Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 513206, 20 pages
doi:10.1155/2012/513206
Research Article
On the Convergence of Absolute Summability for
Functions of Bounded Variation in Two Variables
Ying Mei1 and Dansheng Yu2
1
2
Department of Mathematics, Lishui University, Lishui, Zhejiang 323000, China
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
Correspondence should be addressed to Dansheng Yu, danshengyu@yahoo.com.cn
Received 25 September 2012; Accepted 12 November 2012
Academic Editor: Jaume Giné
Copyright q 2012 Y. Mei and D. Yu. 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.
By adopting some new ideas, we obtain the estimates of an absolute convergence for the functions
of the bounded variation in two variables. Our results generalize the related results of Humphreys
and Bojanic 1999 and Wang and Yu 2003 from one dimension to two dimensions and can be
applied to several summability methods.
1. Introduction
Let f be 2π-periodic functions integrable on −π, π. Denote its Fourier series by
∞
a0 ak cos kx bk sin kx,
S f, x ∼
2 k1
1.1
where the Fourier coefficients ak and bk are defined as follows:
ak :
1
π
π
−π
ft cos kt dt,
bk :
1
π
π
−π
ft sin kt dt.
1.2
Denoted by Sn f, x the nth partial sums of the Fourier series 1.1, that is,
n
a0 ak cos kx bk sin kx.
Sn f, x :
2 k1
1.3
2
Abstract and Applied Analysis
When f is of bounded variation, Bojanić 1 obtained the following result on the rate
of the convergence of approximation by Sn f, x.
Theorem A. If a periodic function f is of bounded variation on the interval −π, π, then the
following estimate holds for every x and n 1, 2, . . .:
n
Sn f, x − fx ≤ 3 V π/k ϕx ,
n k1 0
1.4
where ϕx t : fx t fx − t − fx 0 − fx − 0 and V ϕx , 0, t is the total variation of ϕx
on 0, t.
There are a lot of interesting generalizations that have been achieved by many authors
see 1–9. Among them, Jenei 4 and Móricz 5 generalized Theorem A to the double
Fourier series of functions of bounded variation; Humphreys and Bojanic 3, Wang and Yu
7 investigated the absolute convergence of Cesàro means of Fourier series.
α
be the Cesàro matrix of order α, that is,
Let Cα : cnk
α
cnk
:
Aα−1
n−k
Aαn
,
k 0, 1, . . . , n,
1.5
where
Aαn :
Γn α 1
,
Γα 1Γn 1
n 0, 1, . . . .
1.6
Then, the so-called Cesàro means of order α of 1.1 is defined by denote by σnα f, x:
n
1 σnα f, x α Aα−1
Sk f, x .
An k0 n−k
1.7
Humphreys and Bojanic 3 investigated the rate of the absolute convergence of Cesàro
means of the series 1.1. Their result can be read as follows.
Theorem B. Let x ∈ 0, π and f be 2π-periodic functions of bounded variation on −π, π. Then
for α > 0 and n ≥ 2, one has
n
4α Rαn f, x ≤
V π/k ϕx ,
nπ k1 0
1.8
where
∞ σ α f, x − σ α f, x ,
Rαn f, x :
k
k−1
kn1
ϕx t : fx t fx − t − fx 0 − fx − 0
and varba f be the total variation of f on a, b.
1.9
Abstract and Applied Analysis
3
However, Wang and Yu 7 showed that Theorem A is not correct when 0 < α < 1. In
fact, they proved the following.
Theorem C. Suppose 0 < α < 1, x ∈ 0, π and f are 2π-periodic functions of bounded variation on
−π, π. Then for n ≥ 2, one has
n
100 Rαn f, x ≤ 2 α kα−1 V0π/k ϕx ,
α n k1
1.10
and there exists a 2π-periodic function f ∗ of bounded variation on −π, π and a point x ∈ 0, π
such that
Rαn f ∗ , x >
n
1
kα−1 V0π/k ϕx
20000αnα k1
n ≥ 8.
1.11
Motivated by Theorems B and C, and the results of Jenei 4 and Móricz 5 on the
double Fourier series, we will investigate the absolute convergence of a kind of very general
summability of the double Fourier series. Our new results not only generalize Theorems
B and C to the double case, but also can be applied to many other classical summability
methods. We will present our main results on Section 2. Proofs will be given in Section 3. In
Section 4, we will apply our results to some classical summability methods.
2. The Main Results
Let fx, y be a function periodic in each variable with period 2π and integrable on the twodimensional torus T2 T × T in Lebesgue’s sense, in symbol, f ∈ LT2 . The double Fourier
series of a complex-valued function f ∈ LT2 is defined by
leikxly ,
f x, y ∼
fk,
2.1
k∈Z l∈Z
l are the Fourier coefficients of f:
where the fk,
l : 1
fk,
4π 2
T2
fu, ve−ikulv du dv,
k, l ∈ Z2 .
