ON THE RATE OF STRONG SUMMABILITY BY
MATRIX MEANS IN THE GENERALIZED HÖLDER
METRIC
Rate of Strong Summability
by Matrix Means
Bogdan Szal
BOGDAN SZAL
Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
65-516 Zielona Góra, ul. Szafrana 4a, Poland
vol. 9, iss. 1, art. 28, 2008
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Contents
EMail: B.Szal@wmie.uz.zgora.pl
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Received:
10 January, 2007
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Accepted:
23 February, 2008
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Communicated by:
R.N. Mohapatra
2000 AMS Sub. Class.:
40F04, 41A25, 42A10.
Key words:
Strong approximation, Matrix means, Special sequences.
Abstract:
In the paper we generalize (and improve) the results of T. Singh [5], with mediate function, to the strong summability. We also apply the generalization of L.
Leindler type [3].
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Contents
1
Introduction
3
2
Main Results
8
3
Corollaries
10
4
Lemmas
12
5
Proofs of the Theorems
5.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . .
21
21
25
25
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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1.
Introduction
Let f be a continuous and 2π-periodic function and let
∞
(1.1)
f (x) ∼
a0 X
+
(an cos nx + bn sin nx)
2
n=1
be its Fourier series. Denote by Sn (x) = Sn (f, x) the n-th partial sum of (1.1) and
by ω (f, δ) the modulus of continuity of f ∈ C2π .
The usual supremum norm will be denoted by k·kC .
Let ω (t) be a nondecreasing continuous function on the interval [0, 2π] having
the properties
ω (0) = 0, ω (δ1 + δ2 ) ≤ ω (δ1 ) + ω (δ2 ) .
Such a function will be called a modulus of continuity.
Denote by H ω the class of functions
H ω := {f ∈ C2π ; |f (x + h) − f (x)| ≤ Cω (|h|)} ,
where C is a positive constant. For f ∈ H ω , we define the norm k·kω = k·kH ω by
the formula
kf kω := kf kC + kf kC,ω ,
where
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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kf (· + h) − f (·)kC
,
ω (|h|)
h6=0
kf kC,ω = sup
and kf kC,0 = 0. If ω (t) = C1 |t|α (0 < α ≤ 1), where C1 is a positive constant,
then
H α = {f ∈ C2π ; |f (x + h) − f (x)| ≤ C1 |h|α , 0 < α ≤ 1}
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is a Banach space and the metric induced by the norm k·kα on H α is said to be a
Hölder metric.
Let A := (ank ) (k, n = 0, 1, . . . ) be a lower triangular infinite matrix of real
numbers satisfying the following condition:
n
X
ank = 1.
(1.2)
ank ≥ 0 (k, n = 0, 1, . . . ) , ank = 0, k > n and
k=0
Let the A−transformation of (Sn (f ; x)) be given by
n
X
(1.3)
tn (f ) := tn (f ; x) :=
ank Sk (f ; x)
(n = 0, 1, . . . )
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
k=0
and the strong Ar −transformation of (Sn (f ; x)) for r > 0 by
) r1
( n
X
Tn (f, r) := Tn (f, r; x) :=
ank |Sk (f ; x) − f (x)|r
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(n = 0, 1, . . . ) .
k=0
Now we define two classes of sequences ([3]).
A sequence c := (cn ) of nonnegative numbers tending to zero is called the Rest
Bounded Variation Sequence, or briefly c ∈ RBV S, if it has the property
∞
X
(1.4)
|cn − cn+1 | ≤ K (c) cm
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k=m
for all natural numbers m, where K (c) is a constant depending only on c.
A sequence c := (cn ) of nonnegative numbers will be called a Head Bounded
Variation Sequence, or briefly c ∈ HBV S, if it has the property
(1.5)
m−1
X
k=0
|cn − cn+1 | ≤ K (c) cm
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for all natural numbers m, or only for all m ≤ N if the sequence c has only finite
nonzero terms and the last nonzero term is cN .
Therefore we assume that the sequence (K (αn ))∞
n=0 is bounded, that is, that there
exists a constant K such that
0 ≤ K (αn ) ≤ K
holds for all n, where K (αn ) denote the sequence of constants appearing in the
inequalities (1.4) or (1.5) for the sequence αn := (ank )∞
k=0 . Now we can give the
