Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 485705, 19 pages
doi:10.1155/2010/485705
Research Article
On Weighted Lp Integrability of Functions Defined
by Trigonometric Series
Bogdan Szal
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4a,
65-516 Zielona Góra, Poland
Correspondence should be addressed to Bogdan Szal, b.szal@wmie.uz.zgora.pl
Received 26 January 2010; Accepted 13 April 2010
Academic Editor: László Losonczi
Copyright q 2010 Bogdan Szal. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
r
We introduce a new class of sequences called GMθ and give a sufficient and necessary condition
for weighted Lp integrability of trigonometric series with coefficients to belong to the above class.
This is a generalization of the result proved by M. Dyachenko and S. Tikhonov 2009. Then we
discuss the relations among the weighted best approximation and the coefficients of trigonometric
r
series. Moreover, we extend the results of B. Wei and D. Yu 2009 to the class GMθ .
1. Introduction
Let Lp , 1 ≤ p < ∞, be the space of all p-power integrable functions f of period 2π equipped
with the norm
f p L
π
−π
fxp dx
1/p
.
1.1
Write
fx ∞
k1
ak cos kx,
gx ∞
bk sin kx
1.2
k1
for those x s where the series converge. Denote by φ either f or g, and let λn be its associated
coefficients, that is, λn is either an or bn .
2
Journal of Inequalities and Applications
For r ∈ N and a sequence ck , let
Δr ck ck − ckr .
1.3
In Subsection 2.1 we generalize the following result.
Theorem 1.1. Let a nonnegative sequence λn ∈ R, 1 < p < ∞ and 1 − p < α < 1. Then
∞
p
p
x−α φx ∈ L1 ⇐⇒
nαp−2 λn < ∞.
1.4
n1
In the case when R denotes the class M of all decreasing sequences, this theorem was proved
in 1–4 ; for R ≡ QM, the class of quasimonotone sequences, in 5 ; for R ≡ GMβ in 6, 7 ;
for R ≡ GMβ in 8 ; and for R ≡ GMβ∗ in 9 , where
GM β :
cn :
∞
|Δ1 ck | ≤ Cβn ,
kn
GM β :
cn :
2n
|Δ1 ck | ≤ Cβn ,
1.5
kn
βn∗ βn |cn |,
cn
kn/c
|ck |
k
for some c > 1.
Note that see 6, 8, 10–14 M QM ∪ GM β GM β GM β∗ .
1.6
In 15 Dyachenko and Tikhonov extended Theorem 1.1 to the class GMθ :≡ GMβ# , where
θ ∈ 0, 1 and
βn# nθ−1
∞
|ck |
< ∞ for some c > 1.
θ
kn/c k
1.7
We have see 15 GM β∗ GM1 ⊆ GMθ2 ⊆ GMθ1
for 0 < θ1 ≤ θ2 ≤ 1.
1.8
Let γ be a nonnegative function defined on the interval 0, π . Denote by En ϕ, γp the best
approximation of ϕ by trigonometric polynomials of degree at most n in the weighted Lp norm, that is,
En ϕ, γ
π
p
: inf
Pn ∈Πn
p
γxϕx − Pn x dx
1/p
,
0
where Πn denotes the set of all trigonometric polynomials of degree at most n.
1.9
Journal of Inequalities and Applications
3
A sequence cn of nonnegative terms is called almost increasing decreasing if there
exists a constant C > 0 such that
C · cn ≥ c m
cn ≤ C · cm for n ≥ m.
1.10
We say that a weight function γ ∈ Φα, β, α, β be fixed constants, if γ is defined by the
sequence γn as follows: γπ/n : γn , n ∈ N, and there exist positive constants A and B such
that
Aγn ≤ γx ≤ Bγn1
1.11
for all x ∈ π/n1, π/n , and the sequences γn nα , γn nβ are almost decreasing and almost
increasing, respectively.
In Subsection 2.2 we generalize and extend the following results 16 .
Theorem 1.2. Assume that bn ∈ GMβ∗ . If γ ∈ Φ−p − 1 α, p − 1 β for some α, β > 0 and
∞
p−2 p
λn < ∞, then for 1 ≤ p < ∞
k1 γn n
⎛
En g, γ
p
≤
1/p
C⎝γn1 n1−1/p
cn1
∞
p
γk kp−2 λk
kn1
|Δ1 λk | kn1
1/p ⎞
⎠.
