i+1 ii
! !
n=1 i=0 i=0
X X (cid:12) (cid:12)(cid:12) (cid:12) X
m+1 bn−1 (cid:12) (cid:12)(cid:12) (cid:12) b
=O(1) α φk−1 |a a | λk Xk
n n n,i+1 ii i i
!
n=1 i=0
X X (cid:12) (cid:12)(cid:12) (cid:12)
m mb+1 (cid:12) (cid:12)(cid:12) (cid:12)
=O(1) λk Xk |a | α φk−1|a |
i i ii n n n,i+1
i=0 n=i+1
X(cid:12) (cid:12)(cid:12) (cid:12) X
m (cid:12) (cid:12)(cid:12) (cid:12) b
=O(1) α φk−1 λk Xk |a |
n n i i ii
i=0
X (cid:12) (cid:12)(cid:12) (cid:12)
m (cid:12) (cid:12)(cid:12) (cid:12)
=O(1) α φk λk Xk =O(1),
n n i i
i=0
X (cid:12) (cid:12)(cid:12) (cid:12)
where we also used the fact that a(cid:12) (cid:12)=(cid:12) a (cid:12) =O(φ ), which follows from (T2).
nn nn n
By (T1), (T3) and condition (A), we have
m+1 m+1 bn−1 k
α |T |k =O(1) α |∆a λ X |
n n3 n n,i+1 i i
!
n=1 n=1 i=0
X X X
m+1 n−1 b n−1 k−1
=O(1) α |∆a | λk Xk |∆a |
n n,i+1 i i n,i+1
! !
n=1 i=0 i=0
X X (cid:12) (cid:12)(cid:12) (cid:12) X
m+1 n−1b (cid:12) (cid:12)(cid:12) (cid:12) b
=O(1) α φk−1 |∆a | λk Xk
n n n,i+1 i i
n=1 i=0
X X (cid:12) (cid:12)(cid:12) (cid:12)
m m+1 b (cid:12) (cid:12)(cid:12) (cid:12)
=O(1) λk Xk α φk−1|∆a |
i i n n n,i+1
i=0 n=i+1
X(cid:12) (cid:12)(cid:12) (cid:12) X
m (cid:12) (cid:12)(cid:12) (cid:12) b
=O(1) α φk λk Xk =O(1). (2.3)
i i i i
i=0
X (cid:12) (cid:12)(cid:12) (cid:12)
By using a =O(φ ) again, w(cid:12)e h(cid:12)a(cid:12)ve (cid:12)
nn n
m+1 m+1
α |T |k =O(1) α |a λ X |k
n n4 n nn n n
n=1 n=1
X X
m+1
=O(1) α φk λk Xk
n n n n
n=1
X (cid:12) (cid:12)(cid:12) (cid:12)
=O(1). (cid:12) (cid:12)(cid:12) (cid:12) (2.4)
6
Combining (2.2)-(2.4), we have (2.1). This proves Theorem 1.
3. Applications of The Theorems
3.1. Cesàro’s Matrices. We will use the following formula often in the proofs
(see [21] for proof, for example):
nα 1
Aα = 1+O . (3.1)
n Γ(α+1) n
(cid:18) (cid:18) (cid:19)(cid:19)
In this subsection, we set
n−1, α>1
φ :=1, φ := n=1,2,··· .
0 n 1 =c , 0<α≤1,
(cid:26) Aαn nn
By (3.1), we see that
n−1, α>1
φn ∼ n−α, 0<α≤1, n=1,2,··· .
(cid:26)
Recallthata nonnegativesequence{a } issaidto be almostdecreasing,ifthere
n
is a positive constant K such that
a ≥Ka
n m
holds for all n ≤ m, and it is said to be quasi-β-power decreasing, if nβa is
n
almost decreasing.
(cid:8) (cid:9)
It should be noted that every decreasing sequence is an almost decreasing se-
quence, and every almost decreasing sequence is a quasi-β-power decreasing se-
quence for any non-positive index β, but the converse is not true.
Lemma 1. ([20])Let α > 0, and let {α } be a sequence of positive numbers. If
n
α φk−1n−1 is quasi-ε-power decreasing for some ε>0, then C ∈S(α ,φ ).
n n n n
(cid:8) A direct c(cid:9)alculation leads to
n n−1
1 1
c = Aα−1− Aα−1
ni Aα n−j Aα n−1−j
n j=i n−1 j=i
X X
b Aα Aα iAα−1
= n−i − n−1−i = n−i .
Aα Aα nAα
n n−1 n
Thus, for 0<α≤1,
n |c c |=O 1 n (v+1)Aαn−−v1−1
vv n,v+1 n1+α Aα
v=0 (cid:18) (cid:19)v=1 v
X X
b 1 n/2 1 n/2
=O v1−α+O vα−1
n2 n2α
(cid:18) (cid:19)v=1 (cid:18) (cid:19)v=1
X X
=O(φ ). (3.2)
n
Similarly, for α>1,
n
1
|c |=O(φ ). (3.3)
n,v+1 n
v
v=1
X
7
b
Now set
φn , 0<α≤1,
Xn ≡1= (nφcnn)−1, α>1.
