Volume 8 (2007), Issue 3, Article 70, 12 pp.
ON THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE
FUNCTIONS
WŁODZIMIERZ ŁENSKI
U NIVERSITY OF Z IELONA G ÓRA
FACULTY OF M ATHEMATICS , C OMPUTER S CIENCE AND E CONOMETRICS
65-516 Z IELONA G ÓRA , UL . S ZAFRANA 4 A
P OLAND
W.Lenski@wmie.uz.zgora.pl
Received 20 December, 2006; accepted 25 July, 2007
Communicated by L. Leindler
A BSTRACT. We show the strong approximation version of some results of L. Leindler [3] connected with the theorems of P. Chandra [1, 2].
Key words and phrases: Degree of strong approximation, Special sequences.
2000 Mathematics Subject Classification. 42A24, 41A25.
1. I NTRODUCTION
Let Lp (1 < p < ∞) [C] be the class of all 2π–periodic real–valued functions integrable
in the Lebesgue sense with p–th power [continuous] over Q = [−π, π] and let X p = Lp when
1 < p < ∞ or X p = C when p = ∞. Let us define the norm of f ∈ X p as
 1
 R |f (x)|p dx p when 1 < p < ∞,
Q
kf kX p = kf (·)kX p =

supx∈Q |f (x)|
when p = ∞,
and consider its trigonometric Fourier series
∞
ao (f ) X
Sf (x) =
+
(aν (f ) cos νx + bν (f ) sin νx)
2
ν=0
with the partial sums Sk f .
Let A := (an,k ) (k, n = 0, 1, 2, . . . ) be a lower triangular infinite matrix of real numbers and
let the A−transforms of (Sk f ) be given by
n
X
Tn,A f (x) := an,k Sk f (x) − f (x)
(n = 0, 1, 2, . . . )
k=0
323-06
2
W ŁODZIMIERZ Ł ENSKI
and
q
Hn,A
f
(x) :=
( n
X
) 1q
q
an,k |Sk f (x) − f (x)|
(q > 0, n = 0, 1, 2, . . . ) .
k=0
As a measure of approximation, by the above quantities we use the pointwise characteristic
Z δ
p1
1
p
wx f (δ)Lp :=
|ϕx (t)| dt ,
δ 0
where
ϕx (t) := f (x + t) + f (x − t) − 2f (x) .
wx f (δ)Lp is constructed based on the definition of Lebesgue points (Lp −points), and the modulus of continuity for f in the space X p defined by the formula
ωf (δ)X p := sup kϕ · (h)kX p .
0≤|h|≤δ
We can observe that with pe ≥ p, for f ∈ X pe, by the Minkowski inequality
kw · f (δ)p kX pe ≤ ωf (δ)X pe .
The deviation Tn,A f was estimated by P. Chandra [1, 2] in the norm of f ∈ C and for
monotonic sequences an = (an,k ). These results were generalized by L. Leindler [3] who
considered the sequences of bounded variation instead of monotonic ones. In this note we shall
q
f and the integrable functions. We shall also give some results
consider the strong means Hn,A
on norm approximation.
By K we shall designate either an absolute constant or a constant depending on some parameters, not necessarily the same of each occurrence.
2. S TATEMENT OF THE R ESULTS
Let us consider a function wx of modulus of continuity type on the interval [0, +∞), i.e., a
nondecreasing continuous function having the following properties: wx (0) = 0, wx (δ1 + δ2 ) ≤
wx (δ1 ) + wx (δ2 ) for any 0 ≤ δ1 ≤ δ2 ≤ δ1 + δ2 and let
Lp (wx ) = {f ∈ Lp : wx f (δ)Lp ≤ wx (δ)} .
We can now formulate our main results.
To start with, we formulate the results on pointwise approximation.