conditions to be used later on. We assume that for all n and 0 ≤ m ≤ n,
(1.6)
∞
X
|ank − ank+1 | ≤ Kanm
k=m
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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and
(1.7)
m−1
X
|ank − ank+1 | ≤ Kanm
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k=0
hold if αn := (ank )∞
k=0 belongs to RBV S or HBV S, respectively.
Let ω (t) and ω ∗ (t) be two given moduli of continuity satisfying the following
condition (for 0 ≤ p < q ≤ 1):
p
q
(1.8)
(ω (t))
= O (1)
ω ∗ (t)
(t → 0+ ) .
In [4] R. Mohapatra and P. Chandra obtained some results on the degree of approximation for the means (1.3) in the Hölder metric. Recently, T. Singh in [5]
established the following two theorems generalizing some results of P. Chandra [1]
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with a mediate function H such that:
Z π
ω (f ; t)
(1.9)
dt = O (H (u)) (u → 0+ ) , H (t) ≥ 0
t2
u
and
Z t
(1.10)
H (u) du = O (tH (t))
(t → O+ ) .
0
Theorem 1.1. Let A = (ank ) satisfy the condition (1.2) and ank ≤ ank+1 for k =
0, 1, . . . , n − 1, and n = 0, 1, . . . . Then for f ∈ H ω , 0 ≤ p < q ≤ 1,
h
p
(1.11) ktn (f ) − f kω∗ = O {ω (|x − y|)} q {ω ∗ (|x − y|)}−1
p
π p
− pq
π 1− q
H
ann n q + ann
+ O ann H
,
×
n
n
if ω (f ; t) satisfies (1.9) and (1.10), and
h
i
p
(1.12) ktn (f ) − f kω∗ = O {ω (|x − y|)} q {ω ∗ (|x − y|)}−1
p
π 1− pq n π π o
p
π 1− q
×
ω
+ ann n q H
+O ω
+ ann H
,
n
n
n
n
if ω (f ; t) satisfies (1.9), where ω ∗ (t) is the given modulus of continuity.
Theorem 1.2. Let A = (ank ) satisfy the condition (1.2) and ank ≤ ank+1 for k =
0, 1, . . . , n − 1, and n = 0, 1, . . . . Also, let ω (f ; t) satisfy (1.9) and (1.10). Then for
f ∈ H ω , 0 ≤ p < q ≤ 1,
h
p
(1.13) ktn (f ) − f kω∗ = O {ω (|x − y|)} q {ω ∗ (|x − y|)}−1
n
p
oi
p
−p
× (H (an0 ))1− q an0 n q + an0q
+ O (an0 H (an0 )) ,
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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where ω ∗ (t) is the given modulus of continuity.
The next generalization of another result of P. Chandra [2] was obtained by L.
Leindler in [3]. Namely, he proved the following two theorems
Theorem 1.3. Let (1.2) and (1.9) hold. Then for f ∈ C2π
π π (1.14)
ktn (f ) − f kC = O ω
+ O ann H
.
n
n
If, in addition ω (f ; t) satisfies the condition (1.10), then
Rate of Strong Summability
by Matrix Means
Bogdan Szal
ktn (f ) − f kC = O (ann H (ann )) .
(1.15)
vol. 9, iss. 1, art. 28, 2008
Theorem 1.4. Let (1.2), (1.9) and (1.10) hold. Then for f ∈ C2π
ktn (f ) − f kC = O (an0 H (an0 )) .
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In the present paper we will generalize (and improve) the mentioned results of T.
Singh [5] to strong summability with a mediate function H defined by the following
conditions:
Z π r
ω (f ; t)
dt = O (H (r; u))
(u → 0+ ) , H (t) ≥ 0 and r > 0,
(1.17)
t2
u
and
Z t
(1.18)
H (u) du = O (tH (r; t))
(t → O+ ) .
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(1.16)
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0
We also apply a generalization of Leindler’s type [3].
Throughout the paper we shall use the following notation:
φx (t) = f (x + t) + f (x − t) − 2f (x) .
By K1 , K2 , . . . we shall designate either an absolute constant or a constant depending
on the indicated parameters, not necessarily the same at each occurrence.
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2.
Main Results
Our main results are the following.
Theorem 2.1. Let (1.2), (1.7) and (1.8) hold. Suppose ω (f ; t) satisfies (1.17) for
r ≥ 1. Then for f ∈ H ω ,
p
(2.1) kTn (f, r)kω∗ = O {1 + ln (2 (n + 1) ann )} q
n
π o r1 (1− pq ) r−1
.
× ((n + 1) ann ) ann H r;
n
If, in addition ω (f ; t) satisfies the condition (1.18), then
p
(2.2) kTn (f, r)kω∗ = O {1 + ln (2 (n + 1) ann )} q
r−1
× (ln (2 (n + 1) ann ))
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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r1 (1− pq )
.
ann H (r; ann )
Theorem 2.2. Under the assumptions of above theorem, if there exists a real number
s > 1 such that the inequality