2.2
Let R : a1 , b1 × a2 , b2 be a bounded and closed rectangle on the plane. A function
fx, y defined on R is said to be of bounded variation over R in the sense of Hardy-Krause,
in symbol, fx, y ∈ BVH R, if
i the total variation V f, R of fx, y over R is finite, that is,
n m f xj , yk − f xj−1, yk − f xj , yk−1 f xj , yk < ∞,
V f, R : sup
j1 k1
2.3
4
Abstract and Applied Analysis
where the supremum is extended for all finite partitions
a1 x0 < x1 < · · · < xm b1 ,
a2 y0 < y1 < · · · < yn b2
2.4
of the intervals a1 , b1 and a2 , b2 ;
ii the marginal functions f·, a2 and fa1 , · are of bounded variation over the
intervals a1 , b1 and a2 , b2 , respectively.
For any fx, y ∈ BVH T, define
1 f x, y :
f x 0, y 0 f x − 0, y 0 f x 0, y − 0 f x − 0, y − 0 ,
4
ϕxy u, v : f x u, y v f x − u, y v f x u, y − v f x − u, y − v − 4f x, y .
2.5
For convenience, write
st
V00
ϕxy : V ϕxy , 0, s × 0, t .
2.6
For any double sequence {amn }, define
Δ11 amn : amn − am−1,n − am,n−1 am−1,n−1 ,
Δ10 amn : amn − am−1,n ,
2.7
Δ01 amn : amn − am,n−1 .
For any fourfold sequence {amnjk }, write
Δ11 amnjk : amnjk − am,n,j1,k − am,n,j,k1 am,n,j1,k1 ,
Δ01 amnjk : amnjk − am,n,j,k1 , Δ10 amnjk : am,n,j,k − am,n,j1,k .
2.8
A doubly infinite matrix T : tmnjk is said to be doubly triangular if tmnjk 0 for
j > m or k > n. The mnth term of the T -transform of the double Fourier series 2.1 is defined
by
m n
tmnμν Sμν x, y ,
Tmn x, y :
2.9
μ0 ν0
where Sμν x, y is the μνth partial sum of 2.1, that is,
μ μ ν
ν
leikxly :
Sμν x, y :
Akl x, y .
fk,
|k|0 |l|0
|k|0 |l|0
2.10
Abstract and Applied Analysis
5
Write
tmnjk :
m n
tmnjk ,
0 ≤ j ≤ m, 0 ≤ k ≤ n,
μj νk
tmnjk 0,
if j > m or k > n,
2.11
t 1,1 : tmnij − tm−1,n,i,j − tm,n−1,i,j tm−1,n−1,i,j .
mnjk
Set
∞ ∞
1,1 Δ11 Tjk x, y .
Rmn f; x, y :
2.12
jm1 kn1
Our main results are the following Theorems 2.1 and 2.2.
Theorem 2.1. Let fx, y be a periodic function and fx, y ∈ BVH T2 . Assume that T is a lower
doubly triangular matrix satisfying
t 1,1 t 1,1 0,
mnk0
mn0l
k, l 0, 1, . . . ,
n m 1,1 tmnkl O1,
2.13
2.14
k1 l1
and there exist constants α, β 0 < α, β ≤ 1 such that
⎧
−1−β −β π
π
⎪
⎪
×
,π ,
xy ,
x, y ∈ 0,
⎪
⎪O l
m
n
⎪
⎪
⎪
⎪
1,1
n m ⎨ −1−α −α tmnkl
π
π
sin kx sin ly O k
, π × 0,
x y ,
x, y ∈
,
m
n
k1 l1 kl
⎪
⎪
⎪
⎪
⎪
⎪
π
π
⎪
⎪
,π ×
,π .
⎩O k−1−α l−1−β x−α y−β , x, y ∈
m
n
2.15
Then for n ≥ 2, x, y ∈ T2 , one has
1,1 Rmn f; x, y ≤ C
n
m 1 π/kπ/l kα−1 lβ−1 V00
ϕxy .
β
α
m n k1 l1
2.16
Theorem 2.2. Let fx, y and T satisfy all the conditions of Theorem 2.1, except 2.15 is replaced
by
⎛
⎞
n m t 1,1
mnkl
O mτk∗ nτl∗ ,
Δk∗ l∗ ⎝
⎠
∗ σl∗ σk
k
l
k1 l1 2.17
6
Abstract and Applied Analysis
where k∗ , l∗ 0 or 1, k∗ l∗ ≥ 1, τ0 σ0 0, τ1 σ1 1 2. Then for n ≥ 2, x, y ∈ T2 ,
one has
n
m 1 1,1 π/kπ/l V
Rmn f; x, y ≤ C
ϕxy .
mn k1 l1 00
2.18
3. Proofs of Results
Lemma 3.1 see 10. If gx, y is continuous on a rectangle R : a1 , b1 × a2 , b2 and fx, y ∈
BVH R, then
i gx, y is integrable with respect to fx, y over R in the sense of Riemann-Stieltjes
integral;
ii fx, y is integrable with respect to gx, y over R in the sense of Riemann-Stieltjes
integral and
f x, y dx dy g x, y R
g x, y dx dy f x, y
R
b1
fx, b2 dx gx, b2 −
b1
a1
b2
fx, a2 dx gx, a2 a1
b2
f b1 , y dy g b1 , y −
a2
f a1 , y dy g a1 , y
3.1
a2
− fb1 , b2 gb1 , b2 fa1 , b2 ga1 , b2 fa2 , b1 ga2 , b1 − fa1 , a2 ga1 , a2 .