Theorem 2.1. Let an = (an,m ) satisfy the following conditions:
an,m ≥ 0,
(2.1)
n
X
an,k = 1
k=0
and
m−1
X
(2.2)
|an,k − an,k+1 | ≤ Kan,m ,
k=0
where
m = 0, 1, . . . , n
Suppose wx is such that
Z
p
(2.3)
uq
u
π
and
(wx (t))p
p
t1+ q
n = 0, 1, 2, . . .
p1
= O (uHx (u))
dt
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
X−1
k=0
as
=0 .
u → 0+,
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O N THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS
where Hx (u) ≥ 0, 1 < p ≤ q and
Z t
(2.4)
Hx (u) du = O (tHx (t))
as
3
t→0+.
0
If f ∈ Lp (wx ) , then
0
q
Hn,A
f (x) = O (an,n Hx (an,n ))
with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q.
Theorem 2.2. Let (2.1), (2.2) and (2.3) hold. If f ∈ Lp (wx ) then
π
π
q0
Hn,A f (x) = O wx
+ O an,n Hx
n+1
n+1
and if, in addition, (2.4) holds then
q0
Hn,A
f
(x) = O an,n Hx
π
n+1
with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q.
Theorem 2.3. Let (2.1) , (2.3) , (2.4) and
∞
X
(2.5)
|an,k − an,k+1 | ≤ Kan,m ,
k=m
where
m = 0, 1, . . . , n and n = 0, 1, 2, . . .
p
hold. If f ∈ L (wx ) then
0
q
Hn,A
f (x) = O (an,0 Hx (an,0 ))
with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q.
Theorem 2.4. Let us assume that (2.1), (2.3) and (2.5) hold. If f ∈ Lp (wx ) , then
π
π
q0
Hn,A f (x) = O wx
+ O an,0 Hx
.
n+1
n+1
If, in addition, (2.4) holds then
q0
Hn,A
f
(x) = O an,0 Hx
π
n+1
with q 0 ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q.
Consequently, we formulate the results on norm approximation.
Theorem 2.5. Let an = (an,m ) satisfy the conditions (2.1) and (2.2). Suppose ωf (·)X pe is such
that
p1
Z π
p
p
(ωf
(t)
p
e)
X
= O (uH (u)) as u → 0+
dt
(2.6)
uq
p
t1+ q
u
holds, with 1 < p ≤ q and pe ≥ p, where H (≥ 0) instead of Hx satisfies the condition (2.4).
If f ∈ X pe then
0
q
Hn,A f (·) = O (an,n H (an,n ))
e
Xp
0
with q ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q.
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4
W ŁODZIMIERZ Ł ENSKI
Theorem 2.6. Let (2.1) , (2.2) and (2.6) hold. If f ∈ X pe then
0
π
π
q
+ O an,n H
.
Hn,A f (·) = O ωf
n + 1 X pe
n+1
e
Xp
If, in addition, H (≥ 0) instead of Hx satisfies the condition (2.4) then
0
π
q
Hn,A f (·) = O an,n H
n+1
e
Xp
with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q.
Theorem 2.7. Let (2.1), (2.4) with a function H (≥ 0) instead of Hx , (2.5) and (2.6) hold.
If f ∈ X pe then
0
q
Hn,A f (·) = O (an,0 H (an,0 ))
e
Xp
0
with q ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q.
Theorem 2.8. Let (2.1), (2.5) and (2.6) hold. If f ∈ X pe then
0
π
π
q
+ O an,0 H
.
Hn,A f (·) = O ωf
n + 1 X pe
n+1
e
Xp
If, in addition, H (≥ 0) instead of Hx satisfies the condition (2.4) , then
0
π
q
Hn,A f (·) = O an,0 H
n+1
e
Xp
with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q.
3. AUXILIARY R ESULTS
In tis section we denote by ω a function of modulus of continuity type.
Lemma 3.1. If (2.3) with 0 < p ≤ q and (2.4) with functions ω and H (≥ 0) instead of wx
and Hx , respectively, hold then
Z u
ω (t)
dt = O (uH (u))
(u → 0+) .
