On the Degree of Strong Approximation of Continuous Functions by ...

ON THE DEGREE OF STRONG APPROXIMATION
OF CONTINUOUS FUNCTIONS BY SPECIAL
MATRIX
BOGDAN SZAL
Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
65-516 Zielona Góra
ul. Szafrana 4a, Poland
EMail: B.Szal@wmie.uz.zgora.pl
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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Received: 13 May, 2009
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Accepted: 20 October, 2009
Communicated by: I. Gavrea
2000 AMS Sub. Class.: 40F04, 41A25, 42A10.
Key words:
Strong approximation, matrix means, classes of number sequences.
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Abstract:
In the presented paper we will generalize the result of L. Leindler [3] to the
class MRBV S and extend it to the strong summability with a mediate function
satisfying the standard conditions.
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Contents
1 Introduction 3
2 Main Results 7
3 Lemmas 9
4 Proof of Theorem 2.1 14
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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1. Introduction
Let f be a continuous and 2π-periodic function and let
(1.1) f (x) ∼ a ∞
0
2 + ∑
(a n cos nx + b n sin nx)
n=1
be its Fourier series. Denote by S n (x) = S n (f, x) the n-th partial sum of (1.1) and
by ω (f, δ) the modulus of continuity of f ∈ C 2π . The usual supremum norm will
be denoted by ‖·‖ .
Let A := (a nk ) (k, n = 0, 1, ...) be a lower triangular infinite matrix of real numbers
satisfying the following conditions:
(1.2) a nk ≥ 0 (0 ≤ k ≤ n) , a nk = 0, (k > n) and
where k, n = 0, 1, 2, ....
Let the A−transformation of (S n (f; x)) be given by
(1.3) t n (f) := t n (f; x) :=
n∑
a nk = 1,
k=0
n∑
a nk S k (f; x) (n = 0, 1, ...)
and the strong A r −transformation of (S n (f; x)) for r > 0 be given by
T n (f, r) := T n (f, r; x) :=
k=0
{ n∑
k=0
a nk |S k (f; x) − f (x)| r } 1
r
(n = 0, 1, ...) .
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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Now we define two classes of sequences.

A sequence c := (c n ) of nonnegative numbers tending to zero is called the Rest
Bounded Variation Sequence, or briefly c ∈ RBV S, if it has the property
∞∑
(1.4)
|c n − c n+1 | ≤ K (c) c m
n=m
for m = 0, 1, 2, ..., where K (c) is a constant depending only on c (see [3]).
A null sequence c := (c n ) of positive numbers is called of Mean Rest Bounded
Variation, or briefly c ∈ MRBV S, if it has the property
(1.5)
∞∑
n=2m
|c n − c n+1 | ≤ K (c)
1
m + 1
2m∑
c n
for m = 0, 1, 2, ... (see [5]).
Therefore we assume that the sequence (K (α n )) ∞ n=0
is bounded, that is, there
exists a constant K such that
0 ≤ K (α n ) ≤ K
holds for all n, where K (α n ) denotes the sequence of constants appearing in the
inequalities (1.4) or (1.5) for the sequence α n := (a nk ) ∞ k=0
. Now we can give some
conditions to be used later on. We assume that for all n
∞∑
(1.6)
|a nk − a nk+1 | ≤ Ka nm (0 ≤ m ≤ n)
and
(1.7)
∞∑
k=2m
k=m
|a nk − a nk+1 | ≤ K 1
m + 1
2m∑
k=m
n=m
a nk (0 ≤ 2m ≤ n)
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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hold if α n := (a nk ) ∞ k=0
belongs to RBV S or MRBV S, respectively.
In [1] and [2] P. Chandra obtained some results on the degree of approximation
for the means (1.3) with a mediate function H such that:
(1.8)
and
(1.9)
∫ π
u
ω (f; t)
t 2 dt = O (H (u)) (u → 0 + ) , H (t) ≥ 0
∫ t
0
H (u) du = O (tH (t)) (t → O + ) .
In [3], L. Leindler generalized this result to the class RBV S. Namely, he proved
the following theorem:
Theorem 1.1. Let (1.2), (1.6), (1.8) and (1.9) hold. Then for f ∈ C 2π
It is clear that
‖t n (f) − f‖ = O (a n0 H (a n0 )) .