Lemma 3.2 see 10. If fx, y ∈ BVH R and gx, y is absolutely continuous on R, that is,
g x, y x y
a1
hu, vdu dv C,
x, y ∈ R,
3.2
a2
for some function hx, y ∈ LR and constant C, then
f x, y h x, y dx dy.
f x, y dx dy g x, y R
3.3
R
Proof of Theorem 2.1. By the definition of Tmn x, y see 2.9, we have
j i1 i2
k
Tjk x, y tjki1 i2
Aμν x, y
i1 0 i2 0
|μ|0 |ν|0
j
k
Aμν x, y tjk|μ||ν| .
|μ|0 |ν|0
3.4
Abstract and Applied Analysis
7
Therefore,
j
j−1 k
k
Aμν x, y tjk|μ||ν| −
Aμν x, y tj−1,k,|μ|,|ν|
Δ11 Tjk x, y |μ|0 |ν|0
|μ|0 |ν|0
−
j
k−1
j−1 k−1
Aμν x, y tj,k−1,|μ|,|ν| Aμν x, y tj−1,k−1,|μ|,|ν|
|μ|0 |ν|0
|μ|0 |ν|0
j
k
1,1
Aμν x, y tjk μ
| ||ν|
by 2.11
|μ|0 |ν|0
j k
μ1 ν1
1,1
Aμν x, y A−μ,ν x, y Aμ,−ν x, y A−μ,−ν x, y tjkμν
by 2.13 .
3.5
Noting that
Aμν x, y A−μ,ν x, y Aμ,−ν x, y A−μ,−ν x, y
f μ, ν eiμxνy f −μ, ν ei−μxνy f μ, −ν eiμx−νy f −μ, −ν ei−μx−νy
1
fs, t cos μs − x cos ν t − y ds dt,
2
4π T2
3.6
we have
j
k
1 t 1,1
Δ11 Tjk x, y fs, t cos μs − x cos ν t − y ds dt
jkμν
2
4π μ1 ν1
T2
π
j
k
1 t 1,1
ϕxy s, t cos μs cos νtds dt
4π 2 μ1 ν1 jkμν 0
j
k tjkμν π
1 ϕxy s, tds dt sin μs sin νt,
4π 2 μ1 ν1 μν
0
1,1
where in the last equality, Lemma 3.2 is applied.
By 3.7 and Lemma 3.1, we have
1,1 j π
∞
∞ k t
1
jkμν
1,1
Rmn f; x, y ϕ
d
sin
μs
sin
νt
td
s,
xy
s t
2
4π jm1 kn1μ1 ν1 μν
0
1,1 j π
∞
∞ k t
1
jkμν
sin
μs
sin
νtd
d
ϕ
t
s,
s t xy
2
4π jm1 kn1μ1 ν1 μν
0
3.7
8
Abstract and Applied Analysis
1,1
j ∞
∞ π k tjkμν
1 st
sin μs sin νtds dt V00
≤
ϕxy
4π 2 jm1 kn1 0 μ1 ν1 μν
π/j π/k π/j π
π π/k π π ∞
∞
1 4π 2 jm1 kn1 0
0
0
π/k
π/j 0
π/j π/k
1,1
k tjkμν
j st
× sin μs sin νtds dt V00
ϕxy
μ1 ν1 μν
: I1 I2 I3 I4 .
3.8
By 2.14, we have Denote by χa1 ,b1 ×a2 ,b2 x, y and χa1 ,b1 x the characteristic functions of
a1 , b1 × a2 , b2 and a1 , b1 , respectively.
j ∞
∞ π/j π/k k tjkμν
1 ds dt V st ϕxy
I1 sin
μs
sin
νt
00
2
μν
4π jm1 kn1 0
μ1 ν1
0
j ∞
∞ π/j π/k k 1 st
≤
t
ϕxy
stds dt V00
jkμν
2
4π jm1 kn1 0
0
μ1 ν1
≤C
∞ π/j π/k
∞
0
jm1 kn1
≤C
π/m1 π/n1 ∞
∞
0
≤C
0
0
st
stχ0,π/j×0,π/k s, tds dt V00
ϕxy
jm1 kn1
π/m1 π/n1
0
0
st
ds dt V00
ϕxy
π/m1π/n1 ≤ CV00
≤
st
stds dt V00
ϕxy
ϕxy
n
m C π/jπ/k j α−1 kβ−1 V00
ϕxy .