1.12
Theorem 1.3. Assume that an ∈ GMβ∗ . If γ ∈ Φ−1 α, p − 1 β for some α, β > 0 and
∞
p−2 p
λn < ∞, then for 1 ≤ p < ∞
k1 γn n
⎛
En f, γ
p
≤
1/p
C⎝γn1 n1−1/p
cn1
|Δ1 λk | kn1
∞
p
γk kp−2 λk
kn1
1/p ⎞
⎠.
1.13
If γ ≡ 1 and λn ∈ M or λn ∈ GMβ the above theorem has been obtained by
Konyushkov 17 and Leindler 18 for p > 1, respectively.
In order to formulate our new results we define the next class of sequences.
r
Definition 1.4. Let r ∈ N and θ ∈ 0, 1 . One says that a sequence cn belongs to GMθ , if the
relation
∞
∞
|ck |
<∞
|Δr ck | ≤ Cnθ−1
θ
kn
kn/c k
1.14
holds for all n ∈ N.
Note that for r ≥ 2 and θ ∈ 0, 1 see Theorem 2.1i
1
r
GMθ ≡ GMθ GMθ .
1.15
4
Journal of Inequalities and Applications
Throughout this paper, we use C to denote a positive constant independent of the integer n;
C may depend on the parameters such as p, α, r, θ and λ, and it may have different values in
different occurrences.
2. Statement of the Results
We formulate our results as follows.
Theorem 2.1. Suppose that θ ∈ 0, 1 . The following properties are true.
r
i For any r ≥ 2, and θ ∈ 0, 1 there exists a sequence cn ∈ GMθ , which does not belong
1
to the class GMθ ≡ GMθ .
r1
r2
ii Let r1 , r2 ∈ N, r1 ≤ r2 and θ ∈ 0, 1 . If r1 | r2 , then GMθ ⊆ GMθ .
r1
r2
iii Let r1 , r2 ∈ N and θ ∈ 0, 1 . If r1 r2 and r2 r1 , then the classes GMθ and GMθ are not
comparable.
2.1. Weighted Lp -Integrability
Let r ∈ N and α ∈ R. We define on the interval −π, π an even function ωα,r , which is given
on the interval 0, π by the formula
⎧
2lπ −α
⎪
⎪
x−
⎪
⎪
⎪
r
⎪
⎪
−α
⎨
2l 1π
ωα,r x :
−x
⎪
r
⎪
⎪
⎪
⎪
⎪
⎪
⎩0
2lπ 2l 1π
,
for x ∈
and l ∈ U1,
r
r
2l 1π 2l 1π
,
and l ∈ U2 ,
for x ∈
r
r
2lπ
and l ∈ U3 ,
for x r
2.1
where U1 {0, 1, . . . , r/2 } if r is an odd number, and U1 {0, 1, . . . , r/2 − 1} if r is
an even number; U2 {0, 1, . . . , r/2 − 1} for r ≥ 2, and U3 {0, 1, . . . , r/2 } for r ≥ 1.
r
Theorem 2.2. Let a nonnegative sequence λn ∈ GMθ , where r ∈ N, θ ∈ 0, 1 and 1 ≤ p < ∞. If
1 − θp < α < 1,
2.2
∞
p
nαp−2 λn < ∞.
2.3
then ωα,r |φ|p ∈ L1 if and only if
n1
r
Theorem 2.3. Let a nonnegative sequence bn ∈ GMθ r 1, 2, θ ∈ 0, 1 , and 1 ≤ p < ∞. If
1 − θp < α < p 1,
2.4
Journal of Inequalities and Applications
5
then ωα,r |g|p ∈ L1 if and only if
∞
p
nαp−2 bn < ∞.
2.5
n1
Remark 2.4. If we take r 1 and λn an in Theorem 2.2, then the result of Dyachenko and
Tikhonov 15, Theorems 4.2 and 4.3 follows from Theorems 2.2 and 2.3. By the embedding
relations 1.8 and 1.15 we can also derive from Theorem 2.2 the result of You, Zhou, and
Zhou 9 .
