ON THE DEGREE OF STRONG APPROXIMATION
OF INTEGRABLE FUNCTIONS
Strong Approximation of
Integrable Functions
WŁODZIMIERZ ŁENSKI
University of Zielona Góra
Faculty of Mathematics, Computer Science and Econometrics
65-516 Zielona Góra, ul. Szafrana 4a, Poland
EMail: W.Lenski@wmie.uz.zgora.pl
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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Received:
20 December, 2006
Accepted:
25 July, 2007
Communicated by:
L. Leindler
2000 AMS Sub. Class.:
42A24, 41A25.
Key words:
Degree of strong approximation, Special sequences.
Abstract:
We show the strong approximation version of some results of L. Leindler [3]
connected with the theorems of P. Chandra [1, 2].
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Contents
1
Introduction
3
2
Statement of the Results
5
3
Auxiliary Results
9
4
Proofs of the Results
11
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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1.
Introduction
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 = ∞,
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
and consider its trigonometric Fourier series
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∞
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
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and
q
Hn,A
f
Contents
(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
wx f (δ)Lp
Z δ
p1
1
p
:=
|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
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
ωf (δ)X p := sup kϕ · (h)kX p .
vol. 8, iss. 3, art. 70, 2007
0≤|h|≤δ
We can observe that with pe ≥ p, for f ∈ X pe, by the Minkowski inequality
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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.
q
In this note we shall consider the strong means Hn,A
f and the integrable functions.
We shall also give some results 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.
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2.
Statement of the Results
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.
an,m ≥ 0,
n
X
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
Theorem 2.1. Let an = (an,m ) satisfy the following conditions:
(2.1)
Strong Approximation of
Integrable Functions
an,k = 1
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k=0
Contents
and
m−1
X
(2.2)
|an,k − an,k+1 | ≤ Kan,m ,
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k=0
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where
m = 0, 1, . . . , n
and
n = 0, 1, 2, . . .
X−1
k=0
=0 .
Suppose wx is such that
p1
Z π
p
p
(w
(t))
x
(2.3)
uq
dt
= O (uHx (u)) as u → 0+,
p
t1+ q
u
where Hx (u) ≥ 0, 1 < p ≤ q and
Z t
(2.4)
Hx (u) du = O (tHx (t)) as t → 0 + .
0
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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
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
and if, in addition, (2.4) holds then
vol. 8, iss. 3, art. 70, 2007
0
q
Hn,A
f (x) = O an,n Hx
π
n+1
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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 ,
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where
m = 0, 1, . . . , n and n = 0, 1, 2, . . .
hold. If f ∈ Lp (wx ) then
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0
q
f (x) = O (an,0 Hx (an,0 ))
Hn,A
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
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
0
with q ∈ (0, q] and q such that 1 < q (q − 1) ≤ p ≤ q.
vol. 8, iss. 3, art. 70, 2007
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
(2.6)
Z
p
uq
u
π
(ωf (t)X pe )p
p
t1+ q
p1
dt
= O (uH (u))
as
u → 0+
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
with q 0 ≤ q and p ≤ pe such that 1 < q (q − 1) ≤ p ≤ q.
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
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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
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
with q 0 ≤ 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.
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3.
Auxiliary Results
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)
(3.1)
dt = O (uH (u))
(u → 0+) .
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 π
Z u Z π
ω (s)
ω (s)
=u
ds +
ds dt
1+ pq +1− pq
1+ pq +1− pq
u s
0
t s
p1
Z u Z π
Z π
p
p
p
1
ω
(s)
(ω
(s))
ds +
tq
ds dt,
≤ uq
1+p/q
s1+p/q
0 t
u s
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
and thus the proof is completed.
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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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
(
(3.2)
) 1q
Z π
p1
∞
−ξ
p
|ao (g)|q X
q
q
+
(|ak (g)| + |bk (g)| )
≤K
t g (t) dt .
2
−π
k=0
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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4.
Proofs of the Results
q
Since Hn,A
f is the monotonic function of q we shall consider, in all our proofs, the
q
q0
quantity Hn,A f instead of Hn,A
f.
Proof of Theorem 2.1. Let
Z
( n
q ) 1q
1 π
1
X
sin
k
+
t q
2
Hn,A
f (x) =
an,k ϕx (t)
dt
1
π 0
2
sin
t
2
k=0
Z
( n
q ) 1q
1 an,n
1
X
sin
k
+
t 2
≤
an,k dt
ϕx (t)
π 0
2 sin 12 t
k=0
( n
q ) 1q
1 Z π
1
X
sin
k
+
t 2
dt
+
an,k ϕx (t)
π an,n
2 sin 12 t
k=0
= I (an,n ) + J (an,n )
and, by (2.1) , integrating by parts, we obtain,
Z an,n
|ϕx (t)|
I (an,n ) ≤
dt
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
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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π
Z
=K
an,n
an,n
Z
≤K
π
an,n
an,n
ap/q
n,n
≤K
Z
π
an,n
Z
wx f (t)1
dt +
t2
an,n
0
!
wx f (t)1
dt
t
! p1
(wx f (t)L1 )p
dt
t2
Z
an,n
wx f (t)L1
dt
t
an,n
wx f (t)Lp
dt .
t
+K
0
! p1
(wx f (t)Lp )p
dt
t1+p/q
Z
+K
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 )) .