(3.1)
t
0
Proof. Integrating by parts in the above integral we obtain
Z π
Z u
Z u
ω (t)
d
ω (s)
dt =
t
ds dt
t
dt
s2
0
0
t
Z π
u Z u Z π
ω (s)
ω (s)
= −t
ds +
ds dt
s2
s2
t
0
t
0
Z π
Z u Z π
ω (s)
ω (s)
≤u
ds +
ds dt
s2
s2
u
0
t
Z u Z π
Z π
ω (s)
ω (s)
ds dt
ds +
=u
1+ pq +1− pq
1+ pq +1− pq
0
t s
u s
p1
Z π
Z u Z π
p
p
p
ω
(s)
1
(ω
(s))
≤ uq
ds +
tq
ds dt,
1+p/q
s1+p/q
u s
0 t
t
since 1 −
p
q
≥ 0. Using our assumptions we have
Z u
Z u
ω (t)
1
dt = O (uH (u)) +
O (tH (t)) dt = O (uH (u))
t
0
0 t
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
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O N THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS
5
and thus the proof is completed.
Lemma 3.2 ([4, Theorem
5.20 II, Ch. XII]). Suppose that 1 < q (q − 1) ≤ p ≤ q and ξ =
1/p + 1/q − 1. If t−ξ g (t) ∈ Lp then
) 1q
(
Z π
p1
∞
−ξ
p
|ao (g)|q X
q
q
≤K
t g (t) dt .
(3.2)
+
(|ak (g)| + |bk (g)| )
2
−π
k=0
4. P ROOFS OF THE R ESULTS
q
Since Hn,A
f is the monotonic function of q we shall consider, in all our proofs, the quantity
q
q0
Hn,A f instead of Hn,A
f.
Proof of Theorem 2.1. Let
q
Hn,A
f
Z
q ) 1q
1 π
sin k + 12 t (x) =
an,k ϕx (t)
dt
π 0
2 sin 12 t
k=0
Z
( n
q ) 1q
1 an,n
X
sin k + 12 t ≤
an,k ϕx (t)
dt
1
π 0
2
sin
t
2
k=0
Z
( n
q ) 1q
1 π
1
X
sin
k
+
t 2
ϕx (t)
+
an,k dt
1
π an,n
2
sin
t
2
k=0
( n
X
= I (an,n ) + J (an,n )
and, by (2.1) , integrating by parts, we obtain,
Z an,n
|ϕx (t)|
dt
I (an,n ) ≤
2t
0
Z t
Z an,n
1 d
=
|ϕx (s)| ds dt
2t dt
0
0
Z an,n
Z an,n
wx f (t)1
1
=
|ϕx (t)| dt +
dt
2an,n 0
2t
0
Z an,n
wx f (t)1
1
=
wx f (an,n )1 +
dt
2
t
0
!
Z π
Z an,n
wx f (t)1
wx f (t)1
= K an,n
dt +
dt
t2
t
an,n
0
! p1
Z an,n
Z π
(wx f (t)L1 )p
wx f (t)L1
≤ K an,n
dt
+K
dt
t2
t
an,n
0
! p1
Z an,n
Z π
p
(w
f
(t)
)
wx f (t)Lp
p
x
p/q
L
≤ K an,n
dt
+K
dt .
t1+p/q
t
an,n
0
Since f ∈ Lp (wx ) and (2.4) holds, Lemma 3.1 and (2.3) give
I (an,n ) = O (an,n Hx (an,n )) .
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W ŁODZIMIERZ Ł ENSKI
The Abel transformation shows that
q
k Z π
1
X
sin
ν
+
t 1
2
(an,k − an,k+1 )
(J (an,n ))q =
dt
ϕx (t)
1
π
2
sin
t
a
n,n
2
ν=0
k=0
Z
q
n 1
π
X
sin
ν
+
t 1
2
dt ,
+ an,n
ϕx (t)
1
π
2
sin
t
a
n,n
2
ν=0
n−1
X
whence, by (2.2),
q
n Z π
X
sin ν + 12 t 1
ϕ (t)
dt .