2.2. Relations between The Best Approximation and Fourier Coefficients
r
Theorem 2.5. Let a nonnegative sequence λn ∈ GMθ , where r ∈ N, θ ∈ 0, 1 , and 1 ≤ p < ∞. If
1 − θp < α < 1
2.6
and 2.3 holds, then
⎛
En φ, ωα,r ≤ C⎝nα/p1−1/p
cn1
|Δr λk | kn1
∞
p
kαp−2 λk
kn1
1/p ⎞
⎠,
2.7
where c > 1.
r
Theorem 2.6. Let a nonnegative sequence bn ∈ GMθ r 1, 2, θ ∈ 0, 1 and 1 ≤ p < ∞. If
1 − θp < α < p 1,
and 2.5 holds, then
⎛
En g, ωα,r
≤ C ⎝n
α/p1−1/p
p
cn1
2.8
|Δr bk | kn1
∞
p
kαp−2 bk
kn1
1/p ⎞
⎠,
2.9
where c > 1.
Remark 2.7. If we restrict our attention to the class GMβ∗ , then by 1.8 and 1.15 Wei and
Yu’s result 16 follows from Theorems 2.5 and 2.6.
3. Auxiliary Results
Denote, for r ∈ N,
Dk,r x sin k r/2x
,
2 sin rx/2
k,r x cosk r/2x .
D
2 sin rx/2
3.1
6
Journal of Inequalities and Applications
2lπ/r, then for all m ≥ n
Lemma 3.1 see 19 . Let r ∈ N, l ∈ Z, and cn ∈ C. If x /
m
m
mr
nr−1
ck cos kx Δr ck Dk,r x −
ck Dk,−r x ck Dk,−r x,
kn
kn
km1
kn
m
mr
nr−1
m
k,−r x −
k,−r x − Δr ck D
k,r x.
ck sin kx ck D
ck D
kn
km1
kn
3.2
kn
Lemma 3.2 see 20 . Let p ≥ 1, γn > 0 and an ≥ 0, then
∞
γn
n
p
≤p
αk
p
∞
1−p p
γn αn
p
∞
γk ,
n1
k1
n1
kn
n1
kn
n1
k1
p
p
∞
∞
n
∞
1−p p p
γn
αk
≤p
γn αn
γk .
3.3
4. Proofs of The Main Results
4.1. Proof of Theorem 2.1
i Let r ≥ 2, θ ∈ 0, 1 and
⎧
⎪
⎪
⎨0 if r | n,
cn : 1
⎪
⎪
⎩ n if r n.
4.1
r
First, we prove that cn ∈ GMθ . Let
Ar, k, n : {k : n ≤ k and r | k} ,
Br, k, n : {k : n ≤ k and r k}.
4.2
Then for all n
∞
|Δr ck | kn
∞
r
1
≤r
2
kk
r
k
k∈Br,k,n
k∈Br,k,n
≤ rn
θ−1
∞
1
k∈Br,k,n k
1θ
≤ rn
θ−1
∞
ck
θ
kn k
4.3
r
and cn ∈ GMθ . If r ≥ 2 then
∞
|Δ1 ck | ≥
kn
∞
k∈Ar,k,n
|Δ1 ck | ∞
1
≥ C lnn 1
k
1
k∈Ar,k,n
4.4
Journal of Inequalities and Applications
7
and since
nθ−1
∞
∞
∞
ck
1
1
1
θ−1
θ−1
n
≤
n
≤C
θ
1θ
1θ
n
kn/c k
k∈Br,k,n/c k
kn/c k
4.5
the inequality
∞
∞
ck
|Δ1 ck | ≤ Cnθ−1
θ
k
kn
kn/c
4.6
1
does not hold, that is, cn does not belong to GMθ .
ii Let r1 , r2 ∈ N, r1 ≤ r2 and θ ∈ 0, 1 . If r1 | r2 , then exists a natural number p such
r1
that r2 p · r1 . Supposing that cn ∈ GMθ , we have for all n
p−1
p−1
∞ ∞
|Δr2 ck | |Δ c |
Δr1 ckl·r1 ≤
l0 knl·r r1 k
kn
kn l0
∞
1
∞
∞
ck
r2 ,
≤
|Δr1 ck | ≤ Cnθ−1
θ
r1 kn
k
kn/c
r2
r1
4.7
r2
whence cn ∈ GMθ . Thus GMθ ⊆ GMθ .
iii Let r1 , r2 ∈ N and θ ∈ 0, 1 and let
⎧
⎪
⎪
⎨0 if r1 | n,
1
cn : 1
⎪
⎪
⎩ n if r1 n,
⎧
⎪
⎪
⎨0 if r2 | n,
2
cn : 1
⎪
⎪
⎩ n if r2 n.
