Journal of Inequalities in Pure and
Applied Mathematics
CESÁRO MEANS OF N -MULTIPLE
TRIGONOMETRIC FOURIER SERIES
volume 7, issue 3, article 86,
2006.
USHANGI GOGINAVA
Department of Mechanics and Mathematics
Tbilisi State University
Chavchavadze str. 1
Tbilisi 0128, Georgia
Received 06 March, 2006;
accepted 08 March, 2006.
Communicated by: L. Leindler
EMail: z_goginava@hotmail.com
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
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Abstract
Zhizhiashvili proved sufficient condition for the Cesáro summability by negative
order of N-multiple trigonometric Fourier series in the space Lp , 1 ≤ p ≤ ∞. In
this paper we show that this condition cannot be improved .
2000 Mathematics Subject Classification: 42B08.
Key words: Trigonometric system, Cesáro means, Summability.
Cesáro Means of N -multiple
Trigonometric Fourier Series
Ushangi Goginava
N
N
Let R be N -dimensional Euclidean space. The elements of R are denoted by x = (x1 , . . . , xN ), y = (y1 , . . . , yN ), . . . . For any x, y ∈ RN
the vector (x1 + y1 , . . . , xN + yN ) of the space RN is denoted by x + y. Let
P
1/2
N
2
kxk =
x
.
i=1 i
N
Denote by C [0, 2]
the space of continuous on [0, 2π]N , 2π-periodic relative to each variable functions with the following norm
kf kC =
x∈[0,2π]
|f (x)|
sup
N
and Lp [0, 2π]N , (1 ≤ p ≤ ∞) are the collection of all measurable, 2π-periodic
relative to each variable functions f defined on [0, 2π]N , with the norms
Z
p1
p
kf kp =
|f (x)| dx
< ∞.
[0,2π]
N
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J. Ineq. Pure and Appl. Math. 7(3) Art. 86, 2006
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For the case p = ∞, by Lp [0, 2π]N we mean C [0, 2π]N .
Let M := {1, 2, . . . , N } , B := {s1 , . . . , sr } , sk < sk+1 , k = 1, . . . , r −
1, B ⊂ M, B 0 := M \B. Let
∆{si } (f, x, hsi ) := f (x1 , . . . , xsi −1 , xsi + hsi , xsi +1 , . . . , xN )
− f (x1 , . . . , xsi −1 , xsi , xsi +1 , . . . , xN ) .
The expression we get by successive application of operators ∆{s1 } (f, x, hs1 ) ,
. . . , ∆{sr } (f, x, hsr ) will be denoted by ∆B (f, x, hs1 , . . . , hsr ) , i. e.
∆B (f, x, hs1 , . . . , hsr ) := ∆{sr } ∆B\{sr } , x, hsr .
N
p
Let f ∈ L [0, 2π] . The expression
ωB (δs1 , . . . , δsr ; f ) :=
sup
∆B (f, ·, hs1 , . . . , hsr )
|hsi |≤δsi ,i=1,...,r
p
p
is called a mixed or a particular modulus of continuity in the L norm, when
card(B) ∈ [2, N ] or card(B) = 1.
The total modulus of continuity of the function f ∈ Lp [0, 2π]N in the Lp
norm is defined by
ω (δ, f )p = sup kf (· + h) − f (·)kp
(1 ≤ p ≤ ∞) .
khk≤δ
Suppose that f is a Lebesgue integrable function on [0, 2π]N , 2π periodic
relative to each variable.Then its N -dimensional Fourier series with respect to
Cesáro Means of N -multiple
Trigonometric Fourier Series
Ushangi Goginava
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the trigonometric system is defined by
∞
X
i1 =0
···
∞
X
2−λ(i)
iN =0
where
(B)
ai1 ,...,iN
X
Z
f (x)
Y
sin ik xk ,
k∈B
cos ij xj
j∈B 0
[0,2π]N
Y
cos ij xj
j∈B 0
B⊂M
1
= N
π
Y
(B)
ai1 ,...,iN
Y
sin ik xk dx
k∈B
is the Fourier coefficient of f and λ (i) is the number of those coordinates of the
vector i := (i1 , . . . , iN ) which are equal to zero.
Let Sp1 ,...,pN (f, x) denote the (p1 , . . . , pN )-th rectangular partial sums of the
N -dimensional Fourier series with respect to the trigonometric system, i. e.