1
k −1
k −1
2X
 2X
s
1
−1
s
(2.3)
(a )
≤ K1 2k−1 s
ani
 k−1 ni 
k−1
i=2
Rate of Strong Summability
by Matrix Means
i=2
for any k = 1, 2, . . . , m, where 2m ≤ n + 1 < 2m+1 holds, then the following
estimates
n
π o r1 (1− pq ) (2.4)
kTn (f, r)kω∗ = O
ann H r;
n
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and
(2.5)
kTn (f, r)kω∗
p
= O {ann H (r; ann )} (1− q )
1
r
are true.
Theorem 2.3. Let (1.2), (1.6), (1.8) and (1.17) for r ≥ 1 hold. Then for f ∈ H ω
n
π o r1 (1− pq ) (2.6)
kTn (f, r)kω∗ = O
an0 H r;
.
n
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
If, in addition, ω (f ; t) satisfies (1.18), then
p
1
(2.7)
kTn (f, r)kω∗ = O {an0 H (r; an0 )} r (1− q ) .
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Remark 1. We can observe, that for the case r = 1 under the condition (1.8) the
first part of Theorem 1.1 (1.11) and Theorem 1.2 are the corollaries of the first part
of Theorem 2.1 (2.1) and the second part of Theorem 2.3 (2.7), respectively. We
can also note that the mentioned estimates are better in order than the analogical
estimates from the results of T. Singh, since ln (2 (n + 1) ann ) in Theorem 2.1 is
better than (n + 1) ann in Theorem 1.1. Consequently, if nann is not bounded our
estimate (2.7) in Theorem 2.3 is better than (1.13) from Theorem 1.2.
Remark 2. If in the assumptions of Theorem 2.1 or 2.3 we take ω (|t|) = O (|t|q ),
ω ∗ (|t|) = O (|t|p ) with p = 0, then from (2.1), (2.2) and (2.7) we have the same
estimates such as (1.14), (1.15) and (1.16), respectively, but for the strong approximation (with r = 1).
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3.
Corollaries
In this section we present some
special
cases of our results. From Theorems 2.1, 2.2
β
∗
and 2.3, putting ω (|t|) = O |t| , ω (|t|) = O (|t|α ),
 rα−1
t
if αr < 1,



