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GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 4, 1996, 379-400
ESTIMATION OF THE INTEGRAL MODULUS OF
SMOOTHNESS OF AN EVEN FUNCTION OF SEVERAL
VARIABLES WITH QUASICONVEX FOURIER
COEFFICIENTS
T. TEVZADZE
Abstract. The estimate of the modulus of smoothness of an even
function of several variables with quasiconvex Fourier coefficients obtained in this paper extends one result of S. A. Telyakovski.
1. Let (ai,j )i,j≥0 be a double numerical sequence. Denote by ∆2 ai,j the
expression ∆12 (∆12 ai,j ), where
∆12 ai,j = ∆1 (∆2 ai,j ) = ∆1 (ai,j − ai,j+1 ) = ai,j − ai+1,j − ai,j+1 + ai+1,j+1 .
Definition 1. The double sequence (ai,j )i,j≥0 will be called quasiconvex
if the series
∞ X
∞
X
(i + 1)(j + 1)|∆2 ai,j | + (i + 1)|∆1 (∆12 aij )| + (j + 1)|∆2 (∆12 aij )|
i=0 j=0
converges.
As can be easily shown, if the sequence (ai,j )i,j≥0 is quasiconvex and
lim ai,j = 0,
i+j→∞
then the series
∞ X
∞
X
λi,j ai,j cos ix cos jy,
(∗)
i=0 j=0
where λ0,0 = 14 , λ0,j = λi,0 = 21 , i, j = 1, 2, . . . , λi,j = 1 for i, j > 0,
converges on (0, 2π)2 to some function f ∈ L(T 2 ) and is its Fourier series
with T 2 = [0, 2π]2 .
1991 Mathematics Subject Classification. 42B05.
Key words and phrases. Integral modulus of smoothness, quasiconvex Fourier
coefficients.
379
c 1996 Plenum Publishing Corporation
1072-947X/96/0700-0379$09.50/0
380
T. TEVZADZE
The expression
ωm,n (δ, ρ; f )1 =
Z X
n
m X
m n
f [x + (m − 2µ)h, y + (n − 2ν)η]dx dy
(−1)µ+ν
= sup
ν
µ
|h|≤δ
µ=0 ν=0
2
|η|≤ρT
is usually called an integral modulus of smoothness of order (m, n) of a
function f ∈ L(T 2 ). Here and in what follows it is assumed that δ ∈ [0, π],
ρ ∈ [0, π].
Aljanˇciˇc and Tomiˇc [1], [2] considered an estimate of the integral modulus
of continuity in terms of Fourier coefficients of the function f for some classes
of sequences in the one-dimensional case. In 1963 M. and S. Izumi [3] proved
that if the Fourier coefficients an → 0, n → ∞, form a quasiconvex sequence,
then the estimate
X
X
i2 |∆21 ai | +
i|∆12 ai |1
ω(δ; f )1 ≤ cδ
i≤ δ1
i> δ1
holds for the integral modulus of continuity ω(δ; f )1 .
This result was later generalized by Telyakovski [4] who proved a theorem
giving an estimate of an integral modulus of smoothness of order m, m ∈ N.
In this paper an estimate is obtained for the integral modulus of smoothness for a function of several variables which is even with respect to each
variable.
2.
For a further discussion we need
Lemma 1. Let 2mh ≤ t ≤ π. Then
m
X
i m
m
Kp [t + (m − 2i)h] ≤ c(m)hm pm−1 t−2 ,
(−1)
|Tp | =
i
i=0
|Tpm | ≤ c(m)hm pm+1 ,
where Kp (t) is the Fejer kernel, p ∈ N.
Proof. It is easy to show that
m
X
m
Tpm =
(−1)i
Kp [t + (m − 2i)h] =
i
i=0
m−1
X
m−1
(−1)i
Kp [t + (m−2i)h]−Kp [t+(m−2(i+1))h] =
=
i
i=0
1 Here
and in what follows, c, c(m), c(m, n), . . . denote, generally speaking, various
positive constants depending only on the parameters indicated in the brackets.
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
m−1
X
381
m−1
Kp′ [t + (m − 2(i + 1 − Q1 ))h] = · · · =
(−1)
= −2h
i
i=0
i
= (−1)m hm Kp(m) (t + c(m)h),
where c(m) = m − 2(m − Q1 − · · · − Qm ), 0 < Qi < 1, i = 1, . . . , m.
Since −m < c(m) < m, we have
p
Kp(m) (t
1 X (m)
+ c(m)h) =
D (t + c(m)h) =
p + 1 i=0 i
(
p
i
1 X X m cos j(t + c(m)h),
=±
j
p + 1 i=0 j=1
sin j(t + c(m)h),
2|m,
2 ∤ m,
where Di (t) is the Dirichlet kernel.
Let 2|m (the case 2 ∤ m is considered similarly). Then applying twice the
Abel transformation and taking into account the estimates
c
Kp (t) ≤ 2 , 0 < |t| ≤ π,
pt
Kp (t) ≤ p, |t| ≤ π,
we find
p
i
1 X X m
j − 2(j + 1)m +
p + 1 i=0 j=1
+ (j + 2)m jKj (t + c(m)h) +
Kp(m) (t + c(m)h) =
+
+
p
1 X
(i − 1)m − im iKi (t + c(m)h) +
p + 1 i=0
p
1 X m
i − (i + 1)m iKi (t + c(m)h) +
p + 1 i=0
1
· pm+1 Kp (t + c(m)h) ≤
p+1
(
c(m)pm−1 t−2 , 2mh ≤ t ≤ π,
≤
c(m)pm+1 ,
−π ≤ t ≤ π.
+
Therefore
|Tpm | ≤ c(m)hm pm−1 t−2 ,
|Tpm |
m m+1
≤ c(m)h p
,
2mh ≤ t ≤ π,
−π ≤ t ≤ π.
The two-dimensional analogue of Lemma 1 given below is proved similarly.
382
T. TEVZADZE
Lemma 2. Let 2mh ≤ x ≤ π, 2nη ≤ y ≤ π. Then
m,n
|Tp,q
|
m X
n
X
n
µ+ν m
(−1)
Kp [x+(m−2µ)h]Kq [y+(n−2ν)η] ≤
=
µ
ν
µ=0 ν=0
≤ c(m, n)
hm η n pm−1 q n−1
,
(xy)2
m,n
| ≤ c(m, n)hm η n pm+1 q n+1 ,
|Tp,q
(x, y) ∈ T 2 .
Lemma 3 ([5]). Let 2mh ≤ x ≤ π. Then
m
X
m
Dp [x + (m − 2i)h] ≤ c(m)hm pm x−1 ,
(−1)i
i
i=0
where Dp is the Dirichlet kernel, p ∈ N.
Theorem 1. Let a double sequence (ai,j )i,j≥0 be quasiconvex and
lim ai,j = 0.
i+j→∞
Then for the sum f of the series (∗) we have
1
1
]
[ 2nη
[ 2mh
X] X
m n
ωm,n (h, η; f )1 ≤ c(m, n) h η
im+1 j n+1 |∆2 ai,j | +
i=1
1
]
[ 2mh
X
+ hm
∞
X
j=1
im+1 j|∆2 ai,j | +
i=1 j=[ 1 ]+1
2nη
+η
∞
X
n
1
[ 2nη
]
X
ij n+1 |∆2 ai,j | +
1
]+1 j=1
i=[ 2mh
∞
X
+
∞
X
1
1
]+1 j=[ 2nη
]+1
i=[ 2mh
4
X
=
ij|∆2 ai,j | =
Ps (m, n, h, η).
s=1
Proof. We write
A(m, n, f ) = 4
Zπ Zπ
0
0
m,n
|∆
2mh
Z Zπ
Z Z2nη Zπ Z2nη 2mh
+
+
+
f | dx dy = 4
0
0
2mh 0
0
2nη
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
383
Zπ Zπ
4
X
+
As (m, n, f ),
|∆m,n f | dx dy ≡
s=1
2mh 2nη
where
∆m,n f =
m X
n
X
(−1)µ+ν
µ=0 ν=0
=
m X
n
X
µ+ν
(−1)
µ=0 ν=0
∞
X
+
4
X
1
1
1
[ y1 ]
X
[x
] [y]
]
[x
∞
∞
X
X
X
X
X
m n
+
+
+
ν
µ
1
1
j=1
i=1 j=1
i=1
i=[ x ]+1
∞
X
1
]+1 j=[ y1 ]+1
i=[ x
≡
m n
f [x + (m − 2µ)h, y + (n − 2ν)η] =
µ
ν
j=[ y ]+1
ai,j cos i[x + (m − 2µ)h] cos[y + (n − 2ν)η] ≡
Ts (m, n).
s=1
It is easy to show that
[1]
1
T1 ≡ T1 (m, n) = 2
m+n
[x] y
X
X
ai,j sinm ih sinn jη ×
i=1 j=1
Further,
Zπ Zπ
m+n
(−1) 2 cos ix cos jy,
(−1) m+n−1
2
sin ix cos jy,
×
m+n−1
2
(−1)
cos ix sin jy,
m+n−2
(−1) 2 sin ix sin jy,
m n
|T1 |dx dy ≤ c(m, n)h η
2|m, 2|n,
2 ∤ m, 2|n,
2|m, 2 ∤ n,
2 ∤ |m, 2 ∤ n.