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
Title Page
The Abel transformation shows that
q
k Z π
X
sin ν + 12 t 1
q
dt
(J (an,n )) =
(an,k − an,k+1 )
ϕ (t)
π an,n x
2 sin 12 t
ν=0
k=0
Z
q
n π
X
sin ν + 12 t 1
+ an,n
ϕx (t)
dt ,
π an,n
2 sin 12 t
ν=0
n−1
X
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whence, by (2.2),
q
n Z π
1
X
sin
ν
+
t 1
2
(J (an,n ))q ≤ (K + 1) an,n
ϕx (t)
dt .
1
π an,n
2
sin
t
2
ν=0
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
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Integrating by parts, we have
J (an,n ) ≤ K (an,n )
1
q
(
1
p
p
+ 1+
q
(
≤ K (an,n )
(wx f (t)Lp )
tp/q
1
q
π
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),
1
J (an,n ) ≤ K (an,n ) q
Z
p
Z
π
(an,n ) q
an,n
p
π
) p1
(wx (t))
dt
1+p/q
an,n t
) p1
p
(wx (t))
dt
t1+p/q
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and
I
J
I
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Proof of Theorem 2.2. Let, as before,
π
π
q
Hn,A f (x) ≤ I
+J
n+1
n+1
n+1
≤
π
II
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Thus our result is proved.
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= O (an,n Hx (an,n )) .
π
n+1
vol. 8, iss. 3, art. 70, 2007
(wx (π)) +
(
Włodzimierz Łenski
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(
≤K
Strong Approximation of
Integrable Functions
Z
0
π
n+1
|ϕx (t)| dt = wx
π
n+1
Close
.
L1
π
In the estimate of J n+1
we again use the Abel transformation and (2.2). Thus
q
q
n Z π
1
X
t sin
ν
+
π
1
2
J
≤ (K + 1) an,n
ϕx (t)
dt
1
π
n+1
π
2
sin
t
2
ν=0
n+1
and, by inequality (3.2),
J
π
n+1
≤ K (an,n )
1
q
(Z
π
π
n+1
) p1
|ϕx (t)|p
dt
.
t1+p/q
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
Integrating (2.3) by parts, with the assumption f ∈ Lp (wx ) , we obtain
) p1
(
pq Z π
p
1
(w
(t))
π
π
x
dt
J
≤ K ((n + 1) an,n ) q
1+p/q
π
t
n+1
n+1
n+1 1
π
π
= O ((n + 1) an,n ) q
Hx
n+1
n+1
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
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whence
an,j ≤ (K + 1) an,n
and therefore
(K + 1) (n + 1) an,n ≥
n
X
j=0
an,j = 1.
This inequality implies that
π
π
J
= O an,n Hx
n+1
n+1
and the proof of the first part of our statement is complete.
π
To prove of the second part of our assertion we have to estimate the term I n+1
once more.
π
Proceeding analogously to the proof of Theorem 2.1, with an,n replaced by n+1
,
we obtain
π
π
π
I
=O
Hx
.
n+1
n+1
n+1
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 )) .
By the Abel transformation
q
k Z π
X
sin ν + 12 t 1
q
(J (an,0 )) ≤
|an,k − an,k+1 |
dt
ϕ (t)
π an,0 x
2 sin 12 t
ν=0
k=0
q
n Z π
X
sin ν + 12 t 1
ϕ (t)
+ an,n
dt
π an,0 x
2 sin 12 t
ν=0
n−1
X
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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≤
∞
X
q
∞ Z π
1
X
sin
ν
+
t 1
2
ϕx (t)
dt .
1
π an,0
2
sin
t
2
ν=0
!
|an,k − an,k+1 | + an,n
k=0
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
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
|an,k − an,k+1 | + an,n ≤ (2K + 1) an,0
k=0
and
Strong Approximation of
Integrable Functions
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q
∞ Z π
1
X
sin
ν
+
t 1
2
dt .
(J (an,0 ))q ≤ (2K + 1) an,0
ϕx (t)
1
π an,0
2 sin 2 t
ν=0
Finally, by (3.2),
J (an,0 ) ≤ K (an,0 )
1
q
(Z
π
an,0
) p1
|ϕx (t)|p
dt
t1+p/q
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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
Close
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
k Z π
n−1
X
sin ν + 12 t π
1
J
|an,k − an,k+1 |
ϕ (t)
dt
≤
π π x
n+1
2 sin 12 t
ν=0
n+1
k=0
Z
q
n π
X
sin ν + 12 t 1
ϕx (t)
dt
+ an,n
π π
2 sin 12 t
ν=0
n+1
Z
!
q
∞
∞ π
X
X
sin ν + 12 t 1
dt .