ln πt
if αr = 1,
H (r; t) =



K1
if αr > 1
Rate of Strong Summability
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Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
where r > 0 and 0 < α ≤ 1, and replacing p by β and q by α, we can derive
Corollaries 3.1, 3.2 and 3.3, respectively.
Corollary 3.1. Under the conditions (1.2) and (1.7) we have for f ∈ H α , 0 ≤ β <
α ≤ 1 and r ≥ 1,
 β
α−β
1+ r1 (1− α
)

O {ln (2 (n + 1) ann )}
{ann }
if αr < 1,





n oα−β
1+α−β
π
kTn (f, r)kβ =
O {ln (2 (n + 1) ann )}
ln ann ann
if αr = 1,





 O {ln (2 (n + 1) ann )}1+ r1 (1− αβ ) {ann } α−β
αr
if αr > 1.
Corollary 3.2. Under the assumptions of Corollary 3.1 and (2.3) we have
 α−β

O {ann }
if αr < 1,





oα−β
n π
kTn (f, r)kβ =
a
if αr = 1,
O
ln
nn
ann





 O {ann } α−β
αr
if αr > 1.
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Corollary 3.3. Under the conditions (1.2) and (1.6) we have, for f ∈ H α , 0 ≤ β <
α ≤ 1 and r ≥ 1,
 α−β

O {an0 }
if αr < 1,





oα−β
n π
kTn (f, r)kβ =
if αr = 1,
O
ln an0 an0





 O {an0 } α−β
αr
if αr > 1.
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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4.
Lemmas
To prove our theorems we need the following lemmas.
Lemma 4.1. If (1.17) and (1.18) hold with r > 0 then
Z s r
ω (f ; t)
(4.1)
dt = O (sH (r; s)) (s → 0+ ) .
t
0
Proof. Integrating by parts, by (1.17) and (1.18) we get
Z π r
s Z s Z π r
Z s r
ω (f ; t)
ω (f ; u)
ω (f ; u)
dt = −t
du +
dt
du
2
t
u
u2
0
t
0
t
0
Z s
= O (sH (r; s)) + O (1)
H (r; t) dt
Rate of Strong Summability
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Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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0
= O (sH (r; s)) .
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This completes the proof.
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Lemma 4.2 ([7]). If (1.2), (1.7) hold, then for f ∈ C2π and r > 0,
Page 12 of 27
(4.2)
kTn (f, r)kC

1 
r
n+1
[ 4 ]
 
r 
 X
.
≤ O
an,4k Ekr (f ) + E[ n+1 ] (f ) ln (2 (n + 1) ann )


4


 k=0
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If, in addition, (2.3) holds, then
(4.3)

1 
r
n+1
[ 2 ]

 
 X
kTn (f, r)kC ≤ O 
an,2k Ekr (f ) 

.