1
1
] [y]
[x
Zπ Zπ X
X
im j n |ai,j |dx dy ≤
2mh 2nη i=1 j=1
2mh 2nη
m n
≤ c(m, n)h η
2mh
Z Z2nη
(xy)
1
]
[ 2mh
≤ c(m, n)hm η n
1
[ 2mh
]
m n
≤ c(m, n)h η
1
[ 2nη
]
X X
r=1
(rs)−2
s=1
j=1
im j n |ai,j |dx dy ≤
s
r X
X
im j n |ai,j | ≤
i=1 j=1
1
]
[ 2nη
X X
i=1
[x] [y]
X
X
i=1 j=1
1
1
−2
1
1
] [ 2nη ]
[ 2mh
m n
i j |ai,j |
X X
r=i
s=j
(rs)−2 ≤
384
T. TEVZADZE
1
1
[ 2mh
] [ 2nη ]
X X
m n
≤ c(m, n)h η
im−1 j n−1 |ai,j |.
(2.1)
j=1
i=1
1
1
], q = [ 2nη
], which
Applying twice the Hardy transformation [6], p = [ 2mh
is a two-dimensional analog of the Abel transformation, we find
p X
q
X
m−1 n−1
j
i
p
X
|ai,j | =
i=1 j=1
+
q
X
i
i=1
n−1
|ai,q |
s
s=1
=
q−1
X
j
p
X
i=1
≤
q−1
X
j
n
j=1
n
X
p−1
i=1
j=1
+
m−1
j=1
X
p
i
|∆2 ai,j | +
p
X
i
|∆2 ai,j | − |∆2 ai+1,j |
i
X
q |ai,q | =
m−1 n
i=1
rm−1 +
r=1
r=1
i=1
+pm
p
X
im−1 ≤
q−1
X
p−1 X
q−1
X
|∆12 ai,j |im j n +
i=1 j=1
j n |∆2 ap,j | + q n
j=1
p−1
X
im |∆1 ai,q | + pm q n |ap,q | ≤
i=1
im+1 j n+1 |∆2 ai,j | + q n+1
p−2
X
im+1 |∆12 ai,q − ∆12 ai+1,q | +
i=1
i=1 j=1
+pm+1
s=1
i=1
i=1
≤
m−1
j
X
sn−1 +
|ai,j | − |ai,j+1 |
p−1
i
X
X
m−1
|ai,q | − |ai+1,q |
rm−1 +
i
|∆2 ap,j | + q n
+q n |ap,q |
p−1 X
q−1
X
X
q−1
q−2
X
j n+1 |∆12 ap,j − ∆12 ap,j+1 | + pm+1 q n+1 |∆12 ap,q | +
j=1
+q n
p−2
X
im+1 |∆1 (ai,q − ai+1,q )| + pm
q−2
X
j n+1 |∆2 (ap,j − ap,j+1 )| +
j=1
i=1
n m+1
+q p
m n+1
|∆1 ap,q | + p q
≡
9
X
|∆2 ap,q | + pm q n |ap,q | ≡
Iα (m, n, p, q).
(2.2)
α=1
For I2 we obtain
I2 = q n
p−2
X
i=1
im+1 q|∆1 (∆12 ai,q )| = q n
p−2
X
i=1
∞
X 2
∆ ai,j ≤
im+1 q
j=q
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
≤q
n
p−2
X
im+1
i=1
∞
X
j|∆2 ai,j |.
385
(2.3)
j=q
Similarly,
I3 ≤ pm
q−2
∞ X
X
ij n+1 |∆2 ai,j |.
(2.4)
i=p j=1
Let us now estimate I5 . We have
I5 ≤ q n
p−2
X
im+1 |∆21 ai,q |,
i=1
where
∆21 ai,q = ai,q − ai+1,q + ai+2,q .
It is easy to show that
∆21 ai,q =
∞
X
∆2 (∆21 ai,s ) =
s=q
∞
X
∆2
∞
X
s=j
j=q
∞
∞ X
X
∆2 (∆2 ai,s ) =
∆2 ai,s .
j=q s=j
Therefore
|∆21 ai,q | ≤
∞
∞ X
X
|∆2 ai,s | ≤
∞
X
s|∆2 ai,s |.
(2.5)
s=q
j=q s=j
Thus
I5 ≤ q n
p−2 X
∞
X
im+1 s|∆2 ai,s |.
(2.6)
i=1 s=q
Similarly,
I6 ≤ pm
q−2
∞ X
X
ij n+1 |∆2 ai,j |.
(2.7)
i=p j=1
Next,
I7 = pm q n · p|∆1 ap,q |.
Since
∞
∞
X
X
i|∆21 ai,q |,
p|∆1 ap,q | = p
∆21 ai,q ≤
i=p
i=p
by repeating the arguments used for I5 we obtain
I7 ≤ pm q n
∞
∞ X
X
i=p j=q
ij|∆2 ai,j |.
(2.8)
386
T. TEVZADZE
The same estimate holds for I8 . For I4 we have
I4 ≤ pm q n
∞ X
∞
X
ij|∆2 ai,j |.
(2.9)
i=p j=q
In the same way,
∞
∞ X
∞
∞ X
∞ X
∞ X
X
X
rs|∆2 ar,s |. (2.10)
∆2 ar,s ≤ pm q n
I9 ≤ pm q n
r=p s=q
i=p j=q r=i s=j
Using (2.2)–(2.10), we find
Zπ Zπ
|T1 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.11)
α=1
2mh 2nη
If p = [ x1 ]+ 1, q = [ y1 ]+ 1, we again apply twice the Hardy transformation
for T4 and obtain
T4 =
m X
n
X
(−1)µ+ν
µ=0 ν=0
n
m n
ap,q Dp [x+(m−2µ)h]Dq [y+(n−2ν)η]−
µ
ν
− Dp [x + (m − 2µ)h]∆2 ap,q (q + 1)Kq [y + (n − 2ν)η] −
− Dq [y + (n − 2ν)η]∆1 ap,q (p + 1)Kp [x + (m − 2µ)h] +
+ Dp [x + (m − 2µ)h]
∞
X
∆22 ap,j (j + 1)Kj [y + (n − 2ν)η] +
j=q
+ Dq [y + (n − 2ν)η]
∞
X
∆12 ai,q (i + 1)Ki [x + (m − 2µ)h] +
i=p
+
∆212 ap,q (p
+ 1)(q + 1)Kp [x + (m − 2µ)h]Kq [y + (n − 2ν)η] −
− (p + 1)Kp [x+(m − 2µ)h]
− (q + 1)Kq [y+(n − 2ν)η]
∞
X
∆2 (∆12 ap,j )(j + 1)Kj [y+(n − 2ν)η] −
j=q
∞
X
∆1 (∆12 ai,q )(i + 1)Ki [x+(m − 2µ)h] +
i=p
+
∞ X
∞
X
i=p j=q
=
9
X
α=1
∆2 ai,j (i + 1)(j + 1)Ki [x+(m − 2µ)h]Kj [y+(n − 2ν)η] =
(α)
T4
(m, n, p, q).
(2.12)
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
387
By Lemma 2 (see (2.1)) we have
Zπ Zπ
1
1
2nη
2mh Z
Z
(1)
|T4 | dx dy ≤ c(m, n)hm η n
1
2mh 2nη
|a[x],[y] |xm−1 y n−1 dx dy ≤
1
1
1
[ 2mh
] [ 2nη ]
X X
m n
≤ c(m, n)h η
j=1
i=1
≤
4
X
im−1 j n−1 |ai,j | dx dy ≤
(2.13)
Ps (m, n, h, η).
s=1
(2)
Applying Lemmas 3 and 1 for T4
Zπ Zπ
we find
1
1
(2)
|T4 | dx dy ≤ c(m, n)hm η n
2mh 2nη
2nη
2mh Z
Z
1
xm−1 y n |∆2 a[x]+1,[y]+1 | dx dy ≤
1
1
1
[ 2mh
] [ 2nη ]
≤ c(m, n)hm η n
X X
i=1
im−1 j n |∆2 ai,j |.
j=1
Using the Abel transformation leads to
Zπ Zπ
(2)
|T4 | dx dy
1
]
[ 2nη
m n
≤ c(m, n) h η
2mh 2nη
X
+h η
X
j
j=1
1
[ 2mh
]
m −1
1
[ 2mh
]
n+1
i
m−1
i=1
X
im−1 |∆22 ai,j | +
i=1
1
|∆2 ai,[ 2nη
]| .
Again applying twice the Abel transformation we obtain
Zπ Zπ
(2)
|T4 | dx dy
1
1
]
[ 2nη
[ 2mh
X] X
m n
≤ c(m, n) h η
im+1 j n+1 |∆2 ai,j | +
i=1
2mh 2nη
1
[ 2nη
]
+η n
X
j=1
1
[ 2nη
]
+η
n
X
j=1
j=1
∞
X
∆1 ∆1 (∆22 ai,j ) +
j n+1 h−1
1
]
i=[ 2mh
∞
∞
X
X
j n+1
∆1 ∆1 (∆22 ai,j ) +
1
r=[ 2mh
] i=r
388
T. TEVZADZE
1
]
[ 2mh
X
m
+h
i=1
∞
∞
X 2
X
1
∆2 a[ 2mh
∆2 ai,j + η −1
im+1 η −1
],j ≤
j=q
1
]
j=[ 2nη
1
1
]
[ 2nη
[ 2mh
X] X
m n
im+1 j n+1 |∆2 ai,j | +
≤ c(m, n) h η
i=1
1
]
[ 2nη
+η
n
X
j
j=1
2
i|∆ ai,j | + η
n
1
i=[ 2mh
]+1
1
[ 2mh
]
+hm
1
[ 2nη
]
∞
X
n+1
j=1
∞
X
X
im+1 j|∆2 ai,j | +
i=1 j=[ 1 ]+1
2nη
X
∞
X
j n+1
j=1
1
i=[ 2mh
]+1
∞
X
∞
X
1
1
i=[ 2mh
]+1 j=[ 2nη
]+1
≤
4
X
i|∆2 ai,j | +
ij|∆2 ai,j | ≤
Ps (m, n, h, η).