(1.10) RBV S ⊆ MRBV S.
In [7], we proved that RBV S ≠ MRBV S. Namely, we showed that the sequence
⎧
⎨ 1 if n = 1,
d n :=
⎩
if µ m ≤ n < µ m+1 ,
1+m+(−1) n m
(2 µm ) 2 m
where µ m = 2 m for m = 1, 2, 3, ..., belongs to the class MRBV S but it does not
belong to the class RBV S.
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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In the present paper we will generalize the mentioned result of L. Leindler [3] to
the class MRBV S and extend it to strong summability with a mediate function H
defined by the following conditions:
(1.11)
and
(1.12)
∫ π
u
ω r (f; t)
t 2 dt = O (H (r; u)) (u → 0 + ) , H (t) ≥ 0 and r > 0,
∫ t
0
H (r; u) du = O (tH (r; t)) (t → O + ) .
By K 1 , K 2 , . . . we shall denote either an absolute constant or a constant depending
on the indicated parameters, not necessarily the same in each occurrence.
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Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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2. Main Results
Our main results are the following.
Theorem 2.1. Let (1.2), (1.7) and (1.11) hold. Then for f ∈ C 2π and r > 0
( { (
(2.1) ‖T n (f, r)‖ = O a n0 H r; π )} )
1
r
.
n
If, in addition (1.12) holds, then
(2.2) ‖T n (f, r)‖ = O
Using the inequality
we can formulate the following corollary.
(
)
{a n0 H (r; a n0 )} 1 r
.
‖t n (f) − f‖ ≤ ‖T n (f, 1)‖ ,
Corollary 2.2. Let (1.2), (1.7) and (1.11) hold. Then for f ∈ C 2π
( (
‖t n (f) − f‖ = O a n0 H 1; π ))
.
n
If, in addition (1.12) holds, then
‖t n (f) − f‖ = O (a n0 H (1; a n0 )) .
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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Remark 1. By the embedding relation (1.7) we can observe that Theorem 1.1 follows
from Corollary 2.2.

For special cases, putting
⎧
⎪⎨
t rα−1 if αr < 1,
H (r; t) = ln π if αr = 1,
t
⎪⎩
K 1 if αr > 1,
where r > 0 and 0 < α ≤ 1, we can derive from Theorem 2.1 the next corollary.
Corollary 2.3. Under the conditions (1.2) and (1.7) we have, for f ∈ C 2π and r > 0,
⎧
O ({a n0 } α ) if αr < 1,
⎪⎨ ({ ( ) } α )
π
‖T n (f, r)‖ = O ln
a n0
a n0 if αr = 1,
( )
⎪⎩ O {a n0 } 1 r
if αr > 1.
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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3. Lemmas
To prove our main result we need the following lemmas.
Lemma 3.1 ([6]). If (1.11) and (1.12) hold, then for r > 0
∫ s
0
ω r (f; t)
dt = O (sH (r; s)) (s → 0 + ) .
t
Lemma 3.2. If (1.2) and (1.7) hold, then for f ∈ C 2π and r > 0
⎛{ n∑
} ⎞
1
r
(3.1) ‖T n (f, r)‖ C
≤ O ⎝ a nk Ek r (f) ⎠ ,
k=0
where E n (f) denotes the best approximation of the function f by trigonometric
polynomials of order at most n.
Proof. It is clear that (3.1) holds for n = 0, 1, ..., 5. Namely, by the well known
inequality [8]
(3.2) ‖σ n,m − f‖ ≤ 2 n + 1
m + 1 E n (f) (0 ≤ m ≤ n) ,
where
for m = 0, we obtain
σ n,m (f; x) = 1
m + 1
n∑
k=n−m
S k (f; x) ,
n∑
{T n (f, r; x)} r ≤ 12 r a nk Ek r (f)
k=0
Degree of Strong Approximation
Bogdan Szal
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and (3.1) is obviously valid, for n ≤ 5.
Let n ≥ 6 and let m = m n be such that
Hence
2 m+1 + 4 ≤ n < 2 m+2 + 4.