Sp1 ,...,pN (f, x) :=
p1
X
···
i1 =0
pN
X
X
Ai1 ,...,iN (f, x) ,
iN =0
(B)
ai1 ,...,iN
B⊂M
Y
p1 =0
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cos ij xj
j∈B 0
Y
sin ik xk .
k∈B
The Cesáro (C; α1 , . . . , αN )-means of N -multiple trigonometric Fourier series defined by
!−1 m
mN Y
N
N
1
X
Y
X
α1 ,...,αN
αi
−1
Aαmjj−
σm1 ,...,mN (f, x) =
Ami
···
pj Sp1 ,...,pN (f, x) ,
i=1
Ushangi Goginava
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where
Ai1 ,...,iN (f, x) := 2−λ(i)
Cesáro Means of N -multiple
Trigonometric Fourier Series
pN =0 j=1
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where
Aαn =
(α + 1) (α + 2) · · · (α + n)
,
n!
α 6= −1, −2, . . . ,
n = 0, 1, . . . .
It is well-known that [4]
(1)
c1 (α) nα ≤ Aαn ≤ c2 (α) nα .
For the uniform summability of Cesáro means of negative order of onedimensional trigonometric Fourier series the following result of Zygmund [3] is
well-known: if
ω (δ, f )C = o (δ α )
Cesáro Means of N -multiple
Trigonometric Fourier Series
Ushangi Goginava
and α ∈ (0, 1), then the trigonometric Fourier series of the function f is uniformly (C, −α) summable to f.
In [2] Zhizhiashvili proved sufficient conditions for the convergence of Cesáro
means
order of N -multiple trigonometric Fourier series in the space
of negative
N
p
L [0, 2π] , (1 ≤ p ≤ ∞). The following is proved.
N
p
Theorem A (Zhizhiashvili). Let f ∈ L [0, 2π]
for some p ∈ [1, +∞] and
α1 + · · · + αN < 1, where αi ∈ (0, 1), i = 1, 2, . . . , N . If
ω (δ, f )p = o δ α1 +···+αN ,
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then
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−α1 ,...,−αN
σm
(f ) − f
1 ,...,mN
p
→0
as
mi → ∞, i = 1, . . . , N.
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In case p = ∞ the sharpness of Theorem A has been proved by Zhizhiashvili
[2]. The following theorem shows that Theorem A cannot be improved in cases
1 ≤ p < ∞. Moreover, we prove the following
Theorem 1 (for N = 1 see [1]). Let α1 + · · · + αN < 1 and
αi ∈ (0, 1),
N
i = 1, 2, . . . , N, then there exists the function f0 ∈ C [0, 2π]
for which
(2)
ω (δ, f0 )C = O δ α1 +···+αN
and
lim
m→∞
−α1 ,...,−αN
σm,...,m
(f0 ) − f0
1
Cesáro Means of N -multiple
Trigonometric Fourier Series
> 0.
Ushangi Goginava
Proof. We can define the sequence {nk : k ≤ 1} satisfying the properties
∞
X
1
1
(3)
,
α1 +···+αN = O
α1 +···+αN
n
n
j
k
j=k+1
(4)
k−1
X
1−(α1 +···+αN )
nj
1−(α1 +···+αN )
= O nk
j=1
(5)
,
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nk−1
< .
nk
k
Consider the function f0 defined by
f0 (x1 , . . . , xN ) :=
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∞
X
j=1
fj (x1 , . . . , xN ) ,
J. Ineq. Pure and Appl. Math. 7(3) Art. 86, 2006
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where
1
N
Y
sin nj xi .
nαj 1 +···+αN i=1
N
From (3) it is easy to show that f0 ∈ C [0, 2π] . First we shall prove that
fj (x1 , . . . , xN ) :=
ωi (δ, f )C = O δ α1 +···+αN ,
(6)
Let
1
nk
≤δ<
1
.
nk−1
i = 1, . . . , N.
Then from (3) and (4) we can write that
|f0 (x1 , . . . , xi−1 , xi + δ, xi+1 , . . . , xN ) − f0 (x1 , . . . , xi−1 , xi , xi+1 , . . . , xN )|
∞
X
1
≤
α1 +···+αN |sin nj (xi + δ) − sin nj xi |
n
j=1 j
≤
≤
k−1
X
1
j=1
nαj 1 +···+αN
k−1
X
nj δ
+O
1
nαk 1 +···+αN
1
1−(α1 +···+αN )
= O δnk−1
+O
nαk 1 +···+αN
= O δ α1 +···αN ,
j=1
nαj 1 +···+αN
|sin nj (xi + δ) − sin nj xi | + 2
which proves (6).
∞
X
1
j=k
nαj 1 +···+αN
Cesáro Means of N -multiple
Trigonometric Fourier Series
Ushangi Goginava
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Since
ω (δ, f )C ≤
N
X
ωi (δ, f )C ,
i=1
we obtain the proof of estimation (2).