(2.14)
s=1
(3)
One can estimate T4
quite similarly.
(4)
Let us now consider T4 . We have
(4)
T4
=
m
X
(−1)µ
µ=0
×
1
[X
2nη ]
n
X
m
n
D[ x1 ] [x + (m − 2µ)h]
(−1)ν
×
µ
ν
ν=0
+
j=[ y1 ]+1
∞
X
1
j=[ 2nη
]+1
(j + 1)∆22 a[ x1 ],j Kj [y + (n − 2ν)η] =
(4)
(4)
= T4 (1) + T4 (2).
(2.15)
Since
1
[ 2nη
]
(4)
|T4 (1)|
X
m n −m−1 −2
≤ c(m, n)h η x
y
j n |∆22 a[ x1 ],j |,
j=[ y1 ]+1
we obtain
Zπ Zπ
1
1
] [ 2nη
]
[ 2nη
1
[ 2mh
]
(4)
|T4 (1)| dx dy
m n
≤ c(m, n)h η
X
m−1
i
1
]
[ 2nη
m n
≤ c(m, n)h η
X
j=1
j n |∆22 ai,j | ≤
s=1 j=s+1
i=1
2mh 2nη
X X
1
]
[ 2mh
j
n+1
X
i=1
im−1 |∆22 ai,j |.
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
389
(2)
Repeating the arguments used in estimating T4 , we find
Zπ Zπ
(4)
|T4 (1)| dx dy ≤ P1 (m, n, h, η) + P3 (m, n, h, η).
(2.16)
2mh 2nη
Similarly,
Zπ Zπ
1
[ 2mh
]
(4)
|T4 (2)| dx dy
m −1
≤ c(m, n)h η
X
m−1
i
i=1
2mh 2nη
≤
4
X
∞
X
|∆22 ai,j | ≤
1
]+1
j=[ 2nη
Ps (m, n, h, η).
(2.17)
s=1
By (2.16) and (2.17) we conclude that
Zπ Zπ
(4)
|T4 | dx dy ≤
4
X
Ps (m, n, h, η).
(2.18)
4
X
Ps (m, n, h, η).
(2.19)
s=1
2mh 2nη
Quite similarly we obtain
Zπ Zπ
(5)
|T4 | dx dy ≤
s=1
2mh 2nη
By Lemma 1 and analysis of the arguments used in estimating I2 we
have
Zπ Zπ
1
1
] [ 2nη ]
[ 2mh
(6)
|T4 | dx dy
m n
≤ c(m, n)h η
X X
i=1
2mh 2nη
1
]
[ 2mh
m n
≤ c(m, n)h η
j=1
1
]
[ 2nη
X X
i=1
im j n |∆12 ai,j | ≤
im+1 j n+1 |∆2 ai,j | +
j=1
1
]
[ 2nη
+h−m−1
X
1
j n+1 |∆2 (∆12 a[ 2mh
],j )| +
j=1
1
[ 2mh
]
+η −n−1
X
−m−1 −n−1
1
1
1
η
|∆12 a[ 2mh
im+1 |∆1 (∆12 ai,[ 2nη
]| + h
],[ 2nη
]| ≤
i=1
1
1
]
[ 2nη
[ 2mh
X] X
m n
≤ c(m, n) h η
im+1 j n+1 |∆2 ai,j | +
i=1
j=1
390
T. TEVZADZE
∞
X
+hm η −1
1
im+1 |∆1 (∆12 ai,[ 2nη
] )| +
1
]+1
i=[ 2mh
1
[ 2nη
]
+h−1 η n
X
1
j n+1 |∆2 (∆12 a[ 2mh
],j )| +
j=1
4
X
1
1
+(hη)−1 |∆12 a[ 2mh
], 2nη
] | ≤≤
Ps (m, n, h, η).
(2.20)
s=1
(8)
For T4
Zπ Zπ
2mh 2nη
we obtain
1
1
1
2nη
2mh Z
[ 2mh
Z
X]
(8)
n
m n
im |∆1 (∆12 ai,[y] )|dxdy+
y
|T4 | dx dy ≤ c(m, n)h η
1
1
2nη
+c(m, n)η
n
Z
y
n
1
i=[x]+1
1
π+mh
Z
∞
X
x−2
|∆1 (∆12 ai,[y] )|dxdy ≤
1
]+1
i=[ 2mh
mh
1
1
1
[ 2nη
]
[ 2mh
[ 2mh
X] X
X]
m n
n
≤ c(m, n) h η
im |∆1 (∆12 ai,j )| +
j
r=1
1
[ 2nη
]
+ ηn
X
1
[ mh
]
jn
∞
X
X
i=r+1
j=1
|∆1 (∆12 ai,j )|
r=1 i=[ 1 ]+1
2mh
j=1
≤
1
1
[ 2nη
]
[ 2mh
X
X]
m n
n
≤ c(m, n) h η
j
im+1 |∆1 (∆12 ai,j )| +
r=1
j=1
+η
n
∞
X
1
[ 2nη
]
X
n
ij |∆1 (∆12 ai,j )|
1
]+1 j=1
i=[ 2mh
≤ c(m, n) hm η n
+η
n
∞
X
1
i=[ 2mh
]+1
X
im+1
r=1
1
[ 2nη
]
i
X
j=1
=
1
[ 2nη
]
1
[ 2mh
]
X
j n |∆1 (∆12 ai,j )| +
j=1
j |∆1 (∆12 ai,j )| .
n
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
391
Using the Abel transformation leads to
1
[ 2nη
]
1
]
[ 2nη
X
n
j |∆1 (∆12 ai,j )| ≤
j=1
X
1
j n+1 |∆2 ai,j | + η −n−1 |∆1 (∆12 ai,[ 2nη
] )|.
j=1
Hence (see (2.8)) it is easy to show that
Zπ Zπ
(8)
|T4 | dx dy
Zπ Zπ
|T4 | dx dy ≤
≤
4
X
Pα (m, n, h, η).
(2.21)
4
X
Pα (m, n, h, η).
(2.22)
α=1
2mh 2nη
Similarly,
(7)
α=1
2mh 2nη
Further,
(9)
T4
1
]
[ 2nη
1
[ X
2mh ]
n
µ+ν m
=
(−1)
ν
µ
1
µ=0 ν=0
n
m X
X
1
[ 2mh
]
+
i=[ x ]+1 j=[ y1 ]+1
∞
X
X
X
∞
X
∞
X
∞
X
+
1
]
[ 2nη
X
+
1
i=[ 2mh
]+1 j=[ y1 ]
(i + 1)(j + 1) ×
1
1
1
1
i=[ x
]+1 j=[ 2nη
]+1 i=[ 2mh
]+1 j=[ 2nη
]+1
× Ki [x + (m − 2µ)h]Kj [y + (n − 2ν)η] =
4
X
(9)
T4 (s).
(2.23)
s=1
By Lemma 2 we have
(9)
|T4 (1)|
m n
≤ c(m, n)h η
1
[ 2mh
]
1
[ 2nη
]
X
X
im j n (xy)−2 |∆2 ai,j |.
1
]+1 j=[ y1 ]+1
i=[ x
Therefore
Zπ Zπ
1
1
1
1
[ 2mh
] [ 2nη ] [ 2mh
] [ 2nη ]
(9)
|T4 (1)|dxdy
X X X X
m n
≤ c(m, n)h η
r=1
2mh 2nη
s=1
i=r
im j n |∆2 ai,j | ≤
j=s
1
1
[ 2mh
] [ 2nη ]
m n
≤ c(m, n)h η
X X
i=1
j=1
im+1 j n+1 |∆2 ai,j |.
(2.24)
392
T. TEVZADZE
The remaining terms of (2.22) are estimated by similar arguments. Thus
we conclude that
Zπ Zπ
(9)
|T4 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.25)
α=1
2mh 2nη
Taking into account (2.13), (2.14), (2.18)–(2.22), and (2.25) we obtain
Zπ Zπ
|T4 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.26)
α=1
2mh 2nη
For T2 we have
[1]
|T2 | ≤ c(n)η
n
y
X
j=1
X
X
∞
m
µ m
(−1)
j
ai,j cos i[x + (m − 2µ)h].
µ
n
µ=0
1
i=[ x
]+1
Applying twice the Abel transformation, we find
X
∞
m
(−1)
ai,j cos i[x + (m − 2µ)h] =
µ
1
µ=0
m
X
µ
i=[ x ]+1
X
∞
m
(i + 1)∆21 ai,j Ki [x + (m − 2µ)h] −
=
(−1)µ
µ
1
µ=0
m
X
−∆1 a[ x1 ],j
i=[ x ]+1
h1i
x
K[ x1 ] [x + (m − 2µ)h] − a[ x1 ],j D[ x1 ] [x + (m − 2µ)h] .