β
α
m n j1 k1
For I2 , by the first inequality of 2.15,
j ∞
∞ π/j π k tjkμν
1 st
I2 sin μs sin νtds dt V00
ϕxy
μν
4π 2 jm1 kn1 0
π/k μ1 ν1
≤C
∞
∞
k
jm1 kn1
−1−β
π/j π
0
π/k
st
ϕxy
st−β ds dt V00
3.9
Abstract and Applied Analysis
π/j π/n1 π/j π
∞
∞
−1−β
≤C
k
0
jm1 kn1
≤C
π/m1 π/n1 ∞
0
C
0
0
π/m1 π
∞
0
0
sχ0,π/j s
st
ϕxy
k−1−β t−β ds dt V00
∞
st
ϕxy
k−1−β t−β ds dt V00
kn1
st
ds dt V00
π/m1π/n1 −β
ϕxy Cn
ϕxy Cn−β
≤ CV00
st
st−β ds dt V00
ϕxy
k>π/t
π/n1 jm1
π/m1 π/n1
π/n1
∞
sχ0,π/j s
jm1
0
≤C
π/k
9
π/m1 π
0
π/n1
n π/m1 π/k
k1
0
π/k1
st
t−β ds dt V00
ϕxy
st
t−β ds dt V00
ϕxy
n
π/m1π/k ϕxy Cn−β kβ−1 V00
ϕxy
π/m1π/n1 ≤ CV00
k1
≤
n
m C
π/jπ/k j α−1 kβ−1 V00
ϕxy .
β
α
m n j1 k1
3.10
Similarly, we also have
I3 ≤
m n
C π/jπ/k j α−1 kβ−1 V00
ϕxy .
β
α
m n j1 k1
3.11
By 2.15, we deduce that
I4 ≤ C
∞ π π
∞
π/j
jm1 kn1
π/k
st
j −1−α k−1−β s−α t−β ds dt V00
ϕxy
π/m1 π/n1 π
∞
∞
≤C
π/j
jm1 kn1
π/k
st
π/n1 π/m1 π
π/m1
π/k
π/j
π/n1
× j −1−α k−1−β s−α t−β ds dt V00 ϕxy
⎛
⎞⎛
⎞
π/m1 π/n1
∞
∞
−1−α
−α
−1−β
−β
st
⎝
≤C
j
s ⎠⎝
k
t ⎠ds dt V00
ϕxy
0
C
0
j>π/s
π/m1 π
0
π/n1
k>π/t
⎛
⎞
∞
∞
−1−α
−α
−1−β
−β
st
⎝
ds dt V00
j
s ⎠
k
t
ϕxy
j>π/s
kn1
π
π/m1
π
π/n1
10
Abstract and Applied Analysis
C
π
π/n1
π/m1
C
π/m1
⎝
0
⎞⎛
∞
j
−1−α −α ⎠⎝
s
jm1
∞
⎞
k
st
ds dt V00
ϕxy
−1−β −β ⎠
t
k>π/t
⎛
⎞
∞
∞
−1−α
−α
−1−β
−β
st
⎝
ds dt V00
j
s ⎠
k
t
ϕxy
π
π
⎛
π/n1
jm1
kn1
n
π/m1π/k ϕxy Cn−β kβ−1 V00
ϕxy
π/m1π/n1 ≤ CV00
k1
Cm−α
m
π/jπ/n1 j α−1 V00
ϕxy
j1
n
m C π/jπ/k j α−1 kβ−1 V00
ϕxy
mα nβ j1 k1
≤C
n
m C π/m1π/n1 j α−1 kβ−1 V00
ϕxy .
β
α
m n j1 k1
3.12
We get 2.16 by combining 3.8–3.12.
Proof of Theorem 2.2. Set
i t D
i
sin it j1
cos1/2t − cosi 1/2t
,
2 sin1/2t
i 1, 2, . . . .
3.13
By Abel’s transformation, we have
1,1
n m tmnkl
k1 l1
kl
sin ks sin lt m
n
t 1,1
sin ks l t
Δ01 mnkl D
k l1
l
k1
n
m
t 1,1
sin lt k s
Δ10 mnkl D
l k1
k
l1
n
m k1 l1
Δ11
t 1,1
mnkl
kl
3.14
l t.
k sD
D
i t| Ot−1 , | sin t| ≤ |t|, and
Therefore, by the following well-known inequalities: |D
condition 2.17, we see that 2.15 holds with α β 1, and thus we get Theorem 2.2 from
Theorem 2.1.
Abstract and Applied Analysis
11
4. Applications
4.1. Cesàro’s Means
Let T C, γ, δ Cγ × Cδ be the double Cesàro matrix of order γ, δ with γ, δ > −1, that is,
T has entries
γ−1
tmnjk γ
Cmj
×
δ
Cnk
Am−j Aδ−1
n−k
γ
Am Aδn
.