 k=0

Lemma 4.3 ([7]). If (1.2), (1.6) hold, then for f ∈ C2π and r > 0,
(
) r1 
n
X
(4.4)
kTn (f, r)kC ≤ O 
ank Ekr (f )  .
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
k=0
Lemma 4.4. If (1.2), (1.7) hold and ω (f ; t) satisfies (1.17) with r > 0 then
n+1
4
(4.5)
[X]
an,4k ω
r
k=0
π
f;
k+1
π = O ann H r;
.
n
(4.6)
k=0
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an,4k ω r
π
f;
k+1
= O (ann H (r; ann )) .
anµ − anm ≤ |anµ − anm | ≤
m−1
X
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Proof. First we prove (4.5). If (1.7) holds then
Close
|ank − ank+1 | ≤ Kanm
k=µ
for any m ≥ µ ≥ 0, whence we have
(4.7)
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If, in addition, ω (f ; t) satisfies (1.18) then
n+1
[X
4 ]
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anµ ≤ (K + 1) anm .
From this and using (1.17) we get
n+1
[X
4 ]
an,4k ω
k=0
r
π
f;
k+1
n
X
π
≤ (K + 1) ann
ω f;
k+1
k=0
Z n+1 π
≤ K1 ann
ωr f ;
dt
t
1
Z π r
ω (f ; u)
= πK1 ann
du
2
π
u
n+1
π = O ann H r;
.
n
r
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
Title Page
Now we prove (4.6). Since
Contents
(K + 1) (n + 1) ann ≥
n
X
ank = 1,
k=0
we can see that
(4.8)
k=0
an,4k ω
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n+1
4
[X]
JJ
r
π
f;
k+1
[
≤
1
4(K+1)ann
X
]−1
r
an,4k ω
k=0
+
π
f;
k+1
n
X
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an,4k ω r f ;
1
k=[ 4(K+1)a
nn
]−1
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π
k+1
= Σ 1 + Σ2 .
Using again (4.7), (1.2) and the monotonicity of the modulus of continuity, we can
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estimate the quantities Σ1 and Σ2 as follows
(4.9)
Σ1 ≤ (K + 1) ann
1
[ 4(K+1)a
]−1
Xnn
k=0
Z
ω
r
π
f;
k+1
1
4(K+1)ann
π
ωr f ;
dt
t
1
Z π
ω r (f ; u)
= πK2 ann
du
u2
4π(K+1)ann
Z π r
ω (f ; u)
≤ πK2 ann
du
u2
ann
≤ K2 ann
Rate of Strong Summability
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Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
Title Page
and
(4.10)
Contents
r
Σ2 ≤ K3 ω (f ; 4π (K + 1) ann )
n
X
an,4k
1
k=[ 4(K+1)a
nn
]−1
≤ K3 (8π (K + 1))r ω r (f ; ann )
a nn
≤ K3 (32π (K + 1))r ω r f ;
2
Z ann r
ω (f ; t)
r
≤ 2K3 (32π (K + 1))
dt
ann
t
2
Z ann r
ω (f ; t)
dt.
≤ K4
t
0
If (1.17) and (1.18) hold then from (4.8) – (4.10) we obtain (4.6). This completes
the proof.
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Lemma 4.5. If (1.2), (1.7) hold and ω (f ; t) satisfies (1.17) with r ≥ 1 then
π
(4.11)
ω f,
ln (2 (n + 1) ann )
n+1
n
π o r1 1− r1
= O {(n + 1) ann }
ann H r;
.
n
If, in addition, ω (f ; t) satisfies (1.18) then
π
(4.12)
ω f,
ln (2 (n + 1) ann )
n+1
1
1
= O {ln (2 (n + 1) ann )}1− r {ann H (r; ann )} r .
Proof. Let r = 1. Using the monotonicity of the modulus of continuity
π
π
ω f,
ln (2 (n + 1) ann ) ≤ 2ann ω f,
(n + 1)
n+1
n+1
Z n+1
π
≤ 4ann ω f,
dt
n+1
1
Z n+1 π
dt
≤ 4ann
ω f,
t
1
Z π
ω (f, u)
= 4πann
du
π
u2
n+1
and by (1.17) we obtain that (4.11) holds. Now we prove (4.12). From (1.2) and
(1.7) we get
Z π(K+1)ann
π
π
1
ω f,
ln (2 (n + 1) ann ) ≤ K1 ω f,
dt,
π
n+1
n+1
t
n+1
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
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Z
π(K+1)ann
K1
π
n+1
Z ann
ω (f, t)
ω (f, u)
dt ≤ 2K1 (K + 1) π
du
1
t
u
(K+1)(n+1)
Z ann
ω (f, u)
≤ K2
du
u
0
and by Lemma 4.1 we obtain (4.12).
Assuming r > 1 we can use the Hölder inequality to estimate the following