By virtue of Lemmas 3 and 1 we obtain
m n
|T2 | ≤ c(m, n)h η
[ y1 ]
X
j
n
j=1
+x
−m−2
+ c(n)η
im |∆21 ai,j |x−2 +
1
i=[ x
]+1
−m−1
|∆1 a[ x1 ],j | + |a[ x1 ],j |x
[1]
n
1
[X
2mh ]
y
X
j=1
j
n
X
∞
i|∆12 ai,j | ×
1
]
i=[ 2mh
m
X
m
Ki [x + (m − 2µ)h].
(−1)µ
×
µ
µ=0
+
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
Hence
σm,n ≡
Zπ Zπ
|T2 | dx dy ≤
2mh 2nη
Zπ
≤ c(m, n)hm η n
x−2
m n
+ c(m, n)h η
−m−2
x
Zπ
+ c(m, n)h η
−m−1
x
[ y1 ]
+ c(n)η
im |∆21 ai,j | dx dy +
1
i=[ x
]+1
[ y1 ]
Zπ X
j n |∆1 a[ x1 ],j | dx dy +
[ y1 ]
Zπ X
j n |a[ x1 ],j | dx dy +
2nη j=1
2mh
n
X
jn
2nη j=1
2mh
m n
1
]
[ 2mh
2nη j=1
2mh
Zπ
[ y1 ]
Zπ X
Zπ Zπ X
∞
X
jn
2mh 2nη j=1
i|∆21 ai,j | ×
1
i=[ 2mh
]+1
m
X
m
Ki [x + (m − 2µ)h] dx dy ≤
×
(−1)µ
µ
µ=0
1
1
m n
≤ c(m, n)h η
2nη
2mh Z
Z
1
1
2nη
1
2mh
+
Z
1
xm
Z
Z
1
2nη
xm−1
1
Z
y −2
1
[ 2mh
]
j
n
X
im |∆21 ai,j | dx dy +
i=[x]+1
j n |∆1 a[x],j | dx dy +
[y]
X
j n |a[x],j | dx dy
j=1
1
2nη
2mh Z
Z
1
+
1
1
+ c(m, n)η n
j=1
1
[y]
X
[y]
X
j=1
1
1
2mh
+
y −2
y
−2
y −2
[y]
X
jn
j=1
1
∞
X
|∆21 ai,j | dx dy.
1
]+1
i=[ 2mh
Next, as is easy to show,
σm,n
1
1
1
]
[ 2nη
[ 2mh
[ 2mh
s
X] X
X]
X
m n
−2
n
≤ c(m, n) h η
im |∆21 ai,j | +
s
j
r=1
s=1
j=1
i=r+1
393
394
T. TEVZADZE
1
]
[ 2nη
1
[ 2mh
]
m n
+h η
r
X
rm−1
X
s−2
m
r=1
s
X
X
s−2
s=1
1
[ 2nη
]
s=1
s
X
j n |∆1 ar,j | +
j=1
1
[ 2nη
]
1
[ 2mh
]
+ η n h−1
s−2
s=1
r=1
+ hm η n
X
X
j n |ar,j | +
j=1
∞
X
jn
j=1
s
X
1
i=[ 2mh
]+1
|∆21 ai,j | .
Therefore, after simple calculations, we find
σm,n ≤ c(m, n)hm η n
1
[X
2mh ]
1
]
[ 2nη
im+1
+
1
[ 2nη
]
X
X
r
m
r=1
j n−1 |∆12 ai,j | +
j=1
i=1
1
[ 2mh
]
X
1
1
] [ 2nη ]
[ 2mh
j
n−1
|∆1 ar,j | +
X X
r=1
j=1
+cm,n η n
∞
X
r
m−1 n−1
j
j=1
|ar,j | +
1
[ 2nη
]
X
i
j n−1 |∆21 ai,j |.
j=1
1
]+1
i=[ 2mh
The analysis of estimates (2.14) and (2.18) gives
Zπ Zπ
|T2 | dx dy ≤
Zπ Zπ
|T3 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.27)
4
X
Pα (m, n, h, η).
(2.28)
α=1
2mh 2nη
Similarly,
2mh 2nη
α=1
Therefore by virtue of (2.11), (2.26), (2.27) and (2.28) we obtain
A4 (m, n; f ) ≤
4
X
Pα (m, n, h, η).
α=1
For A1 (m, n; f ) we have
A1 (m, n; f ) =
2mh
Z Z2nη
|∆m,n f | dx dy =
0
0
(2.29)
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
=
2mh
Z Z2nη X
s
r X
0
0
∞
X
+
+
+
i=r+1 j=1
i=1 j=1
∞
X
∞ X
s
X
r
∞
X
X
+
i=1 j=s+1
(i + 1)(j + 1)∆2 ai,j ×
i=r+1 j=s+1
n
m X
X
m n
Ki [x + (m − 2µ)h] ×
µ
ν
µ=0 ν=0
4
X
(α)
×Kj [y+(n−2ν)η]dx dy =
A1 (m, n, h, η),
×
395
(−1)µ+ν
(2.30)
α=1
1
1
], s = [ 2nη
]. By Lemma 2 we obtain
where r = [ 2mh
(1)
A1
2mh
Z Z2nηX
s
r X
m n
≤ c(m, n)h η
i=1 j=1
0
0
1
[ 2mh
]
1
[ 2nη
]
X X
≤ c(m, n)hm η n
i=1
im+2 j n+2 |∆2 ai,j |dx dy ≤
im+1 j n+1 |∆2 ai,j |.
(2.31)
j=1
Further,
(4)
A1
=
2mh
Z Z2nη
0
0
∞
X
∞
X
i=r+1 j=s+1
X X
+
(i + 1)(j + 1)∆2 ai,j
m>2µ n>2ν
X X
n
µ+ν m
×
(−1)
+
+
+
µ
ν
m>2µ n≤2ν
m≤2µ n>2ν
m≤2µ n≤2ν
4
X
(4)
× Ki [x + (m − 2µ)h]Kj [y + (n − 2ν)η] dx dy =
A1 (α).
X X
X X
α=1
(4)
is sufficient to estimate A1 (4). For
n
µ ([ m
2 ] ≤ µ ≤ m) and ν ([ 2 ] ≤ ν ≤ n)
the term of this sum
Obviously, it
(4)
A1;µ,ν (4), where
are fixed, a = 2µ−m,
b = 2ν − n, we obtain, after passing to the variables t = x − ah, τ = y − bη,
(4)
Ai;µ,ν (4) ≤ c(m, n)
∞
X
∞
X
1
1
i=[ 2mh
]+1 j=[ 2nη
]+1
|∆2 ai,j |
3
3 X
X
r=1 s=1
where
Fr,s =
Z Z
ur vs
(i + 1)(j + 1)Ki (t)Kj (τ ) dt dτ,
Fr,s ,
396
T. TEVZADZE
h 1 1i
h1
h
i
1i
u1 = − ah, − , u2 = − , , u3 = , (2m − a)h ,
i
i i
i
h 1 1i
h1
i
h
1i
v1 = − bη, − , v2 = − , , v3 = , (2n − b)η ,
j
j j
j
3
[
ur × vs .
[−ah, −(2m − a)h] × [−bη, (2n − b)η =
r,s=1
Taking into account the properties of Fejer kernels in the respective intervals,
we have
Z
Z Z
dt dτ
dτ
2 2 2
, F2,1 ≤ i j
,
F1,1 ≤ ij
ij(tτ )2
i
jτ 2
v1
u 1 v1
Z
Z Z
dt
dt dτ
2
,
,
F
≤
ij
F3,1 ≤ ij
1,2
2
ij(tτ )
j
it2
u1
u 3 v1
Z
4
2
dt
F2,2 ≤ i2 j 2 , F3,2 ≤ ij 2
,
ij
j
it2
u3
Z Z
Z
dτ
dt dτ
2 2
F1,3 ≤ ij
,
F
,
≤
i
j
2,3
ij(tτ )2
i
jτ 2
u1 v 3
v3
Z Z
dt dτ
F3,3 ≤ ij
.
ij(tτ )2
u3 v 3
Therefore
∞
X
(4)
A1 (4) ≤ c(m, n)
1
i=[ 2mh
]+1
(2)
∞
X
ij|∆2 ai,j |.
(2.32)
1
j=[ 2nη
]+1
(3)
A1 and A1 are estimated by the same scheme as used in deriving (2.31)
and (2.32). Therefore for A1 we have
A1 ≤
4
X
Ps (m, n, h, η).
(2.33)
s=1
It remains to consider A2 (m, n; f ) (A3 is estimated similarly). We have
1 [ 1 ]
] 2nη
[x
Zπ Z2nη
Zπ Z2nη X
∞
X
X
m,n
A2 =
|∆ f | dx dy =
+
2mh 0
1
]
[x
+
X
2mh 0
∞
X
i=0 j=[ 1 ]+1
2nη
+
∞
X
∞
X
1
1
]+1 j=[ 2nη
i=[ x
]+1
i=0 j=0
1
[ 2nη
]
X
i=[ r1 ]+1 j=0
(i + 1)(j + 1)∆2 ai,j ×
+
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
m X
n
X
m n
Ki [x + (m − 2µ)h] ×
µ
ν
µ=0 ν=0
4
X
(α)
× Kj [y + (n − 2ν)η] dx dy =
A2 (m, n, h, η).