4.8
r1
Supposing that r1 r2 and r2 r1 , we can prove, similarly as in i, that cn1 ∈ GMθ ,
r2
r2
r1
r1
r2
c1 /
∈ GMθ , c2 ∈ GMθ and c2 /
∈ GMθ . Therefore the classes GMθ and GMθ are not
comparable.
4.2. Proof of Theorem 2.2
We prove the theorem for the case when φx gx. The case when φx fx can be
proved similarly.
Sufficiency. Suppose that 2.3 holds. Then
ωα,r g p L1
2
π
0
p
ωα,r xgx dx.
4.9
8
Journal of Inequalities and Applications
It is clear that for an odd r
π
2lπ/rπ/r
r/2
p
ωα,r xgx dx 0
2lπ/r
l0
p
∞
ωα,r x bk sin kx dx
k1
p
∞
ωα,r x bk sin kx dx
2lπ/rπ/r
k1
r/2
−1 2l1π/r
l0
4.10
for r 1 the last sum should be omitted, and for an even r
π
r/2
p
ωα,r xgx dx 0
2lπ/rπ/r
2lπ/r
l0
p
∞
ωα,r x bk sin kx dx.
k1
2l1π/r 2lπ/rπ/r
4.11
First, we estimate the following integral:
2lπ/rπ/r
2lπ/r
≤C
p
∞
ωα,r x bk sin kx dx
k1
2lπ/rπ/r
2lπ/r
p
p 2lπ/rπ/r
n
∞
ωα,r x bk sin kx dx ωα,r x
bk sin kx dx
k1
2lπ/r
kn1
: CI1 I2 .
4.12
By 3.3, for α < 1, we have
I1 ∞ 2lπ/rπ/n
2lπ/r π/n1
nr
≤C
∞
nr
2lπ
x−
r
p
−α n
bk sin kx dx
k1
p
p
n
∞
n
∞
p
α−2
α−2
n
bk
≤C n
bk
≤ C nαp−2 bn .
k1
n1
k1
4.13
n1
Using 3.2 with m → ∞ and the inequality
r
rx 2lπ 2lπ π
x − 2l ≤ sin for x ∈
,
,
,
π
2
r
r
r
4.14
Journal of Inequalities and Applications
9
we get
I2 ∞ 2lπ/rπ/n
nr
2lπ/rπ/n1
∞ 2lπ/rπ/n
nr
2lπ/rπ/n1
2lπ
x−
r
2lπ
x−
r
2lπ/rπ/n
∞
≤ C nα
2lπ/rπ/n1
nr
2lπ/rπ/n
∞
α
≤C n
2lπ/rπ/n1
nr
p
−α ∞
nr
k,r x k,−r x dx
Δr bk D
bk D
kn1
kn1
1
2|sin rx/2|p
∞
|Δr bk | kn1
1
r/πx − 2lp
p
∞
∞
αp−2
≤C n
|Δr bk |
nr
p
−α ∞
bk sin kx dx
kn1
∞
nr
p
bk
dx
4.15
kn1
p
|Δr bk |
dx
kn1
.
kn
r
If bn ∈ GMθ , then by 3.3, for 1 − θp < α < p 1, we obtain
⎛
⎞p
∞
b
k⎠
I2 ≤ C nαθp−2 ⎝
θ
k
nr
kn/c
∞
⎞p
p ⎞
n
∞
∞
b
b
k⎠
k
⎠
≤ C⎝ nαθp−2 ⎝
nαθp−2
θ
θ
k
k
nr
nr
kn/c
kn
⎛
≤C
≤C
⎛
∞
∞
n
n1
k1
nα−p−2
∞
p
kbk
∞
nαθp−2
n1
p ∞
bk
kn k
4.16
θ
p
nαp−2 bn .
n1
Now, we estimate the following integral:
p
∞
ωα,r x bk sin kx dx
2lπ/rπ/r
k1
2l1π/r
≤C
p
p 2l1π/r
n
∞
ωα,r x bk sin kx dx ωα,r x
bk sin kx dx
2l1π/r−π/r
2l1π/r−π/r
k1
kn1
2l1π/r
: CI3 I4 .