4.1
Then for 0 ≤ j ≤ m − 1, 0 ≤ k ≤ n − 1,
t 1,1 Δ11 tm−1,n−1,j,k
mnjk
⎛
m−1
Aγ−1
m−1−u
⎝
γ
A
uj
m−1
⎛
⎝
γ
Am
γ
uj
γ
γ
Am−j
−
γ−1
m
Am−u
−
Am−1−j
γ
Am−1
Am
⎞
⎠
⎞
n−1 Aδ−1
n
δ−1
A
n−v
n−1−v
⎠
−
δ
δ
vk An−1
vk An
Aδn−k
Aδn
−
Aδn−1−k
4.2
Aδn−1
γ−1
jkAm−j Aδ−1
n−k
γ
mnAm Aδn
,
1,1
tmnmn
tmnmn 1
β
Aαm An
.
In particular,
t 1,1 t 1,1 0,
mnj0
mn0k
j, k 0, 1, . . . .
4.3
In other words, T satisfies condition 2.13 of Theorem 2.1.
By the well-known inequality e.g., see 11
1
nγ
γ
An ,
1O
n
Γ γ 1
we deduce that
⎛
⎞
γ−1
n m m
n kAδ−1
1 ⎠ 1
1,1 ⎝ jAm
n−k
δ
tmnjk γ γ
δ
An
Am
j1 k1
j1 mAm
k1 nAn
⎛
⎞
m
m/2
1
γ−1
j O m−γ
Am−j γ ⎠
⎝O m−2
A
m
j1
jm/2
4.4
12
Abstract and Applied Analysis
n/2
n
1
−2
−δ
δ−1
kO n
An−k δ
× O n
An
k1
kn/2
O1,
γ, δ > 0,
4.5
which means condition 2.14 of Theorem 2.1.
For 0 < γ < 1, we have
n Aγ−1
n−k
γ
k1
nAn
sin kx Im
n
1 γ
nAn k1
γ−1
An−k eikx
Im
n−1
einx γ
nAn k0
γ−1
Ak e−ikx
4.6
: ImJ1 J2 ,
where
J1 :
einx γ
nAn
1 − e−ix
−γ
,
J2 :
∞
einx γ−1
A e−ikx .
γ
nAn kn k
4.7
By 4.4, we observe that
|J1 | O n−1−γ x−γ .
γ−1
Since Ak
4.8
decreases monotonically to 0, J2 converges for 0 < x ≤ π, and
|J2 | O
γ−1
−1
An −ix γ 1 − e
nAn
O n−2 x−1 O n−1−γ x−γ
4.9
for t ∈ π/n, π.
By 4.4–4.9, we obtain that
n Aγ−1
n−k
γ
k1
nAn
sin kx O n−1−γ x−γ ,
x∈
π
, π , 0 < γ ≤ 1.
n
4.10
On the other hand, we have
n Aγ−1
n−k
n kAγ−1
n−k
γ sin kx Ox
γ Ox,
nA
nA
n
n
k1
k1
π
, γ > 0.
x ∈ 0,
n
4.11
Abstract and Applied Analysis
13
For γ ≥ 1, we have that
⎛
⎞
γ−1
n−1 n−1 An−k 1
γ−1
γ−1
Δ⎝
⎠ 1
A
−
A
O
n−2 .
γ
γ
γ
n−k
n−k−1
nA
nA
nA
n
n
n
k1
k1
4.12
Now, by 4.10 and 4.11, we deduce that
γ−1
1,1
n tmnjk
m m Am−j
n Aδ−1
n−k
sin jx sin ky ≤ sin ky
γ sin jx
δ
j1 k1 jk
j1 mAm
k1 nAn
⎧
−1−δ −δ π
π
⎪
⎪
×
,π ,
xy ,
x, y ∈ 0,
⎪O k
⎪
⎪
m
n
⎪
⎪
⎪
⎨ π
π
−1−γ −γ
O j
, π × 0,
,
x y ,
x, y ∈
⎪
m
n
⎪
⎪
⎪
⎪
⎪
π
π
⎪
⎪
,π ×
,π
⎩O j −1−γ k−1−δ x−γ y−δ , x, y ∈
m
n
4.13
for 0 < γ, δ ≤ 1.
When γ > 1, 0 < δ ≤ 1, by 4.10 and 4.11 again,
γ−1
1,1
n tmnjk
m m Am−j
n Aδ−1
n−k
−1−δ
−δ
O
k
≤
sin
jx
sin
ky
,
sin
jx
sin
ky
xy
γ
j1 k1 jk
j1 mAm
k1 nAδn
4.14
for x, y ∈ 0, π/m × π/n, π. By 4.10–4.12, we have
⎛ γ−1 ⎞
1,1
n tmnjk
Am−j
m m
n Aδ−1
n−k
⎠Dj x
sin jx sin ky ≤ Δ⎝
sin
ky
γ
δ
jk
mAm
j1 k1
j1
k1 nAn
⎧
−2 −1 π
π
⎪
⎪
O
j
,
π
×
0,
,
x
y
,
x,
y
∈
⎪
⎨
m
n
⎪
π
π
⎪
⎪
,π ×
,π .
