ON THE DEGREE OF STRONG APPROXIMATION
OF CONTINUOUS FUNCTIONS BY SPECIAL
MATRIX
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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
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Received:
13 May, 2009
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Accepted:
20 October, 2009
Page 1 of 17
Communicated by:
I. Gavrea
2000 AMS Sub. Class.:
40F04, 41A25, 42A10.
Key words:
Strong approximation, matrix means, classes of number sequences.
Abstract:
In the presented paper we will generalize the result of L. Leindler [3] to the
class M RBV 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
Degree of Strong Approximation
4
Proof of Theorem 2.1
14
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) ∼
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·k .
Let A := (ank ) (k, n = 0, 1, ...) be a lower triangular infinite matrix of real numbers satisfying the following conditions:
(1.2)
ank ≥ 0 (0 ≤ k ≤ n) ,
ank = 0, (k > n)
and
n
X
where k, n = 0, 1, 2, ....
Let the A−transformation of (Sn (f ; x)) be given by
(1.3)
tn (f ) := tn (f ; x) :=
ank Sk (f ; x)
(n = 0, 1, ...)
) r1
ank |Sk (f ; x) − f (x)|r
k=0
Now we define two classes of sequences.
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and the strong Ar −transformation of (Sn (f ; x)) for r > 0 be given by
Tn (f, r) := Tn (f, r; x) :=
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k=0
( n
X
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
ank = 1,
k=0
n
X
Degree of Strong Approximation
(n = 0, 1, ...) .
Close
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
n=m
for m = 0, 1, 2, ..., where K (c) is a constant depending only on c (see [3]).
A null sequence c := (cn ) of positive numbers is called of Mean Rest Bounded
Variation, or briefly c ∈ M RBV S, if it has the property
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
∞
X
2m
1 X
|cn − cn+1 | ≤ K (c)
cn
m + 1 n=m
n=2m
(1.5)
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for m = 0, 1, 2, ... (see [5]).
Therefore we assume that the sequence (K (αn ))∞
n=0 is bounded, that is, there
exists a constant K such that
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0 ≤ K (αn ) ≤ K
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holds for all n, where K (αn ) denotes the sequence of constants appearing in the
inequalities (1.4) or (1.5) for the sequence αn := (ank )∞
k=0 . Now we can give some
conditions to be used later on. We assume that for all n
∞
X
(1.6)
|ank − ank+1 | ≤ Kanm
(0 ≤ m ≤ n)
and
(1.7)
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k=m
∞
X
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2m
1 X
|ank − ank+1 | ≤ K
ank
m + 1 k=m
k=2m
(0 ≤ 2m ≤ n)
Close
hold if αn := (ank )∞
k=0 belongs to RBV S or M RBV 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:
Z π
ω (f ; t)
(1.8)
dt = O (H (u))
(u → 0+ ) , H (t) ≥ 0
t2
u
and
Degree of Strong Approximation
Z
t
Bogdan Szal
H (u) du = O (tH (t))
(1.9)
(t → O+ ) .
vol. 10, iss. 4, art. 111, 2009
0
In [3], L. Leindler generalized this result to the class RBV S. Namely, he proved
the following theorem:
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Theorem 1.1. Let (1.2), (1.6), (1.8) and (1.9) hold. Then for f ∈ C2π
ktn (f ) − f k = O (an0 H (an0 )) .
It is clear that
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(1.10)
RBV S ⊆ M RBV S.
In [7], we proved that RBV S 6= M RBV S. Namely, we showed that the sequence

if n = 1,
 1
dn :=
 1+m+(−1)n m if µ ≤ n < µ
m
m+1 ,
(2µm )2 m
where µm = 2m for m = 1, 2, 3, ..., belongs to the class M RBV S but it does not
belong to the class RBV S.
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In the present paper we will generalize the mentioned result of L. Leindler [3] to
the class M RBV S and extend it to strong summability with a mediate function H
defined by the following conditions:
Z π r
ω (f ; t)
(1.11)
dt = O (H (r; u))
(u → 0+ ) , H (t) ≥ 0 and r > 0,
t2
u
and
Degree of Strong Approximation
Z
Bogdan Szal
t
H (r; u) du = O (tH (r; t))
(1.12)
(t → O+ ) .
vol. 10, iss. 4, art. 111, 2009
0
By K1 , K2 , . . . 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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2.
Main Results
Our main results are the following.
Theorem 2.1. Let (1.2), (1.7) and (1.11) hold. Then for f ∈ C2π and r > 0
n
π o r1 (2.1)
kTn (f, r)k = O
an0 H r;
.
n
If, in addition (1.12) holds, then
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
(2.2)
1
kTn (f, r)k = O {an0 H (r; an0 )} r .
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Using the inequality
Contents
ktn (f ) − f k ≤ kTn (f, 1)k ,
we can formulate the following corollary.
Corollary 2.2. Let (1.2), (1.7) and (1.11) hold. Then for f ∈ C2π
π ktn (f ) − f k = O an0 H 1;
.
n
If, in addition (1.12) holds, then
ktn (f ) − f k = O (an0 H (1; an0 )) .
Remark 1. By the embedding relation (1.7) we can observe that Theorem 1.1 follows
from Corollary 2.2.
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For special cases, putting
 rα−1
if αr < 1,