(cid:26) n
Then X satisfies (1.2) and (1.4) for 0 < α ≤ 1 and α > 1 respectively. Now,
n
applying Lemma 1, (3.2), (3.3), Theorem 1 and Theorem 2, we have the following
Theorem 3. Let α > 0, {α } be sequences of positive numbers. Let {λ } ∈ BV
n n
be a sequence of complex numbers such that λ =O(|λ |) for n=1,2,··· , and
n+1 n
(a) ∞ α φk λk <∞,
n=1 n n n
(b) ∞ α φk−1|∆λ |<∞.
If α φPk−n=1n1−1n nis(cid:12)(cid:12)qu(cid:12)(cid:12)asin-ε-power decreasing, then the summability of |C,α | for
nPn n k
k ≥1, of the series C (t)λ at any point is a local property of f.
n n
(cid:8) (cid:9)
As examples, wePgive two corollaries of Theorem 3.
Corollary 1. Let {λ }∈BV be a sequence of complex numbers such that λ =
n n+1
O(|λ |) for n=1,2,··· , and
n
(c) ∞ nδk−1logγn λk <∞,
n=1 n
(d) ∞ nδklogγn|∆λ |<∞,
then thPe snu=m1mability of C(cid:12)(cid:12),nn(cid:12)(cid:12)δk+k−1logγn , for α≥1, γ ∈R, k ≥1 and 0≤δ <
P k
1, of the series C (t)λ at any point is a local property of f.
k n (cid:12) n (cid:12)
(cid:12) (cid:12)
Proof. Let α =Pnδk+k−1logγn, n=1,2,··· , α =1. Since α≥1, then φ =n−1.
n 0 n
Itisthen obviousthat(c)implies (a), and(d) implies (b).Fromthe conditionthat
0 ≤ δ < 1, we see that there exists an ε > 0 such that δk−1+ε < 0, and thus
k
nδk−1+εlogγn is quasi decreasing for γ ∈ R. In other words, α φk−1n−1 is
n n
quasi-ε-powerdecreasing. Therefore, by Theorem 3, we have Corollary 1. (cid:3)
(cid:8) (cid:9) (cid:8) (cid:9)
Corollary 2. Let {λ }∈BV be a sequence of complex numbers such that λ =
n n+1
O(|λ |) for n=1,2,··· , and
n
(c′) ∞ nδk+(1−α)k−1logγn λk <∞,
n=1 n
(d′) ∞ nδk+(1−α)(k−1)logγnX |∆λ |<∞,
Pn=1 (cid:12) n(cid:12) n
then the summability of C,nδk+k(cid:12)−1(cid:12)logγn , for 0 < α < 1, γ ∈ R, k ≥ 1 and
P k
0≤δ < 2−α+(1−α)k, of the series C (t)λ at any point is a local property of f.
k (cid:12) n (cid:12) n
(cid:12) (cid:12)
Proof. Note that φ =n−α for 0<Pα<1. Then the proof of Corollary 2 is similar
n
to that of Corollary 1. (cid:3)
3.2. Rhaly’s Generalized Cesàro Matrices. Let D be the Rhaly generalized
Cesàromatrix(see[12]),thatis,Dhasentriesoftheformd =tn−k/(n+1), k =
nk
0,1,··· ,n, n=1,2,··· .Whent=1,theRhalygeneralizedCesàromatrixreduces
tothe Cesàromatrixoforder1.Weshallrestrictourattentionto0<t<1.Inthis
case, D does not satisfy condition (ii) of Theorem A. It is routine to deduce that
n tn−r n−1tn−1−r 1 1−tn−v+1 1−tn−v
d = − = − . (3.4)
nv
n+1 n 1−t n+1 n
r=v r=v (cid:18) (cid:19)
X X
8
b
Set φ =1, φ =n−1,n=1,2,··· . By (3.4), we have
0 n
1 1−tn−v+1 1−tn−v
d = −
nv
1−t n+1 n
(cid:18) (cid:19)
b 1−tn−v−ntn−v(1−t)
=−
(1−t)n(n+1)
1 ntn−v
=O + .
n(n+1) n(n+1)
(cid:18) (cid:19)
Thus
n n n
1 1 1
d d =O +O tn−v =O(φ ). (3.5)
vv n,v+1 n2 v n n
Xv=0(cid:12) (cid:12) (cid:18) (cid:19)Xv=1 (cid:18) (cid:19)Xv=1
(cid:12) b (cid:12)
Lemma 2. ([2(cid:12)0])Let 0 <(cid:12) t < 1, and {α } be a sequences of positive numbers. If
n
α φk−1n−1 is quasi-ε−power decreasing for some ε>0, then D ∈S(α ,φ ).
n n n n
(cid:8) Now set (cid:9)
φ n+1
n
X = = .