⎩O j −2 k−1−δ x−1 y−δ , x, y ∈
m
n
4.15
Similarly, we have
⎧
⎪
⎪
⎪O k−2 xy−1 ,
⎪
⎪
⎪
1,1
⎪
⎪
n tmnjk
m ⎨ −1−γ −γ sin jx sin ky O j
x y ,
j1 k1 jk
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩O j −1−γ k−2 x−γ y−1 ,
π
π
,π ,
x, y ∈ 0,
×
m
n
π
π
, π × 0,
x, y ∈
,
m
n
π
π
,π ×
,π ,
x, y ∈
m
n
4.16
14
Abstract and Applied Analysis
for δ > 1, 0 < γ ≤ 1, and
⎧
−2 −1 π
π
⎪
⎪
,π ,
x, y ∈ 0,
×
⎪
⎪O k xy ,
m
n
⎪
⎪
⎪
1,1
⎪
n tmnjk
m ⎨ −2 −1 π
π
O j x y ,
sin
jx
sin
ky
,
π
×
0,
,
x,
y
∈
⎪
m
n
j1 k1 jk
⎪
⎪
⎪
⎪
⎪
π
π
⎪
⎪
,π ×
,π ,
⎩O j −2 k−2 x−1 y−1 , x, y ∈
m
n
4.17
for γ, δ > 1.
Setting
γ :
γ, 0 < γ ≤ 1,
1,
γ > 1,
δ :
δ, 0 < δ ≤ 1,
1,
δ > 1,
4.18
then we have
⎧ ⎪
⎪
O k−1−δ xy−1 ,
⎪
⎪
⎪
⎪
1,1
⎪
⎪
n
m
t
⎨ −1−γ −1 mnjk
sin jx sin ky O j
x y ,
j1 k1 jk
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩O j −1−γ k−1−δ x−1 y−1 ,
π
π
,π ,
x, y ∈ 0,
×
m
n
π
π
, π × 0,
,
x, y ∈
m
n
π
π
x, y ∈
,π ×
,π ,
m
n
4.19
by 4.13–4.17.
From 4.19, we have the following corollary of Theorem 2.1, which generalizes
Theorem B from one dimension to two dimensions.
Theorem 4.1. Let fx, y be a periodic function and fx, y ∈ BVH T2 and T C, γ, δ be the
double Cesàro matrix of order γ, δ with γ, δ > 0. Then for n ≥ 2, x, y ∈ T2 , one has
1,1 Rmn f; x, y ≤ C
1
n
m π/kπ/l γ −1 δ−1
k
l
V
ϕ
.
xy
00
mγ nδ k1
4.20
l1
4.2. Bernstein-Rogosinski’s Means
The so-called Bernstein-Rogosinski means of Fourier series 1.1 are defined by
1
1
1
Sn f, x π Sn f, x −
π
.
Bn f, x :
2
2n 1
2n 1
4.21
Abstract and Applied Analysis
15
Now, we introduce the following Bernstein-Rogosinski means of double Fourier series 2.1:
π
π
π
π
1
Smn f; x ,y Smn f; x ,y −
Bmn f, x :
4
2m 1
2n 1
2m 1
2n 1
π
π
π
π
Smn f; x −
,y Smn f; x −
,y −
.
2m 1
2n 1
2m 1
2n 1
4.22
Direct calculations yield that
m n
Bmn f, x cos
j0 k0
m−1
n−1 j0 k0
m−1
j 1
j
π − cos
π
cos
2m 1
2m 1
cos
j 1
j
π − cos
π
2m 1
2m 1
cos
k1
k
π − cos
π
2n 1
2n 1
j0
n−1 k0
cos
jπ
kπ
cos
Δ11 Sjk x, y
2m 1
2n 1
k
k1
π − cos
π Sjk x, y
cos
2n 1
2n 1
cos
cos
nπ
Sjn x, y
2n 1
mπ
Smk x, y
2m 1
nπ
mπ
cos
Smn x, y ,
2m 1
2n 1
4.23
Thus, Bmn f, x can be regarded as a T -transformation of Fourier series 1.1 with T of entries
defined as
tmnjk cos
j 1
j
k
k1
π − cos
π
cos
π − cos
π , 0 ≤ j < m, 0 ≤ k < n,
2m 1
2m 1
2n 1
2n 1
j 1
j
nπ
tmnjn cos
π − cos
π cos
, 0 ≤ j < m,
2m 1
2n 1
2n 1
k1
mπ
k
tmnmk cos
π − cos
π cos
, 0 ≤ k < n,
2n 1
2n 1
2m 1
tmnmn cos
nπ
mπ
cos
.