 t
ln πt
if αr = 1,
H (r; t) =


K1
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 ∈ C2π and r > 0,


O ({an0 }α )
if αr < 1,



oα  n if αr = 1,
O ln aπn0 an0
kTn (f, r)k =



1

 O {an0 } 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
Z s r
ω (f ; t)
dt = O (sH (r; s))
(s → 0+ ) .
t
0
Lemma 3.2. If (1.2) and (1.7) hold, then for f ∈ C2π and r > 0
(
) r1 
n
X
(3.1)
kTn (f, r)k ≤ O 
ank Ekr (f )  ,
C
k=0
where En (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)
n+1
kσn,m − f k ≤ 2
En (f )
m+1
where
(0 ≤ m ≤ n) ,
for m = 0, we obtain
r
{Tn (f, r; x)} ≤ 12
n
X
k=0
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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n
X
1
σn,m (f ; x) =
Sk (f ; x) ,
m + 1 k=n−m
r
Degree of Strong Approximation
ank Ekr (f )
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and (3.1) is obviously valid, for n ≤ 5.
Let n ≥ 6 and let m = mn be such that
2m+1 + 4 ≤ n < 2m+2 + 4.
Hence
r
{Tn (f, r; x)} ≤
3
X
Degree of Strong Approximation
ank |Sk (f ; x) − f (x)|r
Bogdan Szal
k=0
+
k+1
m−1
X 2 X+4
vol. 10, iss. 4, art. 111, 2009
n
X
r
ani |Si (f ; x) − f (x)| +
ank |Sk (f ; x) − f (x)|r .
k=2m +5
k=1 i=2k +2
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Applying the Abel transformation and (3.2) to the first sum we obtain
Contents
{Tn (f, r; x)}r
r
≤8
3
X
ank Ekr
(f ) +
m−1
X


k=1
k=0
+an,2k+1 +4
2k+1
X+3
2k+1
X+4
(ani − an,i+1 )
i=2k +2
i
X
r
|Sl (f ; x) − f (x)|
l=2k +2
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
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r
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|Si (f ; x) − f (x)|
i=2k +2
+
n−1
X
(ank − an,k+1 )
k=2m +2
+ ann
n
X
k=2m +2
k
X
Close
r
|Sl (f ; x) − f (x)|
l=2m−1
|Sk (f ; x) − f (x)|r
r
≤8
3
X
ank Ekr
(f ) +
k=0
m−1
X


k=1
+an,2k+1 +4
2k+1
X+3
2k+1
X+4
|ani − an,i+1 |
i=2k +2
2k+1
X+3
|Sl (f ; x) − f (x)|r
l=2k +2

|Si (f ; x) − f (x)|r 
i=2k +2
+
n−1
X
|ank − an,k+1 |
k=2m +2
+ ann
2m+2
X+3
Degree of Strong Approximation
r
|Sl (f ; x) − f (x)|
2m+2
X+4
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
l=2m +2
|Sk (f ; x) − f (x)|r .
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k=2m +2
Using the well-known Leindler’s inequality [4]
) 1s
(
n
X
1
|Sk (f ; x) − f (x)|s
≤ K1 En−m (f )
m + 1 k=n−m
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for 0 ≤ m ≤ n, m = O (n) and s > 0, we obtain
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3
X
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{Tn (f, r; x)}r ≤ 8r
ank Ekr (f )



 k=0
k+1 +3
2
m−1
X
X
 2k + 3 E2rk +2 (f ) 
+ K2
|ani − an,i+1 | + an,2k+1 +4 

k=1
i=2k +2
!)
n−1
X
3 (2m + 1) E2rm +2
|ank − an,k+1 | + ann
.
k=2m +2
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Using (1.7) we get
r
r
{Tn (f, r; x)} ≤ 8
3
X
ank Ekr (f )
k=0