×
397
(−1)µ+ν
(2.34)
α=1
Again, by Lemma 2 we obtain
(1)
A2
≤ c(m, n)hm η n
1 [ 1 ]
] 2nη
[x
Zπ Z2nηX
X
2mh 0
i=1 j=1
1
2mh
≤ c(m, n)hm η n
Z
x−2
1
]
[ 2mh
(2)
(2)
A2
X
im+2
j n+1 |∆2 ai,j |dx ≤
j=1
im+1 j n+1 |∆2 ai,j |.
(2.35)
j=1
we have
1
Zπ Z2nη [ X
2mh ]
+
=
2mh 0
n
m X
X
Further,
1
[ 2nη
]
1
]
[ 2nη
X X
i=1
For A2
[x]
X
i=1
1
≤ c(m, n)hm η n
im+2 j n+2 |∆2 ai,j |dx dy ≤
∞
X
1
i=[ 2mh
]+1
1
]+1
i=[ x
1
[X
2nη ]
(i + 1)(j + 1)|∆2 ai,j | ×
j=0
m n
(−1)µ+ν
×
Ki [x + (m − 2µ)h] ×
µ
ν
µ=0 ν=0
(2)
(2)
× Kj [y + (n − 2ν)η] dx dy = A2 (1) + A2 (2).
1
1
[ 2nη
]
(2)
A2 (1)
≤ c(m, n)hm η n
X
j n+1
j=1
≤ c(m, n)h η
X
j
1
[ 2mh
]
n+1
X
r=1
1
]
[ 2nη
X X
i=1
x−2
j=1
X
im+2 |∆2 ai,j |dx dy ≤
i=[x]+1
1
]
[ 2mh
1
]
[ 2mh
j=1
≤ c(m, n)hm η n
1
[ 2mh
]
1
1
[ 2nη
]
m n
2mh
Z
(2.36)
r
−2
X
im+2 |∆2 ai,j |dx ≤
i=r+1
im+1 j n+1 |∆2 ai,j |.
(2.37)
398
T. TEVZADZE
Similarly (see (2.32)) we find
∞
X
(2)
A2 (2) ≤ c(m, n)η n
1
]+1
i=[ 2mh
1
[ 2nη
]
X
ij n+1 |∆2 ai,j |.
(2.38)
j=1
By (2.35)–(2.37) we obtain
(2)
A2 ≤ P1 (m, n, h, η) + P3 (m, n, h, η).
(2.39)
It is likewise easy to show that
(3)
(4)
A2 ≤ P2 (m, n, h, η) + P4 (m, n, h, η), A2 ≤ P4 (m, n, h, η). (2.40)
Keeping in mind (2.33), (2.34), (2.38)–(2.40), we find
A2 ≤
4
X
Ps (m, n, h, η).
(2.41)
s=1
With regard to (2.29), (2.33), and (2.41) we conclude that
A≤
4
X
Ps (m, n, h, η).
s=1
3. Let now Rk be a k-dimensional Euclidean space of points x = (x1 ,. . . ,xk )
with ordinary linear operations and the Euclidean norm kxk. The product
of two vectors x, y ∈ Rk is understood componentwise and so are the
inequalities x < y, x ≤ y. It is assumed that T k = [0, 2π]k .
If n = (n1 , . . . , nk ) is a multi-index with non-negative integral components, 0 = (0, . . . , 0), 1 = (1, . . . , 1), and (an )n≥0 is a k-multiple numerical
sequence, then
∆i an = an − an+(δi1 ,...,δik ) ,
where δij is the Kronecker symbol, i, j ∈ M = {1, . . . , k}.
We introduce the notation
∆M an ≡ ∆1···k an = ∆1 (∆2 (· · · (∆k an ) · · · )),
for each B, ∅ 6= B ⊂ M , ∆B an is defined similarly, and assume that λ(n)
is the number of zero coordinates of the vector n (see [7]). We will also use
the notation B ′ = M \ B, where B ⊂ M .
Definition 2. A sequence (an )n≥0 will be called quasiconvex if the series
X
Y
X
(ni + 1)|∆B (∆M an )|
n≥0 B⊂M, B6=∅ i∈B
converges.
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
399
If the sequence (an )n≥0 is quasiconvex and
lim an = 0,
knk→∞
then the series
X
2−λ(n) an
k
Y
(∗∗)
cos ni xi
i=1
n≥0
converges on (0, 2π)k to some function f : Rk → R summable on T k and is
its Fourier series.
Let m, i, δ, h ∈ Rk , where m and i have non-negative integral components, while δ has positive components. Consider
Z X
Pk
k
Y
mj
ij
j=1
(−1)
ωm (δ; f )1 = sup
f [x + (m − 2i)h] dx
ij
−δ≤h≤δ
j=1
0≤i≤m
Tk
as an integral modulus of smoothness of order m of the function f ∈ L(T k ).
The validity of the following analogue of Lemmas 1 and 2 is obvious.
Lemma 4. Let p ∈ Rk have natural components, h > 0 and 2mh ≤
x ≤ π1. Then
X
Pk
k h
i
Y
mj
i
Kpj [xj + (mj − 2ij )hj ] ≤
|Tpm | =
(−1) j=1 j
ij
j=1
0≤i≤m
≤ c(m)
k
Y
m
mj −1
j
x−2
j hj Pj
,
|Tpm | ≤ c(m)
j=1
k
Y
m
mj +1
hj j Pj
.
j=1
We have
Theorem 2. Let a sequence (an )n≥0 be quasiconvex and
Then for the sum f of the series (∗∗) we have
X X
(m)
ω1 (δ; f ) ≤ c(m)
B:B⊂M
×
Y
ν∈B
where Nν = [ δ1ν ], ν ∈ M .
X
lim an = 0.
knk→∞
×
1≤nν ≤Nν Nµ +1≤nµ ≤∞
ν∈B
µ∈B ′
ν +1
Nν−mν im
ν
Y
µ∈B ′
iµ |∆B (∆M an )|
400
T. TEVZADZE
(As usual, the empty product is assumed to be zero.)
To prove the theorem note that the Hardy transformation is defined in Rk
for k > 2 too and its structure becomes more complicated as the dimension
increases. Nevertheless, the estimates from the proof of Theorem 1 can hold
for the case k > 2 as well.
References
1. S. Aljanˆciˆc and M. Tomiˆc, Sur le module de continuite integral des
s´eries de Fourier a coefficients convexes. C. R. Acad. Sci. Paris 259(1964),
No. 9, 1609–1611.
¨
2. S. Aljanˆciˆc and M. Tomiˆc, Uber
den Stetigkeitsmodul von FourierReihen mit monotonen Koeffizienten. Math. Z. 88(1965), No. 3, 274–284.
3. M. and S. Izumi, Modulus of continuity of functions defined by trigonometric series. J. Math. Anal. and Appl. 24(1968), 564–581.
4. S. A. Telyakovski, The integrability of trigonometric series. Estimation of the integral modulus of continuity. (Russian) Mat. Sbornik
91(133)(1973), No. 4(8), 537–553.
5. T. Sh. Tevzadze, Some classes of functions and trigonometric Fourier
series. (Russian) Some questions of function theory (Russian), v. II, 31–92,
Tbilisi University Press, 1981.
6. E. H. Hardy, On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters.
Quart. J. Math. 37(1906), 53–79.
7. L. V. Zhizhiashvili, Some questions of the theory of trigonometric
Fourier series and their conjugates. (Russian) Tbilisi University Press, 1993.
(Received 29.12.1993; revised version 15.06.1995)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043
Republic of Georgia
ESTIMATION OF THE INTEGRAL MODULUS OF
SMOOTHNESS OF AN EVEN FUNCTION OF SEVERAL
VARIABLES WITH QUASICONVEX FOURIER
COEFFICIENTS
T. TEVZADZE
Abstract. The estimate of the modulus of smoothness of an even
function of several variables with quasiconvex Fourier coefficients obtained in this paper extends one result of S. A. Telyakovski.
1. Let (ai,j )i,j≥0 be a double numerical sequence. Denote by ∆2 ai,j the
expression ∆12 (∆12 ai,j ), where
∆12 ai,j = ∆1 (∆2 ai,j ) = ∆1 (ai,j − ai,j+1 ) = ai,j − ai+1,j − ai,j+1 + ai+1,j+1 .
Definition 1. The double sequence (ai,j )i,j≥0 will be called quasiconvex
if the series
∞ X
∞
X
(i + 1)(j + 1)|∆2 ai,j | + (i + 1)|∆1 (∆12 aij )| + (j + 1)|∆2 (∆12 aij )|
i=0 j=0
converges.
As can be easily shown, if the sequence (ai,j )i,j≥0 is quasiconvex and
lim ai,j = 0,
i+j→∞
then the series
∞ X
∞
X
λi,j ai,j cos ix cos jy,
(∗)
i=0 j=0
where λ0,0 = 14 , λ0,j = λi,0 = 21 , i, j = 1, 2, . . . , λi,j = 1 for i, j > 0,
converges on (0, 2π)2 to some function f ∈ L(T 2 ) and is its Fourier series
with T 2 = [0, 2π]2 .
1991 Mathematics Subject Classification. 42B05.
Key words and phrases. Integral modulus of smoothness, quasiconvex Fourier
coefficients.