2m 1
2n 1
4.24
By direct calculations, we have
t 1,1 mnjk
cos
j
j
π − cos
π
2m − 1
2m 1
cos
k
k
π − cos
π ,
2n − 1
2n 1
0 ≤ j < m, 0 ≤ k < n,
4.25
16
Abstract and Applied Analysis
j
j
nπ
π − cos
π cos
,
2m − 1
2m 1
2n 1
k
mπ
k
π − cos
π cos
,
cos
2n − 1
2n 1
2m 1
t 1,1 mnjn
t 1,1
mnmk
cos
0 ≤ j < m,
4.26
0 ≤ k < n,
4.27
nπ
mπ
1,1
tmnmn
cos
.
cos
2m 1
2n 1
4.28
Therefore, 4.26 and 4.27 imply 2.13, while from 4.25–4.28, we get
j
j
k
k
π − cos
π cos
π − cos
π 2m − 1
2m 1
2n − 1
2n 1
n m m−1 n−1 1,1 tmnjk ≤
cos
j1 k1
j1 k1
j1
j
j
cos nπ
cos
π
−
cos
π
2m − 1
2m 1 2n 1
m−1
n−1 cos k π − cos k π cos mπ cos mπ cos nπ
2n − 1
2n 1 2m 1
2m 1
2n 1
k1
n−1
m−1
≤4
sin
j1 k1
m−1
2
sin
j1
≤ 4π 4
n−1
m−1
j
2mj
2nk
k
π sin
π sin 2
π sin 2
π
4m2 − 1
4m2 − 1
4n − 1
4n − 1
n−1
j
2mj
k
2nk
π
sin
π
2
π sin 2
π 1
sin 2
4m2 − 1
4m2 − 1
4n
−
1
4n
−1
k1
2mj 2
j1 k1 4m
2
2nk2
2
− 1 4n2 − 1
2
2π 2
m−1
j1
2mj 2
4m2 − 1
2
2π 2
n−1
2nk2
2
k1 4n
− 12
1
O1,
4.29
which implies 2.14.
Finally, we verify that T satisfies condition 2.17; thus T satisfies all the conditions of
Theorem 2.2. By direct calculations, we deduce that
⎛
⎞
t 1,1 m m n j
j
mπ
mnjk
⎝ cos
Δ01
⎠
π − cos
π cos
k 2m − 1
2m 1 2m 1
j1 k1 j1
n−1 cosk/2n − 1π − cosk/2n 1π
×
k
k1
cosk1/2n − 1π − cosk1/2n 1π 1
nπ
−
n cos 2n 1
k1
Abstract and Applied Analysis
n−1 cosk/2n − 1π − cosk/2n 1π
O
k
k1
17
cosk 1/2n − 1π − cosk 1/2n 1π −2
n
k1
n−1 k
k1
k1
1
k
π −cos
π − cos
π −cos
π
O1
cos
k
2n − 1
2n 1
2n − 1
2n 1
k1
−
O1
k1
k1
1
k 1 cos
−
π − cos
π
k k1 2n − 1
2n 1 n−1 1
n−1 2nkπ
2nk 1π
kπ
1 sin
−
sin
sin 2
2
2
k
4n − 1
4n − 1
4n − 1
k1
kπ
k 1π
sin 2
− sin
4n − 1
4n2 − 1
2nk 1π sin
4n2 − 1 n−1
2nk 1π
1
k 1π
sin
sin
2−1
kk
1
4n
4n2 − 1
k1
O1
n−1
1
k1
O1
k
sin
n−1
nπ
kπ
1
2nk 1π
k 1π
sin
O1
sin
sin
2
2
2
2
4n − 1
4n − 1
k
4n − 1
4n2 − 1
k1
n−1
n−3 O n−2 .
k1
4.30
Analogously,
t 1,1 n m mnjk
O m−2 ,
Δ10
j j1 k1 t 1,1 n m mnjk
O m−2 n−2 .
Δ11
j j1 k1 4.31
Now, 4.30 and 4.31 imply 2.17.
4.3. Riesz’s Means
Let {pn } and {qn } be positive sequences such that Pn p0 p1 · · ·pn → ∞, Qn q0 q1 · · ·
qn → ∞, and let T : tmnjk be a lower triangular matrix with entries tmnjk pj qk /Pm Qn,
j 0, 1, . . . m, k 0, 1, . . . , n; m, n 1, 2, . . .. Then the T -transformation of Fourier series 2.1
is known as Riesz’s mean.
18
Abstract and Applied Analysis
Proposition 4.2. Let {pn } and {qn } be positive nondecreasing sequences such that (one says An ∼ Bn ,
if there are two positive constants C1 and C2 such that C1 ≤ An /Bn ≤ C2 .)
npn ∼ Pn ,
n
pk
k1
k
O pn ,
nqn ∼ Qn ,
n
qk
k1
k
4.32
O qn .