4.17
10
Journal of Inequalities and Applications
By 3.3, for α < 1, we have
∞ 2l1π/r−π/n1 2l 1π
I3 nr
≤C
∞
r
2l1π/r−π/n
p
−α n
−x
bk sin kx dx
k1
p
n
∞
p
α−2
n
bk
≤ C nαp−2 bn .
n1
4.18
n1
k1
Using 3.2 with m → ∞ and the inequality
2l 1 −
r
rx x ≤ sin π
2
for x ∈
2l 1π 2l 1π
,
,
r
r
4.19
we obtain
I4 ∞ 2l1π/r−π/n1 2l 1π
2l1π/r−π/n
nr
r
p
−α ∞
−x
bk sin kx dx
kn1
p
2l1π/r−π/n1 ∞
nr
k,r x k,−r x dx
≤C n
Δr bk D
bk D
kn1
2l1π/r−π/n
nr
kn1
∞
≤C
α
∞
α
2l1π/r−π/n1
n
2l1π/r−π/n
nr
≤C
∞
α
2l1π/r−π/n1
n
2l1π/r−π/n
nr
≤C
∞
αp−2
n
nr
1
rx p
2sin 2
∞
|Δr bk | kn1
1
2l 1 − r/πxp
p
∞
|Δr bk | .
nr
p
bk
dx
kn1
∞
4.20
p
|Δr bk |
dx
kn1
kn
r
If bn ∈ GMθ , then by 3.3, for 1 − θp < α < p 1, we obtain
⎛
⎞p
∞
∞
b
k
⎠
I4 ≤ C nαθp−2 ⎝
θ
nr
kn/c k
p
p ∞
n
∞
∞
bk
α−p−2
αθp−2
≤C
n
kbk
n
θ
n1
n1
k1
kn k
≤C
∞
p
nαp−2 bn .
n1
4.21
Journal of Inequalities and Applications
11
Thus, combining 4.9, 4.12–4.13, 4.16–4.18, 4.21, and 4.10 or 4.11, we
obtain that
π
−π
∞
p
p
ωα,r xgx dx ≤ C nαp−2 bn .
4.22
n1
Necessity. We follow the method adopted by S. Tikhonov 15 . Note that if 1 − p < α, then
g ∈ L1 . Integrating g, we have
x
Fx :
gtdt 0
∞
bn
n1
∞
bn 2 nx
sin
1 − cos nx 2
n
n
2
n1
4.23
and consequently
F
k
π
bn
≥
.
k
n
nk/2
4.24
r
If bn ∈ GMθ , then using 4.24,
bv ≤
vr−1
∞
lv
lv
bl Δr bl ≤
∞
|Δr bl | ≤ Cvθ−1
lv
∞ 2 v/c
−1 bl
s1
Cv
θ−1
s0 l2s v/c
≤ Cvθ−1
lθ
≤ Cv
θ−1
∞
∞
bl
.
θ
lv/c l
2 v/c s
s0
π
s1
2 v/c
2s1
v/c
l2s v/c
s0
∞
2s v/c 1−θ F
1−θ
bl
l
≤ Cv
θ−1
2 v/c
∞
−1 π F
2s v/c −θ
l
s0
l2s v/c
≤ Cv
θ−1
∞ 2s1 v/c
∞
−1 1 π 1 π θ−1
≤
Cv
.
F
F
θ
θ
l
l
s0 l2s v/c l
lv/c l
s1
4.25
12
Journal of Inequalities and Applications
Using this and 3.3, for 1 − θp < α < p 1, we obtain
⎛
∞
∞
∞
αp−2 p
αp−2θ−1p ⎝
k
bk ≤ C k
k1
k1
vk/c
⎞p
1 π ⎠
F
v
vθ
⎛
∞
k
≤ C⎝ kαp−2θ−1p ⎝
⎛
k1
vk/c
⎞p
1 π ⎠
F
v
vθ
⎞
∞
∞
π p
1
⎠
kαp−2θ−1p
F
θ
v
k1
vk v
4.26
∞
k
π p
α−2−p
≤C
k
vF
v
v1
k1
p ∞
∞
1 π αθp−2
k
F
θ
v
k1
vk v
p
∞
π
αp−2
≤C k
.