379
c 1996 Plenum Publishing Corporation
1072-947X/96/0700-0379$09.50/0
380
T. TEVZADZE
The expression
ωm,n (δ, ρ; f )1 =
Z X
n
m X
m n
f [x + (m − 2µ)h, y + (n − 2ν)η]dx dy
(−1)µ+ν
= sup
ν
µ
|h|≤δ
µ=0 ν=0
2
|η|≤ρT
is usually called an integral modulus of smoothness of order (m, n) of a
function f ∈ L(T 2 ). Here and in what follows it is assumed that δ ∈ [0, π],
ρ ∈ [0, π].
Aljanˇciˇc and Tomiˇc [1], [2] considered an estimate of the integral modulus
of continuity in terms of Fourier coefficients of the function f for some classes
of sequences in the one-dimensional case. In 1963 M. and S. Izumi [3] proved
that if the Fourier coefficients an → 0, n → ∞, form a quasiconvex sequence,
then the estimate
X
X
i2 |∆21 ai | +
i|∆12 ai |1
ω(δ; f )1 ≤ cδ
i≤ δ1
i> δ1
holds for the integral modulus of continuity ω(δ; f )1 .
This result was later generalized by Telyakovski [4] who proved a theorem
giving an estimate of an integral modulus of smoothness of order m, m ∈ N.
In this paper an estimate is obtained for the integral modulus of smoothness for a function of several variables which is even with respect to each
variable.
2.
For a further discussion we need
Lemma 1. Let 2mh ≤ t ≤ π. Then
m
X
i m
m
Kp [t + (m − 2i)h] ≤ c(m)hm pm−1 t−2 ,
(−1)
|Tp | =
i
i=0
|Tpm | ≤ c(m)hm pm+1 ,
where Kp (t) is the Fejer kernel, p ∈ N.
Proof. It is easy to show that
m
X
m
Tpm =
(−1)i
Kp [t + (m − 2i)h] =
i
i=0
m−1
X
m−1
(−1)i
Kp [t + (m−2i)h]−Kp [t+(m−2(i+1))h] =
=
i
i=0
1 Here
and in what follows, c, c(m), c(m, n), . . . denote, generally speaking, various
positive constants depending only on the parameters indicated in the brackets.
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
m−1
X
381
m−1
Kp′ [t + (m − 2(i + 1 − Q1 ))h] = · · · =
(−1)
= −2h
i
i=0
i
= (−1)m hm Kp(m) (t + c(m)h),
where c(m) = m − 2(m − Q1 − · · · − Qm ), 0 < Qi < 1, i = 1, . . . , m.
Since −m < c(m) < m, we have
p
Kp(m) (t
1 X (m)
+ c(m)h) =
D (t + c(m)h) =
p + 1 i=0 i
(
p
i
1 X X m cos j(t + c(m)h),
=±
j
p + 1 i=0 j=1
sin j(t + c(m)h),
2|m,
2 ∤ m,
where Di (t) is the Dirichlet kernel.
Let 2|m (the case 2 ∤ m is considered similarly). Then applying twice the
Abel transformation and taking into account the estimates
c
Kp (t) ≤ 2 , 0 < |t| ≤ π,
pt
Kp (t) ≤ p, |t| ≤ π,
we find
p
i
1 X X m
j − 2(j + 1)m +
p + 1 i=0 j=1
+ (j + 2)m jKj (t + c(m)h) +
Kp(m) (t + c(m)h) =
+
+
p
1 X
(i − 1)m − im iKi (t + c(m)h) +
p + 1 i=0
p
1 X m
i − (i + 1)m iKi (t + c(m)h) +
p + 1 i=0
1
· pm+1 Kp (t + c(m)h) ≤
p+1
(
c(m)pm−1 t−2 , 2mh ≤ t ≤ π,
≤
c(m)pm+1 ,
−π ≤ t ≤ π.
+
Therefore
|Tpm | ≤ c(m)hm pm−1 t−2 ,
|Tpm |
m m+1
≤ c(m)h p
,
2mh ≤ t ≤ π,
−π ≤ t ≤ π.
The two-dimensional analogue of Lemma 1 given below is proved similarly.
382
T. TEVZADZE
Lemma 2. Let 2mh ≤ x ≤ π, 2nη ≤ y ≤ π. Then
m,n
|Tp,q
|
m X
n
X
n
µ+ν m
(−1)
Kp [x+(m−2µ)h]Kq [y+(n−2ν)η] ≤
=
µ
ν
µ=0 ν=0
≤ c(m, n)
hm η n pm−1 q n−1
,
(xy)2
m,n
| ≤ c(m, n)hm η n pm+1 q n+1 ,
|Tp,q
(x, y) ∈ T 2 .
Lemma 3 ([5]). Let 2mh ≤ x ≤ π. Then
m
X
m
Dp [x + (m − 2i)h] ≤ c(m)hm pm x−1 ,
(−1)i
i
i=0
where Dp is the Dirichlet kernel, p ∈ N.
Theorem 1. Let a double sequence (ai,j )i,j≥0 be quasiconvex and
lim ai,j = 0.
i+j→∞
Then for the sum f of the series (∗) we have
1
1
]
[ 2nη
[ 2mh
X] X
m n
ωm,n (h, η; f )1 ≤ c(m, n) h η
im+1 j n+1 |∆2 ai,j | +
i=1
1
]
[ 2mh
X
+ hm
∞
X
j=1
im+1 j|∆2 ai,j | +
i=1 j=[ 1 ]+1
2nη
+η
∞
X
n
1
[ 2nη
]
X
ij n+1 |∆2 ai,j | +
1
]+1 j=1
i=[ 2mh
∞
X
+
∞
X
1
1
]+1 j=[ 2nη
]+1
i=[ 2mh
4
X
=
ij|∆2 ai,j | =
Ps (m, n, h, η).
s=1
Proof. We write
A(m, n, f ) = 4
Zπ Zπ
0
0
m,n
|∆
2mh
Z Zπ
Z Z2nη Zπ Z2nη 2mh
+
+
+
f | dx dy = 4
0
0
2mh 0
0
2nη
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
383
Zπ Zπ
4
X
+
As (m, n, f ),
|∆m,n f | dx dy ≡
s=1
2mh 2nη
where
∆m,n f =
m X
n
X
(−1)µ+ν
µ=0 ν=0
=
m X
n
X
µ+ν
(−1)
µ=0 ν=0
∞
X
+
4
X
1
1
1
[ y1 ]
X
[x
] [y]
]
[x
∞
∞
X
X
X
X
X
m n
+
+
+
ν
µ
1
1
j=1
i=1 j=1
i=1
i=[ x ]+1
∞
X
1
]+1 j=[ y1 ]+1
i=[ x
≡
m n
f [x + (m − 2µ)h, y + (n − 2ν)η] =
µ
ν
j=[ y ]+1
ai,j cos i[x + (m − 2µ)h] cos[y + (n − 2ν)η] ≡
Ts (m, n).
s=1
It is easy to show that
[1]
1
T1 ≡ T1 (m, n) = 2
m+n
[x] y
X
X
ai,j sinm ih sinn jη ×
i=1 j=1
Further,
Zπ Zπ
m+n
(−1) 2 cos ix cos jy,
(−1) m+n−1
2
sin ix cos jy,
×
m+n−1
2
(−1)
cos ix sin jy,
m+n−2
(−1) 2 sin ix sin jy,
m n
|T1 |dx dy ≤ c(m, n)h η
2|m, 2|n,
2 ∤ m, 2|n,
2|m, 2 ∤ n,
2 ∤ |m, 2 ∤ n.
1
1
] [y]
[x
Zπ Zπ X
X
im j n |ai,j |dx dy ≤
2mh 2nη i=1 j=1
2mh 2nη
m n
≤ c(m, n)h η
2mh
Z Z2nη
(xy)
1
]
[ 2mh
≤ c(m, n)hm η n
1
[ 2mh
]
m n
≤ c(m, n)h η
1
[ 2nη
]
X X
r=1
(rs)−2
s=1
j=1
im j n |ai,j |dx dy ≤
s
r X
X
im j n |ai,j | ≤
i=1 j=1
1
]
[ 2nη
X X
i=1
[x] [y]
X
X
i=1 j=1
1
1
−2
1
1
] [ 2nη ]
[ 2mh
m n
i j |ai,j |
X X
r=i
s=j
(rs)−2 ≤
384
T. TEVZADZE
1
1
[ 2mh
] [ 2nη ]
X X
m n
≤ c(m, n)h η
im−1 j n−1 |ai,j |.
(2.1)
j=1
i=1
1
1
], q = [ 2nη
], which
Applying twice the Hardy transformation [6], p = [ 2mh
is a two-dimensional analog of the Abel transformation, we find
p X
q
X
m−1 n−1
j
i
p
X
|ai,j | =
i=1 j=1
+
q
X
i
i=1
n−1
|ai,q |
s
s=1
=
q−1
X
j
p
X
i=1
≤
q−1
X
j
n
j=1
n
X
p−1
i=1
j=1
+
m−1
j=1
X
p
i
|∆2 ai,j | +
p
X
i
|∆2 ai,j | − |∆2 ai+1,j |
i
X
q |ai,q | =
m−1 n
i=1
rm−1 +
r=1
r=1
i=1
+pm
p
X
im−1 ≤
q−1
X
p−1 X
q−1
X
|∆12 ai,j |im j n +
i=1 j=1
j n |∆2 ap,j | + q n
j=1
p−1
X
im |∆1 ai,q | + pm q n |ap,q | ≤
i=1
im+1 j n+1 |∆2 ai,j | + q n+1
p−2
X
im+1 |∆12 ai,q − ∆12 ai+1,q | +
i=1
i=1 j=1
+pm+1
s=1
i=1
i=1
≤
m−1
j
X
sn−1 +
|ai,j | − |ai,j+1 |
p−1
i
X
X
m−1
|ai,q | − |ai+1,q |
rm−1 +
i
|∆2 ap,j | + q n
+q n |ap,q |
p−1 X
q−1
X
X
q−1
q−2
X
j n+1 |∆12 ap,j − ∆12 ap,j+1 | + pm+1 q n+1 |∆12 ap,q | +
j=1
+q n
p−2
X
im+1 |∆1 (ai,q − ai+1,q )| + pm
q−2
X
j n+1 |∆2 (ap,j − ap,j+1 )| +
j=1
i=1
n m+1
+q p
m n+1
|∆1 ap,q | + p q
≡
9
X
|∆2 ap,q | + pm q n |ap,q | ≡
Iα (m, n, p, q).