integrals
π
Z
π
n+1
(Z
π
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and
Z
ann
1
(K+1)(n+1)
(Z
) r1 (Z
ann
ω r (f, u)
du
1
u
(K+1)(n+1)
Z
1− r1
≤ {ln (2 (n + 1) ann )}
ω (f, u)
du ≤
u
0
ann
ann
)1− r1
1
du
1
u
(K+1)(n+1)
r1
ω r (f, u)
du .
u
π
ω f,
n+1
ln (2 (n + 1) ann ) ≤ 4πann
n+1
π
1− r1 (Z
π
π
n+1
ω r (f, u)
du
u2
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From this, if (1.17) holds then
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
)1− r1
) r1 (Z
π
1
ω r (f, u)
du
du
2
2
π
π
u
u
n+1
n+1
) r1
1− r1 (Z π r
ω (f, u)
n+1
du
≤
2
π
u
π
n+1
ω (f, u)
du ≤
u2
Rate of Strong Summability
by Matrix Means
) r1
= O {(n + 1) ann }
1− r1
n
π o r1 ann H r;
n
and if (1.17) and (1.18) hold then
π
ω f,
ln (2 (n + 1) ann )
n+1
ann
ω r (f, u)
≤ 2K1 (K + 1) π {ln (2 (n + 1) ann )}
du
u
0
n
π o r1 1− r1
.
= O {ln (2 (n + 1) ann )}
ann H r;
n
1− r1
Z
r1
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
This ends our proof.
Title Page
Lemma 4.6. If (1.2), (1.6) hold and ω (f ; t) satisfies (1.17) with r > 0 then
n
π X
π
r
(4.13)
ank ω f ;
= O an0 H r;
.
k
+
1
n
k=0
Contents
If, in addition, ω (f ; t) satisfies (1.18), then
n
X
π
r
(4.14)
ank ω f ;
= O (an0 H (r; an0 )) .
k+1
k=0
ann − anm ≤ |anm − ann |
≤
k=m
|ank − ank+1 | ≤
∞
X
k=m
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Proof. First we prove (4.13). If (1.6) holds then
n−1
X
JJ
|ank − ank+1 | ≤ Kanm
for any n ≥ m ≥ 0, whence we have
ann ≤ (K + 1) anm .
(4.15)
From this and using (1.17) we get
n
n
X
X
π
π
r
r
ω f;
ank ω f ;
≤ (K + 1) an0
k
+
1
k+1
k=0
k=0
Z n+1 π
r
dt
≤ K1 an0
ω f;
t
1
Z π r
ω (f ; u)
du
= πK1 an0
π
u2
n+1
π = O an0 H r;
.
n
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
Title Page
Contents
Now, we prove (4.14). Since
(K + 1) (n + 1) an0 ≥
n
X
ank = 1,
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Page 19 of 27
k=0
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we can see that
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h
n
X
k=0
ank ω r f ;
π
k+1
≤
1
(K+1)an0
X
i
−1
ank ω r f ;
k=0
π
k+1
n
X
+
k=
h
1
(K+1)an0
Close
ank ω
i
−1
r
π
f;
k+1
.
Using again (1.2), (1.6) and the monotonicity of the modulus of continuity, we get
n
X
π
r
(4.16)
ank ω f ;
k+1
k=0
h
1
(K+1)an0
i
−1
X
≤ (K + 1) an0
ω
k=0
r
π
f;
k+1
Rate of Strong Summability
by Matrix Means
n
X
r
+ K1 ω (f ; π (K + 1) ano )
k=
Z
h
Bogdan Szal
ank
1
(K+1)an0
1
(K+1)an0
≤ K2 an0
1Z
≤ K3 an0
π
an0
π
dt + K1 ω r (f ; π (K + 1) ano )
ωr f ;
t
ω r (f ; u)
r
du + ω (f ; an0 ) .
u2
According to
ω r (f ; an0 ) ≤ 4r ω r
an0 ≤ 2 · 4r
f;
2
Z
an0
an0
2
ω r (f ; t)
dt ≤ 2 · 4r
t
vol. 9, iss. 1, art. 28, 2008
i
−1
Z
0
an0
Title Page
Contents
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ω r (f ; t)
dt,
t
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(1.17), (1.18) and (4.16) lead us to (4.14).
Close
5.
Proofs of the Theorems
In this section we shall prove Theorems 2.1, 2.2 and 2.3.
5.1.
Proof of Theorem 2.1
Setting
Rn (x + h, x) = Tn (f, r; x + h) − Tn (f, r; x)
Rate of Strong Summability
by Matrix Means
Bogdan Szal
and
gh (x) = f (x + h) − f (x)
vol. 9, iss. 1, art. 28, 2008
and using the Minkowski inequality for r ≥ 1, we get
Title Page
|Rn (x + h, x)|
(
) r1 ) r1 ( n
n
X
X
−
ank |Sk (f ; x) − f (x)|r = ank |Sk (f ; x + h) − f (x + h)|r
k=0
k=0
( n
) r1
X
≤
ank |Sk (gh ; x) − gh (x)|r
.
k=0
By (4.2) we have
|Rn (x + h, x)|
1