F
k
k1
Defining dv :
π/v
π/v1
|gx|dx we get
p
∞
∞
∞
αp−2 p
αp−2
k
bk ≤ C k
dv
k1
k1
4.27
vk
and by 3.3, for α > 1 − θp ≥ 1 − p, we obtain
∞
p
kαp−2 bk ≤
k1
∞
p
kα2p−2 dk .
4.28
k1
Applying Hölder’s inequality, for p > 1, we have
p
dk
π/k
π/k1
gxdx
p
≤C
1
k2p−1
π/k
π/k1
gxp dx.
4.29
Journal of Inequalities and Applications
13
Finally,
∞
p
kαp−2 bk
≤C
k1
r
p
kα2p−2 dk
k1
≤C
r
p
kα2p−2 dk
kr
k
α2p−2
π/k
gxdx
π
gxdx
p
0
≤C
p
π/k1
k1
≤C
∞
π
kr
∞ π/k
p
0
π
gxp dx
π/k1
p
x−α gx dx
4.30
π/k1
kr
gxdx
π/k
∞
α
k
p
ωα,r x gx dx ,
0
which completes the proof.
4.3. Proof of Theorem 2.3
The proof of Theorem 2.3 goes analogously as the proof of Theorem 2.2. The only difference
is that instead of 4.13 for r 1, 2 and 4.18 for r 2 we use the below estimations.
Applying the inequalities | sin kx| ≤ kx for x ∈ 0, π, | sin kx| ≤ kπ − x for x ∈ 0, π
and using 3.3, for α < 1 p, we have
I1 p
p
π/n n
∞
n
2lπ −α α
x kbk dx
x−
bk sin kx dx ≤ C n
k1
r
π/n1
π/n1
nr
k1
∞ π/n
nr
p
∞
n
∞
p
α−p−2
kbk
≤ C nαp−2 bn
≤C n
n1
I3 ∞ π−π/n1
n2
n1
k1
π−π/n
p
p
π−π/n1 n
∞
n
α
π − x bk sin kx dx ≤ C n
π − x kbk dx
k1
π−π/n
n2
k1
−α p
∞
n
∞
p
α−p−2
kbk
≤ C nαp−2 bn .
≤C n
n1
k1
n1
4.31
This ends our proof.
4.4. Proof of Theorem 2.5
We prove the theorem for the case when φx gx. The case when φx fx can be
proved similarly.
14
Journal of Inequalities and Applications
If n ≤ r then by 2.3 we obtain that 2.7 obviously holds. Let n ≥ r. It is clear that if r
is an odd number, then
En g, ωα,r
p
π
≤
p
ωα,r xgx − Sn g, x dx
1/p
0
r/2
2lπ/rπ/r
l0
2lπ/r
p
ωα,r xgx − Sn g, x dx
r/2
−1 2l1π/r
p
ωα,r xgx − Sn g, x dx
4.32
1/p
2lπ/rπ/r
l0
for r 1 the last sum should be omitted, and if r is an even number, then
En g, ωα , r
p
r/2
≤
2lπ/rπ/r
l0
2lπ/r
2l1π/r p
ωα,r xgx − Sn g, x dx
1/p
.
4.33
2lπ/rπ/r
Let
π
2lπ π
2lπ
<x≤
,
r
m1
r
r
4.34
where m : mx ≥ r and l 0, 1, . . . , r/2 − 1 if r is an even number, and l 0, 1, . . . , r/2 if
r is an odd number.
Then, for n ≥ m, by 3.2 and 4.14, we get
2lπ/rπ/r
p
ωα,r xgx − Sn g, x dx
2lπ/rπ/n1
n 2lπ/rπ/m
p
2lπ −α gx − Sn g, x dx
x−
r
mr 2lπ/rπ/m1
p
2lπ/rπ/m
n
∞
nr
α
k,r x k,−r x dx
≤C m
Δr bk D
bk D
2lπ/rπ/m1 kn1
mr
kn1
p
2lπ/rπ/m
n
∞
nr
1
b
bk dx
≤ C mα
|Δ
|
r
k
p
rx 2lπ/rπ/m1 2
mr
kn1
sin kn1
2
p
2lπ/rπ/m
n
∞
1
α
≤C m
|Δr bk | dx
p
2lπ/rπ/m1 r/πx − 2l
mr
kn1
⎛
p
p
cn1
n
∞
n
αp−2
αp−2
≤C m
|Δr bk | dx ≤ C⎝ m
|Δr bk | dx
mr
n
kn1
⎛
mαp−2 ⎝
mr
: CΣ1 Σ2 .