(2.2)
α=1
For I2 we obtain
I2 = q n
p−2
X
i=1
im+1 q|∆1 (∆12 ai,q )| = q n
p−2
X
i=1
∞
X 2
∆ ai,j ≤
im+1 q
j=q
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
≤q
n
p−2
X
im+1
i=1
∞
X
j|∆2 ai,j |.
385
(2.3)
j=q
Similarly,
I3 ≤ pm
q−2
∞ X
X
ij n+1 |∆2 ai,j |.
(2.4)
i=p j=1
Let us now estimate I5 . We have
I5 ≤ q n
p−2
X
im+1 |∆21 ai,q |,
i=1
where
∆21 ai,q = ai,q − ai+1,q + ai+2,q .
It is easy to show that
∆21 ai,q =
∞
X
∆2 (∆21 ai,s ) =
s=q
∞
X
∆2
∞
X
s=j
j=q
∞
∞ X
X
∆2 (∆2 ai,s ) =
∆2 ai,s .
j=q s=j
Therefore
|∆21 ai,q | ≤
∞
∞ X
X
|∆2 ai,s | ≤
∞
X
s|∆2 ai,s |.
(2.5)
s=q
j=q s=j
Thus
I5 ≤ q n
p−2 X
∞
X
im+1 s|∆2 ai,s |.
(2.6)
i=1 s=q
Similarly,
I6 ≤ pm
q−2
∞ X
X
ij n+1 |∆2 ai,j |.
(2.7)
i=p j=1
Next,
I7 = pm q n · p|∆1 ap,q |.
Since
∞
∞
X
X
i|∆21 ai,q |,
p|∆1 ap,q | = p
∆21 ai,q ≤
i=p
i=p
by repeating the arguments used for I5 we obtain
I7 ≤ pm q n
∞
∞ X
X
i=p j=q
ij|∆2 ai,j |.
(2.8)
386
T. TEVZADZE
The same estimate holds for I8 . For I4 we have
I4 ≤ pm q n
∞ X
∞
X
ij|∆2 ai,j |.
(2.9)
i=p j=q
In the same way,
∞
∞ X
∞
∞ X
∞ X
∞ X
X
X
rs|∆2 ar,s |. (2.10)
∆2 ar,s ≤ pm q n
I9 ≤ pm q n
r=p s=q
i=p j=q r=i s=j
Using (2.2)–(2.10), we find
Zπ Zπ
|T1 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.11)
α=1
2mh 2nη
If p = [ x1 ]+ 1, q = [ y1 ]+ 1, we again apply twice the Hardy transformation
for T4 and obtain
T4 =
m X
n
X
(−1)µ+ν
µ=0 ν=0
n
m n
ap,q Dp [x+(m−2µ)h]Dq [y+(n−2ν)η]−
µ
ν
− Dp [x + (m − 2µ)h]∆2 ap,q (q + 1)Kq [y + (n − 2ν)η] −
− Dq [y + (n − 2ν)η]∆1 ap,q (p + 1)Kp [x + (m − 2µ)h] +
+ Dp [x + (m − 2µ)h]
∞
X
∆22 ap,j (j + 1)Kj [y + (n − 2ν)η] +
j=q
+ Dq [y + (n − 2ν)η]
∞
X
∆12 ai,q (i + 1)Ki [x + (m − 2µ)h] +
i=p
+
∆212 ap,q (p
+ 1)(q + 1)Kp [x + (m − 2µ)h]Kq [y + (n − 2ν)η] −
− (p + 1)Kp [x+(m − 2µ)h]
− (q + 1)Kq [y+(n − 2ν)η]
∞
X
∆2 (∆12 ap,j )(j + 1)Kj [y+(n − 2ν)η] −
j=q
∞
X
∆1 (∆12 ai,q )(i + 1)Ki [x+(m − 2µ)h] +
i=p
+
∞ X
∞
X
i=p j=q
=
9
X
α=1
∆2 ai,j (i + 1)(j + 1)Ki [x+(m − 2µ)h]Kj [y+(n − 2ν)η] =
(α)
T4
(m, n, p, q).
(2.12)
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
387
By Lemma 2 (see (2.1)) we have
Zπ Zπ
1
1
2nη
2mh Z
Z
(1)
|T4 | dx dy ≤ c(m, n)hm η n
1
2mh 2nη
|a[x],[y] |xm−1 y n−1 dx dy ≤
1
1
1
[ 2mh
] [ 2nη ]
X X
m n
≤ c(m, n)h η
j=1
i=1
≤
4
X
im−1 j n−1 |ai,j | dx dy ≤
(2.13)
Ps (m, n, h, η).
s=1
(2)
Applying Lemmas 3 and 1 for T4
Zπ Zπ
we find
1
1
(2)
|T4 | dx dy ≤ c(m, n)hm η n
2mh 2nη
2nη
2mh Z
Z
1
xm−1 y n |∆2 a[x]+1,[y]+1 | dx dy ≤
1
1
1
[ 2mh
] [ 2nη ]
≤ c(m, n)hm η n
X X
i=1
im−1 j n |∆2 ai,j |.
j=1
Using the Abel transformation leads to
Zπ Zπ
(2)
|T4 | dx dy
1
]
[ 2nη
m n
≤ c(m, n) h η
2mh 2nη
X
+h η
X
j
j=1
1
[ 2mh
]
m −1
1
[ 2mh
]
n+1
i
m−1
i=1
X
im−1 |∆22 ai,j | +
i=1
1
|∆2 ai,[ 2nη
]| .
Again applying twice the Abel transformation we obtain
Zπ Zπ
(2)
|T4 | dx dy
1
1
]
[ 2nη
[ 2mh
X] X
m n
≤ c(m, n) h η
im+1 j n+1 |∆2 ai,j | +
i=1
2mh 2nη
1
[ 2nη
]
+η n
X
j=1
1
[ 2nη
]
+η
n
X
j=1
j=1
∞
X
∆1 ∆1 (∆22 ai,j ) +
j n+1 h−1
1
]
i=[ 2mh
∞
∞
X
X
j n+1
∆1 ∆1 (∆22 ai,j ) +
1
r=[ 2mh
] i=r
388
T. TEVZADZE
1
]
[ 2mh
X
m
+h
i=1
∞
∞
X 2
X
1
∆2 a[ 2mh
∆2 ai,j + η −1
im+1 η −1
],j ≤
j=q
1
]
j=[ 2nη
1
1
]
[ 2nη
[ 2mh
X] X
m n
im+1 j n+1 |∆2 ai,j | +
≤ c(m, n) h η
i=1
1
]
[ 2nη
+η
n
X
j
j=1
2
i|∆ ai,j | + η
n
1
i=[ 2mh
]+1
1
[ 2mh
]
+hm
1
[ 2nη
]
∞
X
n+1
j=1
∞
X
X
im+1 j|∆2 ai,j | +
i=1 j=[ 1 ]+1
2nη
X
∞
X
j n+1
j=1
1
i=[ 2mh
]+1
∞
X
∞
X
1
1
i=[ 2mh
]+1 j=[ 2nη
]+1
≤
4
X
i|∆2 ai,j | +
ij|∆2 ai,j | ≤
Ps (m, n, h, η).
(2.14)
s=1
(3)
One can estimate T4
quite similarly.
(4)
Let us now consider T4 . We have
(4)
T4
=
m
X
(−1)µ
µ=0
×
1
[X
2nη ]
n
X
m
n
D[ x1 ] [x + (m − 2µ)h]
(−1)ν
×
µ
ν
ν=0
+
j=[ y1 ]+1
∞
X
1
j=[ 2nη
]+1
(j + 1)∆22 a[ x1 ],j Kj [y + (n − 2ν)η] =
(4)
(4)
= T4 (1) + T4 (2).
(2.15)
Since
1
[ 2nη
]
(4)
|T4 (1)|
X
m n −m−1 −2
≤ c(m, n)h η x
y
j n |∆22 a[ x1 ],j |,
j=[ y1 ]+1
we obtain
Zπ Zπ
1
1
] [ 2nη
]
[ 2nη
1
[ 2mh
]
(4)
|T4 (1)| dx dy
m n
≤ c(m, n)h η
X
m−1
i
1
]
[ 2nη
m n
≤ c(m, n)h η
X
j=1
j n |∆22 ai,j | ≤
s=1 j=s+1
i=1
2mh 2nη
X X
1
]
[ 2mh
j
n+1
X
i=1
im−1 |∆22 ai,j |.