n
d n
nn
Then X =O(1) and ∆X =O(φ ). Therefore,by Lemma 2, (3.5) and Theorem
n n n
1, we have
Theorem4. Let0<t<1andlet{α }beasequenceof positive numbers.Assume
n
that {λ } ∈ BV is a sequence of complex numbers such that λ = O(|λ |) for
n n+1 n
n = 1,2,··· , and (A), (B) in Theorem 1 hold. If α φk−1n−1 is quasi-ε-power
n n
decreasing for some ε>0, then the summability of |D,α | for k ≥1, of the series
(cid:8) n k (cid:9)
C (t)n+1λ at any point is a local property of f.
n n n
PObviously,wecanalsohaveacorollaryofTheorem3thatissimilartoCorollary
1. We omit the details here.
3.3. p−Cesáro Matrices. Let E be the p−Cesàro matrix (see [13]), that is, the
entries of E has the form e = 1/(n+1)p, i = 0,1,··· ,n, n = 1,2,··· . When
ni
p=1,the p−Cesàromatrixreducesto theCesàromatrixoforder1again. Also,E
does not satisfy condition (ii) of Theorem A. We restrict our attention to the case
when 1<p≤2.
Set φ =1, φ =n−p,n=1,2,··· . Then
0 n
n−i+1 (n−i)
eni =eni−en−1,i = (n+1)p − np , (3.6)
and
b
1 1
∆ieni =en,i+1−eni =eni−en−1,i = (n+1)p − np. (3.7)
By (3.6), we have
b b b
1 1 1
e =(n−i) − + =O(φ ). (3.8)
ni (n+1)p np (n+1)p n
(cid:18) (cid:19)
Now set Xn =beφnnn. Then direct calculations yield that
∆X =O n−2 =O(φ ), 1<p≤2,
n n
9
(cid:0) (cid:1)
and
n
|e e |=O(φ ).
vv n,v+1 n
v=1
X
Lemma 3. ([20])Let p > 1 and {bα } be a sequences of positive numbers. If
n
α φk−1n−1 is quasi-ε-power decreasing for some ε> 0 such that p−2+ε> 0,
n n
then D ∈S(α ,φ ).
n n
(cid:8) (cid:9)
Therefore, we have
Theorem5. Let1<p≤2andlet{α }beasequenceofpositivenumbers.Assume
n
that {λ } ∈ BV is a sequence of complex numbers such that λ = O(|λ |) for
n n+1 n
n = 1,2,··· , and (A), (B) in Theorem 1 hold. If α φk−1n−1 is quasi-ε-power
n n
decreasing for some ε>0 such that p−2+ε>0, then the summability of |E,α |
(cid:8) (cid:9) n k
for k ≥1, of the series C (t)X λ at any point is a local property of f.
n n n
3.4. Riesz’s matrices.PWe firstly establish a general result, then apply it to the
Riesz’s matrices.
Lemma 4. ([20]) Let A be a lower triangular matrix with nonnegative entries, and
{α } be a sequence of positive numbers. If
n
(I) a =1, n=0,1,··· ,
n0
(II) an−1,v ≥anv for n≥v+1,
(III) na =O(1),
nn
(IV) ∞ α n−k+1|∆ a |=O α a v−k+1 ,
n=v+1 n v nv v vv
(V) ∞ α n−k+1a =O α v−k+1 ,
Pn=v+1 n n,v+1 v(cid:0) (cid:1)
then A∈S α ,n−1 . b
P n (cid:0) (cid:1)
Now, by setting φ =b1,X = 0,φ := n−1,X = (nφ )−1,n = 1,2,··· , and
(cid:0) (cid:1)0 0 n n n
applying Theorem 2, we have
Theorem 6. Let {α } be a sequence of positive numbers, and let A be a lower
n
triangular matrix with nonnegative entries satisfying conditions (I)-(V) of Lemma
4. Assume that {λ } ∈ BV is a sequence of complex numbers such that λ =
n n+1
O(|λ |) for n=1,2,··· , and (A), (B) in Theorem 1 hold. If (1.3) and (1.4) hold,
n
thenthesummabilityof|A,α | for k≥1,of theseries C (t)X λ atanypoint
n k n n n
is a local property of f.
P
We now show that under some necessaryconditions, Riesz matrix R satisfies all
the conditions in Lemma 4. For any positive sequence {p } such that P = p +
n n 0
p +···+p →∞,thecorrespondingRieszmatrixRhastheentriesr := pv, v =
1 n nv Pn
0,1,··· ,n; n = 0,1,2,··· . Now obviously, we have rn0 = 1 and rn−1,v ≥ rnv for
n≥v+1. Direct calculations yield that (set P−1 =0)
Pv−1pn
r = , (3.9)
nv
PnPn−1
and
b p p
n v
|∆ r |= . (3.10)
v nv
PnPn−1
So if np =O(P ) and
n n
b
m+1
p
α n−k+1 n =O α v−k+1P−1 , (3.11)
n PnPn−1 v v
n=v+1
X (cid:0) (cid:1)
10