∞
kcn1
⎞p
mr
⎞
|Δr bk |⎠ dx⎠
kn1
4.35
Journal of Inequalities and Applications
15
We immediately have for α > 1 − θp ≥ 1 − p
Σ1 ≤ Cn
αp−1
cn1
p
|Δr bk |
4.36
.
kn1
r
If bn ∈ GMθ and p > 1, then by Hölder’s inequality we have for α > 1 − θp
Σ2 ≤ C
n
m
αp−2
θ−1
n
mr
≤
p
∞
bk
θ
k
kn1
∞
p
Cnαθp−1
kαp−2 bk
kn1
∞
bk
θ
k
kn1
≤ Cn
αθp−1
∞
k−p−θp−α2/p−1
p−1
kn1
p
4.37
≤
∞
p
Ċ
kαp−2 bk .
kn1
r
When bn ∈ GMθ and p 1, an elementary calculation gives for α > 1 − θ
Σ2 ≤ C
n
mα−1 nθ−1
mr
∞
bk
θ
kn1 k
≤ Cnαθp−1
∞
∞
bk
≤
Ċ
kα−1 bk .
θ
k
kn1
kn1
4.38
If m ≥ n 1 ≥ r 1, then
2lπ/rπ/n1
p
ωα,r xgx − Sn g, x dx
2lπ/r
∞ 2lπ/rπ/m
≤C
2lπ
r
x−
2lπ/rπ/m1
mn1
∞
mα
2lπ/rπ/m1 2lπ/rπ/m
mn1
−α
m
gx − Sn g, x p dx
p
bk sin kx dx
kn1
p
cm1
∞
α 2lπ/rπ/m
m 2lπ/rπ/m1 bk sin kx dx
km1
mn1
∞
mα
mn1
4.39
p ⎞
∞
dx⎠
b
sin
kx
k
2lπ/rπ/m1 kcm1 1
2lπ/rπ/m
: CΣ3 Σ4 Σ5 .
We have
Σ3 ≤ C
∞
m
mn1
α−2
m
p
bk
kn1
,
4.40
16
Journal of Inequalities and Applications
and taking γm mα−2 and αk 0 for k < n 1, αk bk for k ≥ n 1 in 3.3, we get for α < 1
Σ3 ≤
∞
p
C
mα−21−p bm
mn1
∞
p
k
∞
≤C
α−2
p
4.41
mαp−2 bm .
mn1
km
r
If bn ∈ GMθ , then using 3.2 and 4.14, we have
⎛
∞
⎜
Σ4 Σ5 ≤ C⎜
mα−2
⎝
cm1
mn1
⎛
p
bk
∞
mα
mn1
km1
⎛
2lπ/rπ/m
1
r
x − 2l
π
2lπ/rπ/m1
p ⎝
∞
⎞p
⎞
⎟
|Δr bk⎠
| dx⎟
⎠
kcm1 1
cm1
p
p ⎞
∞
∞
bk
b
k
⎠
≤ C⎝
mαθp−2
mαp−2 mθ−1
θ
θ
k
k
mn1
mn1
km1
km1
∞
∞
bk
.
≤C
mαθp−2
θ
mn1
km k
∞
4.42
Set γm mαθp−2 , αk 0 for k < n 1 and αk k−θ bk for k ≥ n 1. Then by 3.3, we
have for α > 1 − θp
∞
Σ4 Σ5 ≤ C
m
αθp−21−p
p
m−θp bm
m
mn1
k
∞
≤C
αθp−2
p
mαp−2 bm .