r
n+1

]
[
4

X
r 
an,4k Ekr (gh ) + E[ n+1 ] (gh ) ln (2 (n + 1) ann )
≤ K1
4



 k=0
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≤ K2

n+1

4 ]
[X


an,4k ω r gh ,
k=0
π
k+1
+ ω gh ,
π
n+1
1
r
r 

ln (2 (n + 1) ann )
.


Since
|gh (x + l) − gh (x)| ≤ |f (x + l + h) − f (x + h)| + |f (x + l) − f (x)|
Rate of Strong Summability
by Matrix Means
Bogdan Szal
and
vol. 9, iss. 1, art. 28, 2008
|gh (x + l) − gh (x)| ≤ |f (x + l + h) − f (x + l)|+|f (x + h) − f (x)| ≤ 2ω (|h|) ,
therefore, for 0 ≤ k ≤ n,
Title Page
π
ω gh ,
k+1
(5.1)
π
≤ 2ω f,
k+1
Contents
and f ∈ H ω
ω gh ,
(5.2)
π
k+1
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Page 22 of 27
≤ 2ω (|h|) .
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From (5.2) and (1.2)
(5.3)
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|Rn (x + h, x)| ≤ 2K2 ω (|h|)

n+1

4 ]
[X


an,4k + (ln (2 (n + 1) ann ))r
k=0
≤ 2K2 ω (|h|) (1 + ln (2 (n + 1) ann )) .
1
r




Close
On the other hand, by (5.1),
|Rn (x + h, x)|

1
r
n+1

]
[
4
r

X
π
π
≤ 2K2
an,4k ω r f,
+ ω f,
ln (2 (n + 1) ann )
.


k+1
n+1

 k=0
(5.4)
Using (5.3) and (5.4) we get
(5.5)
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
kTn (f, r; · + h) − Tn (f, r; ·)kC
ω (|h|)
h6=0
sup
p
p
(kRn (· + h, ·)kC ) q
= sup
(kRn (· + h, ·)kC )1− q
ω (|h|)
h6=0
p
q
≤ K3 (1 + ln (2 (n + 1) ann ))

n+1

4 ]
[X
π
r
×
an,4k ω f,

k+1
 k=0
+ ω f,
π
n+1
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Contents
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Page 23 of 27
r ) r1 (1− pq )
ln (2 (n + 1) ann )
.
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Similarly, by (4.2) we have
(5.6)
kTn (f, r)kC ≤ K4

n+1

4 ]
[X


k=0
an,4k ω
r
π
f,
k+1
+ ω f,
≤ K4

n+1

4 ]
[X


an,4k ω
π
n+1
r
k=0
r ) r1
ln (2 (n + 1) ann )
π
f,
k+1
r ) r1 pq
π
+ ω f,
ln (2 (n + 1) ann )
n+1
×