{T n (f, r; x)} r ≤
+
m−1
∑
3∑
a nk |S k (f; x) − f (x)| r
k=0
2 k+1 ∑+4
k=1 i=2 k +2
a ni |S i (f; x) − f (x)| r +
n∑
k=2 m +5
a nk |S k (f; x) − f (x)| r .
Applying the Abel transformation and (3.2) to the first sum we obtain
{T n (f, r; x)} r
≤ 8 r
3∑
a nk Ek r (f) +
k=0
+a n,2 k+1 +4
+
∑n−1
k=2 m +2
+ a nn
n∑
m−1
∑
k=1
2 k+1 ∑+4
i=2 k +2
⎛
⎝
∑
2 k+1 +3
i=2 k +2
(a nk − a n,k+1 )
k=2 m +2
(a ni − a n,i+1 )
⎞
|S i (f; x) − f (x)| r ⎠
k∑
|S k (f; x) − f (x)| r
i∑
l=2 k +2
l=2 m−1 |S l (f; x) − f (x)| r
|S l (f; x) − f (x)| r
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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≤ 8 r
3∑
a nk Ek r (f) +
k=0
+a n,2 k+1 +4
+
∑n−1
k=2 m +2
2 m+2 ∑+4
+ a nn
m−1
∑
k=1
2 k+1 ∑+4
i=2 k +2
⎛
⎝
∑
2 k+1 +3
i=2 k +2
2 k+1 ∑+3
|a ni − a n,i+1 | |S l (f; x) − f (x)| r
⎞
|S i (f; x) − f (x)| r ⎠
l=2 k +2
2 m+2 ∑+3
|a nk − a n,k+1 | |S l (f; x) − f (x)| r
k=2 m +2
l=2 m +2
|S k (f; x) − f (x)| r .
Using the well-known Leindler’s inequality [4]
{
} 1
s
1
n∑
|S k (f; x) − f (x)| s
m + 1
k=n−m
for 0 ≤ m ≤ n, m = O (n) and s > 0, we obtain
3∑
{T n (f, r; x)} r ≤ 8 r a nk Ek r (f)
+ K 2
⎧
⎨
⎩
m−1
∑
k=1
k=0
⎛
⎛
⎝ ( 2 k + 3 ) E r 2 k +2 (f) ∑
2 k+1 +3
⎝
i=2 k +2
3 (2 m + 1) E r 2 m +2
≤ K 1 E n−m (f)
⎞⎞
|a ni − a n,i+1 | + a n,2
⎠⎠
k+1 +4
( n−1 ∑
k=2 m +2
|a nk − a n,k+1 | + a nn
)}
.
Degree of Strong Approximation
Bogdan Szal
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Using (1.7) we get
3∑
{T n (f, r; x)} r ≤ 8 r a nk Ek r (f)
+ K 2
⎧
⎨
⎩
m−1
∑
k=1
k=0
⎛
⎛
⎝ ( 2 k + 3 ) E r 2 k +2 (f) 1
⎝K
2 k−1 + 2
(
3 (2 m + 1) E2 r 1
m +2 (f) K
2 m−1 + 2
In view of (1.7), we also obtain for 1 ≤ k ≤ m − 1,
and
a n,2 k+1 +4 =
a nn =
≤
≤
∞∑
i=2 k+1 +4
∞∑
i=2 k +2
(a ni − a ni+1 ) ≤
2∑
k +2
i=2 k−1 +1
∞∑
i=2 k+1 +4
1
|a ni − a ni+1 | ≤ K
2 k−1 + 2
∞∑
(a ni − a ni+1 ) ≤
i=n
∞∑
i=2 m +2
∞∑
|a ni − a ni+1 |
i=n
1
|a ni − a ni+1 | ≤ K
2 m−1 + 2
⎞⎞
a ni + a n,2
⎠⎠
k+1 +4
2∑
m +2
i=2 m−1 +1
|a ni − a ni+1 |
2∑
k +2
i=2 k−1 +1
2∑
m +2
i=2 m−1 +1
a ni + a nn
)}
a ni
a ni.
.