4.43
mn1
k1
Let
2l 1π π
2l 1π
π
− ≤x<
−
,
r
r
r
m1
4.44
where m : mx ≥ r and l 0, 1, . . . , r/2 − 1r ≥ 2. Then, for n ≥ m, using 3.2 and 4.19,
we get
2l1π/r−π/n1
p
ωα,r xgx − Sn g, x dx
2l1π/r−π/r
n 2l1π/r−π/m1 2l 1π
mr
≤C
n
2l1π/r−π/m
m
α
2l1π/r−π/m
mr
≤C
n
mα
≤C
2l1π/r−π/m1
2l1π/r−π/m
mr
n
2l1π/r−π/m1
m
αp−2
mr
≤ CΣ1 Σ2 .
∞
−α
1
2|sin rx/2|p
gx − Sn g, x p dx
∞
|Δr bk | kn1
1
2l 1 − r/πxp
p
|Δr bk |
kn1
r
−x
nr
p
bk
kn1
∞
p
|Δr bk |
kn1
dx
4.45
Journal of Inequalities and Applications
17
Therefore,
2l1π/r−π/n1
2l1π/r−π/r
⎛
⎞
p
cn1
∞
p
p
ωα,r xgx − Sn g, x dx ≤ C⎝nαp−1
kαp−2 bk ⎠.
|Δr bk |
kn1
kn1
4.46
If m ≥ n 1 ≥ r 1, then
2l1π/r
p
ωα,r xgx − Sn g, x dx
2l1π/r−π/n1
∞ 2l1π/r−π/m1 2l 1π
mn1
⎛
r
2l1π/r−π/m
−x
−α
gx − Sn g, x p dx
p
2l1π/r−π/m1 m
≤ C⎝
m
bk sin kx dx
2l1π/r−π/m
mn1
kn1
∞
α
p
2l1π/r−π/m1 cm1
∞
α
m
bk sin kx dx
2l1π/r−π/m
mn1
km1
p ⎞
2l1π/r−π/m1 ∞
∞
dx⎠
mα
b
sin
kx
k
kcm1 1
2l1π/r−π/m
mn1
4.47
: CΣ6 Σ7 Σ8 .
Similarly as in the estimation of the quantity Σ3 using 3.3 for α < 1, we have
Σ6 ≤ C
∞
m
α−2
mn1
m
p
bk
kn1
≤C
∞
p
mαp−2 bm .
4.48
mn1
r
If bn ∈ GMθ , then using 3.2 and 4.19, we have
⎛
Σ 7 Σ8 ≤ C ⎝
∞
m
α−2
cm1
mn1
∞
mα
p
bk
km1
2l1π/r−π/m1
2l1π/r−π/m
mn1
⎞p ⎞
⎛
∞
1
⎝
|Δr bk |⎠ dx⎠
2l 1 − r/πxp kcm1 1
cm1
p
p ⎞
∞
∞
bk
b
k
⎠
≤ C⎝
mαθp−2
mαp−2 mθ−1
θ
θ
k
k
mn1
mn1
km1
km1
⎛
≤C
∞
∞
m
mn1
αθp−2
∞
bk
km k
θ
.
4.49
18
Journal of Inequalities and Applications
Further, by 3.3, we have for α > 1 − θp
∞
Σ 7 Σ8 ≤ C
p
mαp−2 bm .
4.50
m−n1
Combining 4.32 or 4.33, 4.35–4.43 and 4.45–4.50 we complete the proof of
Theorem 2.5.
4.5. Proof of Theorem 2.6
The proof of Theorem 2.6 goes analogously as the proof of Theorem 2.5. The only difference
is that instead of 4.41 for r 1, 2 and 4.48 for r 2 we use the below estimations.
Applying the inequalities | sin kx| ≤ kx for x ∈ 0, π and | sin kx| ≤ kπ − x for
x ∈ 0, π and using 3.3, for α < 1 p, we have
Σ3 ∞
m
α
p
p
π/m m
∞
m
α
x
bk sin k x dx ≤
m
kbk dx
π/m1 kn1
π/m1
mn1
kn1
π/m
mn1
≤C
∞
m
α−p−2
mn1
m
p
≤C
kbk
∞
p
mαp−2 bm ,
mn1
kn1
p
π−π/m1 ∞
m
Σ6 mα
bk sin kx dx
π−π/m
mn1
kn1
≤
∞
m
α
π−π/m1 ≤C
∞
π − x
π−π/m
mn1
m
mn1
α−p−2
m
p
kbk
kn1
m
4.51
p
kbk
dx
kn1
≤C
∞
p
mαp−2 bm .
mn1
This completes the proof.
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