n+1

4 ]
[X


an,4k ω
r
k=0
π
f,
k+1
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
Title Page
Contents
r ) r1 (1− pq )
π
ln (2 (n + 1) ann )
+ ω f,
n+1
p
q
≤ K5 (1 + ln (2 (n + 1) ann ))

n+1

4 ]
[X
π
r
×
an,4k ω f,

k+1
 k=0
+ ω f,
Rate of Strong Summability
by Matrix Means
π
n+1
r ) r1 (1− pq )
ln (2 (n + 1) ann )
.
Collecting our partial results (5.5), (5.6) and using Lemma 4.4 and Lemma 4.5 we
obtain that (2.1) and (2.2) hold. This completes our proof.
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5.2.
Proof of Theorem 2.2
Using (4.3) and the same method as in the proof of Lemma 4.4 we can show that
(5.7)
n+1
[X
2 ]
an,2k ω
k=0
r
π
f,
k+1
π = O ann H r;
n
Rate of Strong Summability
by Matrix Means
holds, if ω (t) satisfies (1.17) and (1.18), and
Bogdan Szal
(5.8)
n+1
[X
2 ]
an,2k ω r f,
k=0
π
k+1
vol. 9, iss. 1, art. 28, 2008
= O (ann H (r; ann ))
Title Page
if ω (t) satisfies (1.17).
The proof of Theorem 2.2 is analogously to the proof of Theorem 2.1. The only
difference being that instead of (4.2), (4.5) and (4.6) we use (4.3), (5.7) and (5.8)
respectively. This completes the proof.
5.3. Proof of Theorem 2.3
Contents
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Using the same notations as in the proof of Theorem 2.1, from (4.4) and (5.2) we get
( n
) r1
X
(5.9)
|Rn (x + h, x)| ≤ K1
ank Ekr (gh )
k=0
≤ K2
( n
X
ank ω r
k=0
≤ 2K2 ω (|h|) .
π
gh ,
k+1
) r1
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On the other hand, by (4.4) and (5.1), we have
(5.10)
|Rn (x + h, x)| ≤ K2
( n
X
ank ω r
k=0
≤ 2K2
( n
X
ank ω r
k=0
π
gh ,
k+1
) r1
π
f,
k+1
) r1
.
Rate of Strong Summability
by Matrix Means
Bogdan Szal
Similarly, we can show that
(5.11)
kTn (f, r)kC ≤ K3
vol. 9, iss. 1, art. 28, 2008
( n
X
k=0
ank ω
r
f,
π
k+1
) r1
.
Title Page
Contents
Finally, using the same method as in the proof of Theorem 2.1 and Lemma 4.6, (2.6)
and (2.7) follow from (5.9) – (5.11).
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References
[1] P. CHANDRA, On the degree of approximation of a class of functions by means
of Fourier series, Acta Math. Hungar., 52 (1988), 199–205.
[2] P. CHANDRA, A note on the degree of approximation of continuous function,
Acta Math. Hungar., 62 (1993), 21–23.
[3] L. LEINDLER, On the degree of approximation of continuous functions, Acta
Math. Hungar., 104(1-2), (2004), 105–113.
Rate of Strong Summability
by Matrix Means
Bogdan Szal
vol. 9, iss. 1, art. 28, 2008
[4] R.N. MOHAPATRA AND P. CHANDRA, Degree of approximation of functions
in the Hölder metric, Acta Math. Hungar., 41(1-2) (1983), 67–76.
Title Page
[5] T. SINGH, Degree of approximation to functions in a normed spaces, Publ.
Math. Debrecen, 40(3-4) (1992), 261–271.
[6] XIE-HUA SUN, Degree of approximation of functions in the generalized Hölder
metric, Indian J. Pure Appl. Math., 27(4) (1996), 407–417.
[7] B. SZAL, On the strong approximation of functions by matrix means in the
generalized Hölder metric, Rend. Circ. Mat. Palermo (2), 56(2) (2007), 287–
304.
Contents
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