4.33
Then T satisfies all the conditions of Theorem 2.2.
Proof. Direct calculations yield that set P−1 Q−1 0
⎞
m
n−1
n
n−1
pm qn Pj−1 Qk−1
p
p
q
q
i
i
i
i
⎠
⎝
−
−
,
P
P
Q
Q
Pm Pm−1 Qn Qn−1
ij m
ij m−1
ik n
ik n−1
⎛
t 1,1
mnjk
0 ≤ j < m, 0 ≤ k < n,
4.34
t 1,1 mnmk
pm
Pm
n
n−1
pm qn Qk−1
qi qi
−
,
Q
Q
P
n
n−1
m Qn Qn−1
ik
ik
0 ≤ k < n,
4.35
⎛
t 1,1
mnjn
⎞
m
n−1
p q P
p
p
i
i
⎠ qn m n j−1 ,
⎝
−
P
P
Qn Pm Pm−1 Qn
ij m
ij m−1
1,1
tmnmn
0 ≤ j < m,
pm qn
.
Pm Qn
4.36
It follows from 4.35 that T satisfies condition 2.13.
Since {pn } and {qn } are nondecreasing, by 4.32, 4.34–4.36, we have
n m 1,1 tmnjk j1 k1
n
m pm qn
mnpm qn
Pj−1 Qk−1 O
O1.
Pm Pm−1 Qn Qn−1 j1 k1
Pm Qn
Hence, T satisfies condition 2.14.
Since {pn } and {qn } are nondecreasing, by 4.32–4.36, we have
⎛ 1,1 ⎞ ⎛
⎞
t
n m m−1
p
p
mnjk
m
m
Δ01 ⎝
⎠
⎠ ⎝
Pj−1 k
Pm Pm−1 j1
Pm
j1 k1 ×
n−1 Qk−1
qn
qn Qk −
Qn Qn−1 k1 k
k1
nPn
4.37
Abstract and Applied Analysis
19
O1
O1
qn
Qn Qn−1
n−1
n−1
qk
Qk−1
−2
O n
kk 1 k2 k 1
k1
n−1
qk
1 n−2
nQn−1 k1 k
qn−1
nQn−1
O n−2 .
O
4.38
Similarly,
t 1,1 n m mnjk
O m−2 ,
Δ10
j j1 k1 t 1,1 n m mnjk
O m−2 n−2 .
Δ11
j j1 k1 4.39
Now, by 4.38 and 4.39, we show that T also satisfies condition 2.17.
Acknowledgments
A Project Supported by Scientific Research Fund of Zhejiang Provincial Education
Department Y201223607. Research of the second author is supported by NSF of China
10901044, Qianjiang Rencai Program of Zhejiang Province 2010R10101, SRF for ROCS,
SEM, and Program for excellent Young Teachers in HZNU.
References
1 R. Bojanić, “An estimate of the rate of convergence for Fourier series of functions of bounded variation,” Institut Mathématique, vol. 26, pp. 57–60, 1979.
2 A. S. Belov and S. A. Telyakovskii, “Refinement of the Dirichlet-Jordan and Young’s theorems on
Fourier series of functions of bounded variation,” Sbornik, vol. 198, pp. 777–791, 2007.
3 N. Humphreys and R. Bojanic, “Rate of convergence for the absolutely C, α summable Fourier series
of functions of bounded variation,” Journal of Approximation Theory, vol. 101, no. 2, pp. 212–220, 1999.
4 A. Jenei, “On the rate of convergence of Fourier series of two-dimensional functions of bounded
variation,” Analysis Mathematica, vol. 35, no. 2, pp. 99–118, 2009.
5 F. Móricz, “A quantitative version of the Dirichlet-Jordan test for double Fourier series,” Journal of
Approximation Theory, vol. 71, no. 3, pp. 344–358, 1992.
6 S. A. Telyakovskiı̆, “On the rate of convergence of Fourier series of functions of bounded variation,”
Mathematical Notes, vol. 72, pp. 872–876, 2002.
7 K. Y. Wang and D. Yu, “On the absolute C, α convergence for functions of bounded variation,”
Journal of Approximation Theory, vol. 123, no. 2, pp. 300–304, 2003.
8 C. B. Lu and C. Q. Gu, “The computation of the inverse of block-wise centrosymmetric matrices,”
Publicationes Mathematicae Debrecen, vol. 1-2, no. 82, pp. 1–17, 2013.
20
Abstract and Applied Analysis
9 C. B. Lu and C. Q. Gu, “The computation of the square roots of circulant matrices,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6819–6829, 2011.
10 F. Móricz, “Pointwise behavior of double Fourier series of functions of bounded variation,” Monatshefte für Mathematik, vol. 148, no. 1, pp. 51–59, 2006.
11 A. Zygmund, Trigonometric Series, vol. 1, Cambridge University, New York, NY, USA, 1959.
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