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Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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Hence
3∑
{T n (f, r; x)} r ≤ 8 r a nk Ek r (f)
≤ 8 r
≤ K 4
This ends our proof.
k=0
+ K 3
⎧
⎨
⎩
3∑
k=0
m−1
∑
k=1
2 k +2
E r 2 k +2 (f) ∑
i=2 k−1 +1
2∑
m +2
a nk Ek r (f) + 2K 3
n∑
a nk Ek r (f) .
k=0
k=3
a ni + E r 2 m +2 (f)
a nk E r k (f)
2∑
m +2
i=2 m−1 +1
a ni
⎫
⎬
⎭
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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4. Proof of Theorem 2.1
Using Lemma 3.2 we have
(4.1)
|T n (f, r; x)| ≤ K 1
{ n∑
k=0
a nk E r k (f)
} 1
r
≤ K 2
{ n∑
k=0
a nk ω r (f;
) } 1 r
π
.
k + 1
If (1.7) holds, then, for any m = 1, 2, ..., n,
m−1
∑
a nm − a n0 ≤ |a nm − a n0 | = |a n0 − a nm | =
(a nk − a nk+1 )
∣
∣
whence
≤
m−1
∑
k=0
|a nk − a nk+1 | ≤
k=0
∞∑
|a nk − a nk+1 | ≤ Ka n0 ,
k=0
(4.2) a nm ≤ (K + 1) a n0 .
Therefore, by (1.2),
(4.3) (K + 1) (n + 1) a n0 ≥
n∑
a nk = 1.
k=0
First we prove (2.1). Using (4.2), we get
n∑
)
a nk ω
(f;
r π
≤ (K + 1) a n0
k + 1
k=0
n∑
)
ω
(f;
r π
k + 1
k=0
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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≤ K 3 a n0
∫ n+1
1
= πK 3 a n0
∫ π
π
n+1
and by (4.1), (1.11) we obtain that (2.1) holds.
Now, we prove (2.2). From (4.3) we obtain
n∑
)
a nk ω
(f;
r π
k + 1
k=0
≤
[
1
]−1
(K+1)a n0
∑
k=0
a nk ω r (f;
)
π
+
k + 1
[
k=
(
ω r f; π )
dt
t
ω r (f; u)
du
u 2
n∑
1
(K+1)a n0
]−1
a nk ω r (f;
)
π
.
k + 1
Again using (1.2), (4.2) and the monotonicity of the modulus of continuity, we get
n∑
)
a nk ω
(f;
r π
k + 1
k=0
[
1
]−1
(K+1)a
∑ n0
)
≤ (K + 1) a n0 ω
(f;
r π
k + 1
k=0
n∑
+ K 4 ω r (f; π (K + 1) a no )
[
k=
1
(K+1)a n0
]−1
a nk
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(4.4)
Moreover
(4.5)
≤ K 5 a n0
∫ 1
(K+1)a n0
1
( ∫ π
≤ K 6 a n0
(
ω r f; π )
t
a n0
ω r (f; u)
u 2 du + ω r (f; a n0 )
(
ω r (f; a n0 ) ≤ 4 r ω r f; a )
n0
∫ 2
an0
≤ 2 · 4 r
dt + K 4 ω r (f; π (K + 1) a no )
)
.
a n0
2
≤ 2 · 4 r ∫ an0
0
ω r (f; t)
dt
t
ω r (f; t)
dt.
t
Thus collecting our partial results (4.1), (4.4), (4.5) and using (1.11) and Lemma 3.1
we can see that (2.2) holds. This completes our proof.
□
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Bogdan Szal
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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.
[4] L. LEINDLER, Strong Approximation by Fourier Series, Akadèmiai Kiadò, Budapest
(1985).
[5] L. LEINDLER, Integrability conditions pertaining to Orlicz space, J. Inequal.
Pure and Appl. Math., 8(2) (2007), Art. 38.
[6] B. SZAL, On the rate of strong summability by matrix means in the generalized
Hölder metric, J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 28.
[7] B. SZAL, A note on the uniform convergence and boundedness a generalized
class of sine series, Comment. Math., 48(1) (2008), 85–94.
[8] Ch. J. DE LA VALLÉE - POUSSIN, Leçons sur L’Approximation des Fonctions
d’une Variable Réelle, Paris (1919).
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