(J (an,n )) ≤ (K + 1) an,n
π an,n x
2 sin 12 t
ν=0
q
Using inequality (3.2), we obtain
J (an,n ) ≤ K (an,n )
1
q
(Z
π
an,n
) p1
|ϕx (t)|p
dt
.
t1+p/q
Integrating by parts, we have
J (an,n ) ≤ K (an,n )
1
q
(
1
p
+ 1+
q
(
1
q
π
(wx f (t)Lp )
tp/q
≤ K (an,n )
p
t=an,n
Z
π
an,n
) p1
(wx f (t)Lp )p
dt
t1+p/q
Z
π
(wx f (π)Lp ) +
an,n
) p1
(wx f (t)Lp )p
dt
.
t1+p/q
Since f ∈ Lp (wx ) , by (2.3),
) p1
(wx (t))p
J (an,n ) ≤ K (an,n )
(wx (π)) +
dt
1+p/q
an,n t
(
) p1
Z π
p
p
(wx (t))
≤ K (an,n ) q
dt
1+p/q
an,n t
(
Z
1
q
π
= O (an,n Hx (an,n )) .
Thus our result is proved.
Proof of Theorem 2.2. Let, as before,
q
Hn,A
f
(x) ≤ I
π
n+1
+J
π
n+1
and
Z π
π
n + 1 n+1
π
I
≤
|ϕx (t)| dt = wx
.
n+1
π
n + 1 L1
0
π
we again use the Abel transformation and (2.2). Thus
In the estimate of J n+1
q
q
n Z π
X
sin ν + 12 t π
1
J
≤ (K + 1) an,n
ϕx (t)
dt
π π
n+1
2 sin 12 t
ν=0
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
n+1
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O N THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS
7
and, by inequality (3.2),
J
π
n+1
≤ K (an,n )
1
q
(Z
) p1
|ϕx (t)|p
dt
.
t1+p/q
π
π
n+1
Integrating (2.3) by parts, with the assumption f ∈ Lp (wx ) , we obtain
J
π
n+1
1
q
≤ K ((n + 1) an,n )
(
1
= O ((n + 1) an,n ) q
pq Z
π
n+1
π
π
n+1
π
Hx
n+1
π
n+1
) p1
(wx (t))p
dt
t1+p/q
π
as in the previous proof, with n+1
instead of an,n .
Finally, arguing as in [3, p.110], we can see that, for j = 0, 1, . . . , n − 1,
n−1
n−1
X
X
|an,j − an,n | ≤ (an,k − an,k+1 ) ≤
|an,k − an,k+1 | ≤ Kan,n ,
k=j
k=0
whence
an,j ≤ (K + 1) an,n
and therefore
(K + 1) (n + 1) an,n ≥
n
X
an,j = 1.
j=0
This inequality implies that
J
π
n+1
= O an,n Hx
π
n+1
and the proof of the first part of our statement is complete.
To prove of the second part of our assertion we have to estimate the term I
Proceeding analogously to the proof of Theorem 2.1, with an,n replaced by
I
π
n+1
=O
π
Hx
n+1
π
n+1
π
n+1
π
,
n+1
once more.
we obtain
.
By the inequality from the first part of our proof, the relation (n + 1)−1 = O (an,n ) holds,
whence the second statement follows.
Proof of Theorem 2.3. As usual, let
q
Hn,A
f (x) ≤ I (an,0 ) + J (an,0 ) .
Since f ∈ Lp (wx ) ,by the same method as in the proof of Theorem 2.1, Lemma 3.1 and (2.3)
yield
I (an,0 ) = O (an,0 Hx (an,0 )) .
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W ŁODZIMIERZ Ł ENSKI
By the Abel transformation
q
k Z π
1
X
sin
ν
+
t 1
2
(J (an,0 ))q ≤
|an,k − an,k+1 |
ϕx (t)
dt
1
π an,0
2 sin 2 t
ν=0
k=0
q
n Z π
X
sin ν + 12 t 1
dt
ϕ (t)
+ an,n
1
π an,0 x
2
sin
t
2
ν=0
Z
!
q
∞
∞ 1
π
X
X
sin
ν
+
t 1
2
dt .