ϕx (t)
≤
|an,k − an,k+1 | + an,n
π π
2 sin 12 t
ν=0
n+1
k=0
From the assumption (2.5), arguing as before, we can see that
q ! 1q
∞ Z π
X
sin ν + 12 t π
1
J
≤ K an,0
ϕx (t)
dt
1
π
n+1
π
2
sin
t
2
ν=0
n+1
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Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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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
k=0
an,k = 1,
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whence
J
π
n+1
(
pq Z
p
π
) p1
|ϕx (t)|
dt
π
t1+p/q
n+1
(
) p1
pq Z π
p
π
|ϕx (t)|
≤ K (n + 1) an,0
dt
.
π
n+1
t1+p/q
n+1
≤ K ((n + 1) an,0 )
1
q
π
n+1
.
Since f ∈ Lp (wx ), integrating by parts we obtain
π
π
π
J
≤ K (n + 1) an,0
Hx
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
=O
Hx
.
I
n+1
n+1
n+1
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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From the start of our proof we have (n + 1)
sertion also follows.
−1
= O (an,0 ) , whence the second as-
Proof of Theorem 2.5. We begin with the inequality
q
H f (·) ≤ kI (an,n )k + kJ (an,n )k
n,A
p
e
e
Xp
X
e
Xp
.
Close
By (2.1) , Lemma 3.1 gives
kI (an,n )k
e
Xp
kϕ (t)k
an,n
Z
≤
e
Xp
dt
2t
Z an,n
ωf (t)X pe
≤
dt
2t
0
= O (an,n H (an,n )) .
0
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
As in the proof of Theorem 2.1,
kJ (an,n )k
e
Xp
vol. 8, iss. 3, art. 70, 2007
(
) p1 Z π
p
1 |ϕ
(t)|
≤ K (an,n ) q dt
1+p/q
an,n t
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e
Xp
1
(Z
kϕ (t)k
π
≤ K (an,n ) q
an,n
1
(Z
e
Xp
t1+p/q
dt
) p1
π
≤ K (an,n ) q
an,n
ωf (t)X pe
dt
t1+p/q
e
Xp
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,
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whence, by (2.6) ,
kJ (an,n )k
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) p1
p
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= O (an,n H (an,n ))
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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 ≤ K (an,n )
1
q
(Z
kϕ (t)kp pe
π
X
π
n+1
e
Xp
t1+p/q
) p1
Strong Approximation of
Integrable Functions
dt
Włodzimierz Łenski
(
Z
) p1
π
ωf (t)X pe
dt
1+p/q
π
t
n+1 1
π
π
≤ K ((n + 1) an,n ) q
H
n+1
n+1
π
π
≤ K (n + 1) an,n
H
n+1
n+1
π
.
= O an,n H
n+1
1
≤ K ((n + 1) an,n ) q
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
π
proof of Theorem 2.5, with n+1
instead of an,n ,and the relation
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
e
Xp
e
Xp
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(n + 1)−1 = O (an,n ) .
n,A
vol. 8, iss. 3, art. 70, 2007
e
Xp
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)|
·
≤ K (an,0 ) q dt
1+p/q
an,0 t
Strong Approximation of
Integrable Functions
e
Xp
≤ K (an,0 )
1
q
(Z
π
an,0
) p1
ωf (t)X pe
dt
t1+p/q
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
and, by (2.6) ,
kJ (an,0 )k
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= O (an,0 H (an,0 )) .
e
Xp
Contents
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
and
π
I
n+1 e
Xp
≤ ωf
π
n+1
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Page 21 of 23
e
Xp
X
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.
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X pe
We apply a similar method
as πthat
used in the proof of Theorem 2.4 to obtain an
estimate for the quantity J n+1 ,
e
Xp
(
) p1 pq Z π
p
π
π
|ϕx (t)|
J
≤ K (n + 1) an,0 dt
1+p/q
π
n + 1 pe
n
+
1
t
n+1
X
e
Xp
Close
(
≤ K (n + 1) an,0
(
≤ K (n + 1) an,0
π
n+1
pq Z
π
n+1
pq Z
π
kϕ· (t)kp pe
X
t1+p/q
π
n+1
π
π
n+1
) p1
dt
) p1
ωf (t)X pe
dt
t1+p/q
Strong Approximation of
Integrable Functions
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
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
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Contents
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.
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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 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.
Strong Approximation of
Integrable Functions
Włodzimierz Łenski
vol. 8, iss. 3, art. 70, 2007
[4] A. ZYGMUND, Trigonometric Series, Cambridge, 2002.
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