N
1
1 ,...,−αN
Next we shall prove that σn−α
(f
)
diverge
in
the
metric
of
L
[0,
2π]
.
0
k ,...,nk
It is clear that
(7)
1 ,...,−αN
σn−α
(f0 ) − f0
k ,...,nk
1
1 ,...,−αN
≥ σn−α
(fk )
k ,...,nk
−
−
k−1
X
Cesáro Means of N -multiple
Trigonometric Fourier Series
1
1 ,...,−αN
(fj ) − fj
σn−α
k ,...,nk
j=1
∞
X
1 ,...,−αN
σn−α
(fj )
k ,...,nk
C
j=k+1
∞
X
kfj kC
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It is evident that
1 ,...,−αN
σn−α
(fj ) = 0,
k ,...,nk
−
j=k
= I − II − III − IV.
(8)
Ushangi Goginava
C
j = k + 1, k + 2, . . . .
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Using (3) for IV we have
(9)
IV ≤
∞
X
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1
nα1 +···+αN
j=k j
=O
1
nαk 1 +···+αN
.
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Since [2]
1 ,...,−αN
σn−α
(fj ) − fj
k ,...,nk
C
X
=O
ωB
B⊂M
1
, fj
nk
and
ωi
1
, fj
nk
1
=O
nαj 1 +···+αN
nj
nk
P
αs
!
ns∈B
k
C
!
,
from (4) and (5) we get
(10)
II = O
=O
1
k−1
X
1−(α +···+αN )
nk 1
j=1
1
k−2
X
1−(α +···+αN )
nk 1
j=1
Cesáro Means of N -multiple
Trigonometric Fourier Series
!
1−(α1 +···+αN )
nj
Ushangi Goginava
1−(α1 +···+αN )
1−(α1 +···+αN )
nj
+
nk−1
1−(α1 +···+αN )
nk
!
=O
1−(α +···+αN )
nk−1 1
1−(α +···+αN )
nk 1
1−(α1 +···+αN ) !
1
k
=O
!
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= o (1)
as
k → ∞.
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Since
(B)
ai1 ,...,iN (fk ) = 0,
and
(M )
ai1 ,...,iN (fk ) =
for
B ⊂ M, B 6= M
 −α1 −···−αN
, for
 nk
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i1 = · · · = iN = n k ;
J. Ineq. Pure and Appl. Math. 7(3) Art. 86, 2006

0,
otherwise,
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from (1) we have
1 ,...,−αN
(11) σn−α
(fk )
k ,...,nk
Z 2π
Z 2π 1
1 ,...,−αN
=
···
σn−α
(fk ; x1 , . . . , xN ) dx1 · · · dxN
k ,...,nk
0
0
2π
Z
≥
Z
···
0
=
2π
0
1
1
A−α
nk
Z
×
···
=π
N
(fk ; x1 , . . . , xN )
N
Y
sin nk xi dx1 · · · dxN
i=1
nk
X
1
nk Y
N
X
1 −1
A−α
nk− ij
···
N
A−α
nk
i1 =0
iN =0 j=1
Z
N
2π
2π
Y
···
Si1 ,...,iN (fk ; x1 , . . . , xN )
sin nk xi dx1 · · · dxN
0
= πN
1 ,...,−αN
σn−α
k ,...,nk
1
1
A−α
nk
1
1
A−α
nk
0
···
···
i=1
1
N
A−α
nk
1
N
A−α
nk
)
a(M
nk ,...,nk (fk )
1 −···−αN
n−α
k
≥ c (α1 , . . . , αN ) > 0.
Combining (7) – (11) we complete the proof of Theorem 1.
Cesáro Means of N -multiple
Trigonometric Fourier Series
Ushangi Goginava
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References
[1] U. GOGINAVA, Cesáro means of trigonometric Fourier series, Georg.
Math. J., 9 (2002), 53–56.
[2] L.V. ZHIZHIASHVILI, Trigonometric Fourier Series and their Conjugates,
Kluwer Academic Publishers, Dobrecht, Boston, London, 1996.
[3] A. ZYGMUND, Sur la sommabilite des series de Fourier des functions
verfiant la condition de Lipshitz, Bull. de Acad. Sci. Ser. Math. Astronom.
Phys., (1925), 1–9.
[4] A. ZYGMUND, Trigonometric Series, Vol. 1, Cambridge Univ. Press,
1959.
Cesáro Means of N -multiple
Trigonometric Fourier Series
Ushangi Goginava
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