≤
|an,k − an,k+1 | + an,n
ϕx (t)
1
π an,0
2
sin
t
2
ν=0
k=0
n−1
X
Arguing as in [3, p.110], by (2.5), we have
|an,n − an,0 | ≤
n−1
X
|an,k − an,k+1 | ≤
k=0
∞
X
|an,k − an,k+1 | ≤ Kan,0 ,
k=0
whence an,n ≤ (K + 1) an,0 and therefore
∞
X
|an,k − an,k+1 | + an,n ≤ (2K + 1) an,0
k=0
and
q
∞ Z π
X
sin ν + 12 t 1
(J (an,0 )) ≤ (2K + 1) an,0
ϕ (t)
dt .
π an,0 x
2 sin 12 t
ν=0
q
Finally, by (3.2),
J (an,0 ) ≤ K (an,0 )
1
q
(Z
π
an,0
p
) p1
|ϕx (t)|
dt
t1+p/q
and, by (2.3),
J (an,0 ) = O (an,0 Hx (an,0 )) .
This completes of our proof.
Proof of Theorem 2.4. We start as usual with the simple transformation
π
π
q
Hn,A f (x) ≤ I
+J
.
n+1
n+1
Similarly, as in the previous proofs, by (2.1) we have
π
π
I
≤ wx f
.
n+1
n + 1 L1
We estimate the term J in the following way
q
q X
n−1
k Z π
1
X
t sin
ν
+
π
1
2
J
≤
|an,k − an,k+1 |
ϕx (t)
dt
1
π π
n+1
t
2
sin
2
ν=0
n+1
k=0
Z
q
n π
X
sin ν + 12 t 1
+ an,n
ϕx (t)
dt
π π
2 sin 12 t
ν=0
n+1
Z
!
q
∞ ∞
π
X
X
sin ν + 21 t 1
≤
|an,k − an,k+1 | + an,n
ϕx (t)
dt .
π π
2 sin 12 t
ν=0
n+1
k=0
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
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O N THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS
9
From the assumption (2.5), arguing as before, we can see that
q ! 1q
∞ Z π
1
X
t sin
ν
+
π
1
2
ϕx (t)
J
≤ K an,0
dt
1
π π
n+1
2
sin
t
2
ν=0
n+1
and, by (3.2) ,
J
π
n+1
≤ K (an,0 )
1
q
(Z
π
π
n+1
) p1
|ϕx (t)|p
.
dt
t1+p/q
From (2.5) , it follows that an,k ≤ (K + 1) an,0 for any k ≤ n, and therefore
(K + 1) (n + 1) an,0 ≥
n
X
an,k = 1,
k=0
whence
J
π
n+1
) p1
|ϕx (t)|p
≤ K ((n + 1) an,0 )
dt
.
1+p/q
π
t
n+1
(
) p1
pq Z π
π
|ϕx (t)|p
≤ K (n + 1) an,0
dt
.
1+p/q
π
n+1
t
n+1
1
q
(
π
n+1
pq Z
π
Since f ∈ Lp (wx ), integrating by parts we obtain
π
π
π
≤ K (n + 1) an,0
Hx
J
n+1
n+1
n+1
π
≤ Kan,0 Hx
n+1
and the proof of the first part of our statement is complete.
π
To prove the second part, we have to estimate the term I n+1
once more.
Proceeding analogously to the proof of Theorem 2.1 we obtain
π
π
π
I
=O
Hx
.
n+1
n+1
n+1
From the start of our proof we have (n + 1)−1 = O (an,0 ) , whence the second assertion also
follows.
Proof of Theorem 2.5. We begin with the inequality
q
H f (·) ≤ kI (an,n )k
n,A
e
Xp
e
Xp
+ kJ (an,n )k
e
Xp
.
By (2.1) , Lemma 3.1 gives
Z
kI (an,n )k
e
Xp
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
≤
an,n
kϕ (t)k
p
e
X
dt
2t
0
Z an,n
ωf (t)X pe
≤
dt
2t
0
= O (an,n H (an,n )) .