m−1
X
1
 2k + 3 E2rk +2 (f ) K
+ K2
k−1

2
+2
ani + an,2k+1 +4 
m +2
2X
1
2m−1 + 2
an,2k+1 +4 =
(ani − ani+1 ) ≤
i=2k+1 +4
≤
∞
X
ani + ann
|ani − ani+1 | ≤ K
.
vol. 10, iss. 4, art. 111, 2009
i=2m−1 +1
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∞
X
|ani − ani+1 |
i=2k+1 +4
i=2k +2
Bogdan Szal
!)
In view of (1.7), we also obtain for 1 ≤ k ≤ m − 1,
∞
X
Degree of Strong Approximation
i=2k−1 +1
k=1
3 (2m + 1) E2rm +2 (f ) K

k +2
2X
k +2
2X
1
2k−1 + 2
ani
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i=2k−1 +1
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and
ann =
≤
∞
X
(ani − ani+1 ) ≤
i=n
∞
X
i=2m +2
∞
X
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|ani − ani+1 |
Close
i=n
|ani − ani+1 | ≤ K
1
2m−1 + 2
m +2
2X
i=2m−1 +1
ani.
Hence
r
r
{Tn (f, r; x)} ≤ 8
3
X
ank Ekr (f )
k=0
+ K3

m−1
X

r
≤8
3
X
E2rk +2 (f )
k=1
k +2
2X
ani + E2rm +2 (f )
i=2m−1 +1
i=2k−1 +1
2m +2
ank Ekr
k=0
n
X
≤ K4
(f ) + 2K3
X
m +2
2X
ani



Degree of Strong Approximation
Bogdan Szal
ank Ekr
(f )
vol. 10, iss. 4, art. 111, 2009
k=3
ank Ekr (f ) .
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k=0
This ends our proof.
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4.
Proof of Theorem 2.1
Using Lemma 3.2 we have
(4.1)
|Tn (f, r; x)| ≤ K1
( n
X
) r1
ank Ekr (f )
k=0
≤ K2
( n
X
ank ω
r
k=0
π
f;
k+1
) r1
Degree of Strong Approximation
.
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
If (1.7) holds, then, for any m = 1, 2, ..., n,
anm − an0
m−1
X
≤ |anm − an0 | = |an0 − anm | = (ank − ank+1 )
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k=0
≤
m−1
X
k=0
|ank − ank+1 | ≤
∞
X
|ank − ank+1 | ≤ Kan0 ,
k=0
whence
(4.2)
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anm ≤ (K + 1) an0 .
Therefore, by (1.2),
(4.3)
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(K + 1) (n + 1) an0 ≥
n
X
ank = 1.
k=0
First we prove (2.1). Using (4.2), we get
n
n
X
X
π
π
r
r
ank ω f ;
≤ (K + 1) an0
ω f;
k+1
k+1
k=0
k=0
Close
n+1
Z
π
≤ K3 an0
ω f;
dt
t
1
Z π r
ω (f ; u)
= πK3 an0
du
π
u2
n+1
r
and by (4.1), (1.11) we obtain that (2.1) holds.
Now, we prove (2.2). From (4.3) we obtain
n
X
Degree of Strong Approximation
Bogdan Szal
ank ω
r
k=0
h
≤
π
f;
k+1
1
(K+1)an0
X
vol. 10, iss. 4, art. 111, 2009
i
−1
ank ω r
k=0
π
f;
k+1
n
X
+
k=
h
ank ω r
1
(K+1)an0
i
−1
π
f;
k+1
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.
Again using (1.2), (4.2) and the monotonicity of the modulus of continuity, we get
n
X
π
r
ank ω f ;
k+1
k=0
h
≤ (K + 1) an0
1
(K+1)an0
X
i
−1
ω
r
k=0
π
f;
k+1
+ K4 ω (f ; π (K + 1) ano )
k=
h
1
(K+1)an0
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n
X
r
Contents
ank
i
−1
Z
(4.4)
1
(K+1)an0
≤ K5 an0
1Z
≤ K6 an0
π
an0
π
ωr f ;
dt + K4 ω r (f ; π (K + 1) ano )
t
ω r (f ; u)
r
du + ω (f ; an0 ) .
u2
Moreover
(4.5)
a n0
ω r (f ; an0 ) ≤ 4r ω r f ;
Z an02 r
ω (f ; t)
≤ 2 · 4r
dt
an0
t
Z a2 n0 r
ω (f ; t)
≤ 2 · 4r
dt.
t
0
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.
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
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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).
Degree of Strong Approximation
Bogdan Szal
vol. 10, iss. 4, art. 111, 2009
Title Page
[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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