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10
W ŁODZIMIERZ Ł ENSKI
As in the proof of Theorem 2.1,
kJ (an,n )k
e
Xp
(
) p1 Z π
p
1
|ϕ (t)|
q
≤ K (an,n ) dt
1+p/q
t
an,n
e
Xp
≤ K (an,n )
1
q
(Z
kϕ (t)kp pe
π
X
t1+p/q
an,n
≤ K (an,n )
1
q
(Z
π
an,n
) p1
dt
) p1
ωf (t)X pe
,
dt
t1+p/q
whence, by (2.6) ,
kJ (an,n )k
e
Xp
= O (an,n H (an,n ))
holds and our result follows.
Proof of Theorem 2.6. It is clear that
q
π
H f (·) ≤ I
n,A
e
n+1 Xp
X
π
+ J
n
+
1
p
e
e
Xp
and immediately
π
I
n+1 e
Xp
π
≤
w
·
f
n + 1 1
≤ ωf
e
Xp
π
n+1
X pe
and
π
J
n+1 e
Xp
≤ K (an,n )
1
q
(Z
π
kϕ (t)kp pe
) p1
X
dt
t1+p/q
(
) p1
Z π
1
ωf
(t)
π
p
e
X
≤ K ((n + 1) an,n ) q
dt
π
n + 1 n+1
t1+p/q
1
π
π
q
≤ K ((n + 1) an,n )
H
n+1
n+1
π
π
≤ K (n + 1) an,n
H
n+1
n+1
π
= O an,n H
.
n+1
π
n+1
Thus our first statement holds. The second one follows on using a similar process to that in the
proof of Theorem 2.1. We have to only use the estimates obtained in the proof of Theorem 2.5,
π
with n+1
instead of an,n ,and the relation
(n + 1)−1 = O (an,n ) .
Proof of Theorem 2.7. As in the proof of Theorem 2.5, we have
q
H f (·) ≤ kI (an,0 )k + kJ (an,0 )k
n,A
e
Xp
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
e
Xp
e
Xp
http://jipam.vu.edu.au/
O N THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS
11
and
kI (an,0 )k
= O (an,0 H (an,0 )) .
e
Xp
Also, from the proof of Theorem 2.3,
kJ (an,0 )k
e
Xp
(
) p1 Z π
p
1
|ϕ· (t)|
q
≤ K (an,0 ) dt
1+p/q
t
an,0
e
Xp
≤ K (an,0 )
1
q
(Z
π
an,0
) p1
ωf (t)X pe
dt
t1+p/q
and, by (2.6) ,
kJ (an,0 )k
= O (an,0 H (an,0 )) .
e
Xp
Thus our result is proved.
Proof of Theorem 2.8. We recall, as in the previous proof, that
q
π
π
H f (·) ≤ I
+ J
n,A
e
n+1
n
+
1
Xp
p
e
e
Xp
X
and
π
I
n+1 ≤ ωf
e
Xp
π
n+1
.
X pe
We apply a similar
method
as that used in the proof of Theorem 2.4 to obtain an estimate for
π
,
the quantity J n+1
e
Xp
π
J
n+1 e
Xp
(
) p1 pq Z π
p
π
|ϕ
(t)|
x
≤ K (n + 1) an,0 dt
1+p/q
π
n
+
1
t
n+1
e
Xp
(
≤ K (n + 1) an,0
(
≤ K (n + 1) an,0
π
n+1
pq Z
π
n+1
pq Z
π
kϕ· (t)k
π
n+1
e
Xp
t1+p/q
π
n+1
π
p
) p1
dt
) p1
ωf (t)X pe
dt
t1+p/q
and, by (2.6) ,
π
J
n+1 e
Xp
π
≤ K (n + 1) an,0
H
n+1
π
≤ Kan,0 H
.
n+1
π
n+1
Thus the proof of the first part of our statement is complete.
To prove of the second part, we follow the line of the proof of Theorem 2.6.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
http://jipam.vu.edu.au/
12
W ŁODZIMIERZ Ł ENSKI
R EFERENCES
[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 functions, Acta Math. Hungar.,
62 (1993), 21–23.
[3] L. LEINDLER, On the degree of approximation of continuous functions, Acta Math. Hungar., 104
(2004), 105–113.
[4] A. ZYGMUND, Trigonometric Series, Cambridge, 2002.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70, 12 pp.
http://jipam.vu.edu.au/