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
389
(2)
Repeating the arguments used in estimating T4 , we find
Zπ Zπ
(4)
|T4 (1)| dx dy ≤ P1 (m, n, h, η) + P3 (m, n, h, η).
(2.16)
2mh 2nη
Similarly,
Zπ Zπ
1
[ 2mh
]
(4)
|T4 (2)| dx dy
m −1
≤ c(m, n)h η
X
m−1
i
i=1
2mh 2nη
≤
4
X
∞
X
|∆22 ai,j | ≤
1
]+1
j=[ 2nη
Ps (m, n, h, η).
(2.17)
s=1
By (2.16) and (2.17) we conclude that
Zπ Zπ
(4)
|T4 | dx dy ≤
4
X
Ps (m, n, h, η).
(2.18)
4
X
Ps (m, n, h, η).
(2.19)
s=1
2mh 2nη
Quite similarly we obtain
Zπ Zπ
(5)
|T4 | dx dy ≤
s=1
2mh 2nη
By Lemma 1 and analysis of the arguments used in estimating I2 we
have
Zπ Zπ
1
1
] [ 2nη ]
[ 2mh
(6)
|T4 | dx dy
m n
≤ c(m, n)h η
X X
i=1
2mh 2nη
1
]
[ 2mh
m n
≤ c(m, n)h η
j=1
1
]
[ 2nη
X X
i=1
im j n |∆12 ai,j | ≤
im+1 j n+1 |∆2 ai,j | +
j=1
1
]
[ 2nη
+h−m−1
X
1
j n+1 |∆2 (∆12 a[ 2mh
],j )| +
j=1
1
[ 2mh
]
+η −n−1
X
−m−1 −n−1
1
1
1
η
|∆12 a[ 2mh
im+1 |∆1 (∆12 ai,[ 2nη
]| + h
],[ 2nη
]| ≤
i=1
1
1
]
[ 2nη
[ 2mh
X] X
m n
≤ c(m, n) h η
im+1 j n+1 |∆2 ai,j | +
i=1
j=1
390
T. TEVZADZE
∞
X
+hm η −1
1
im+1 |∆1 (∆12 ai,[ 2nη
] )| +
1
]+1
i=[ 2mh
1
[ 2nη
]
+h−1 η n
X
1
j n+1 |∆2 (∆12 a[ 2mh
],j )| +
j=1
4
X
1
1
+(hη)−1 |∆12 a[ 2mh
], 2nη
] | ≤≤
Ps (m, n, h, η).
(2.20)
s=1
(8)
For T4
Zπ Zπ
2mh 2nη
we obtain
1
1
1
2nη
2mh Z
[ 2mh
Z
X]
(8)
n
m n
im |∆1 (∆12 ai,[y] )|dxdy+
y
|T4 | dx dy ≤ c(m, n)h η
1
1
2nη
+c(m, n)η
n
Z
y
n
1
i=[x]+1
1
π+mh
Z
∞
X
x−2
|∆1 (∆12 ai,[y] )|dxdy ≤
1
]+1
i=[ 2mh
mh
1
1
1
[ 2nη
]
[ 2mh
[ 2mh
X] X
X]
m n
n
≤ c(m, n) h η
im |∆1 (∆12 ai,j )| +
j
r=1
1
[ 2nη
]
+ ηn
X
1
[ mh
]
jn
∞
X
X
i=r+1
j=1
|∆1 (∆12 ai,j )|
r=1 i=[ 1 ]+1
2mh
j=1
≤
1
1
[ 2nη
]
[ 2mh
X
X]
m n
n
≤ c(m, n) h η
j
im+1 |∆1 (∆12 ai,j )| +
r=1
j=1
+η
n
∞
X
1
[ 2nη
]
X
n
ij |∆1 (∆12 ai,j )|
1
]+1 j=1
i=[ 2mh
≤ c(m, n) hm η n
+η
n
∞
X
1
i=[ 2mh
]+1
X
im+1
r=1
1
[ 2nη
]
i
X
j=1
=
1
[ 2nη
]
1
[ 2mh
]
X
j n |∆1 (∆12 ai,j )| +
j=1
j |∆1 (∆12 ai,j )| .
n
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
391
Using the Abel transformation leads to
1
[ 2nη
]
1
]
[ 2nη
X
n
j |∆1 (∆12 ai,j )| ≤
j=1
X
1
j n+1 |∆2 ai,j | + η −n−1 |∆1 (∆12 ai,[ 2nη
] )|.
j=1
Hence (see (2.8)) it is easy to show that
Zπ Zπ
(8)
|T4 | dx dy
Zπ Zπ
|T4 | dx dy ≤
≤
4
X
Pα (m, n, h, η).
(2.21)
4
X
Pα (m, n, h, η).
(2.22)
α=1
2mh 2nη
Similarly,
(7)
α=1
2mh 2nη
Further,
(9)
T4
1
]
[ 2nη
1
[ X
2mh ]
n
µ+ν m
=
(−1)
ν
µ
1
µ=0 ν=0
n
m X
X
1
[ 2mh
]
+
i=[ x ]+1 j=[ y1 ]+1
∞
X
X
X
∞
X
∞
X
∞
X
+
1
]
[ 2nη
X
+
1
i=[ 2mh
]+1 j=[ y1 ]
(i + 1)(j + 1) ×
1
1
1
1
i=[ x
]+1 j=[ 2nη
]+1 i=[ 2mh
]+1 j=[ 2nη
]+1
× Ki [x + (m − 2µ)h]Kj [y + (n − 2ν)η] =
4
X
(9)
T4 (s).
(2.23)
s=1
By Lemma 2 we have
(9)
|T4 (1)|
m n
≤ c(m, n)h η
1
[ 2mh
]
1
[ 2nη
]
X
X
im j n (xy)−2 |∆2 ai,j |.
1
]+1 j=[ y1 ]+1
i=[ x
Therefore
Zπ Zπ
1
1
1
1
[ 2mh
] [ 2nη ] [ 2mh
] [ 2nη ]
(9)
|T4 (1)|dxdy
X X X X
m n
≤ c(m, n)h η
r=1
2mh 2nη
s=1
i=r
im j n |∆2 ai,j | ≤
j=s
1
1
[ 2mh
] [ 2nη ]
m n
≤ c(m, n)h η
X X
i=1
j=1
im+1 j n+1 |∆2 ai,j |.
(2.24)
392
T. TEVZADZE
The remaining terms of (2.22) are estimated by similar arguments. Thus
we conclude that
Zπ Zπ
(9)
|T4 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.25)
α=1
2mh 2nη
Taking into account (2.13), (2.14), (2.18)–(2.22), and (2.25) we obtain
Zπ Zπ
|T4 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.26)
α=1
2mh 2nη
For T2 we have
[1]
|T2 | ≤ c(n)η
n
y
X
j=1
X
X
∞
m
µ m
(−1)
j
ai,j cos i[x + (m − 2µ)h].
µ
n
µ=0
1
i=[ x
]+1
Applying twice the Abel transformation, we find
X
∞
m
(−1)
ai,j cos i[x + (m − 2µ)h] =
µ
1
µ=0
m
X
µ
i=[ x ]+1
X
∞
m
(i + 1)∆21 ai,j Ki [x + (m − 2µ)h] −
=
(−1)µ
µ
1
µ=0
m
X
−∆1 a[ x1 ],j
i=[ x ]+1
h1i
x
K[ x1 ] [x + (m − 2µ)h] − a[ x1 ],j D[ x1 ] [x + (m − 2µ)h] .
By virtue of Lemmas 3 and 1 we obtain
m n
|T2 | ≤ c(m, n)h η
[ y1 ]
X
j
n
j=1
+x
−m−2
+ c(n)η
im |∆21 ai,j |x−2 +
1
i=[ x
]+1
−m−1
|∆1 a[ x1 ],j | + |a[ x1 ],j |x
[1]
n
1
[X
2mh ]
y
X
j=1
j
n
X
∞
i|∆12 ai,j | ×
1
]
i=[ 2mh
m
X
m
Ki [x + (m − 2µ)h].
(−1)µ
×
µ
µ=0
+
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
Hence
σm,n ≡
Zπ Zπ
|T2 | dx dy ≤
2mh 2nη
Zπ
≤ c(m, n)hm η n
x−2
m n
+ c(m, n)h η
−m−2
x
Zπ
+ c(m, n)h η
−m−1
x
[ y1 ]
+ c(n)η
im |∆21 ai,j | dx dy +
1
i=[ x
]+1
[ y1 ]
Zπ X
j n |∆1 a[ x1 ],j | dx dy +
[ y1 ]
Zπ X
j n |a[ x1 ],j | dx dy +
2nη j=1
2mh
n
X
jn
2nη j=1
2mh
m n
1
]
[ 2mh
2nη j=1
2mh
Zπ
[ y1 ]
Zπ X
Zπ Zπ X
∞
X
jn
2mh 2nη j=1
i|∆21 ai,j | ×
1
i=[ 2mh
]+1
m
X
m
Ki [x + (m − 2µ)h] dx dy ≤
×
(−1)µ
µ
µ=0
1
1
m n
≤ c(m, n)h η
2nη
2mh Z
Z
1
1
2nη
1
2mh
+
Z
1
xm
Z
Z
1
2nη
xm−1
1
Z
y −2
1
[ 2mh
]
j
n
X
im |∆21 ai,j | dx dy +
i=[x]+1
j n |∆1 a[x],j | dx dy +
[y]
X
j n |a[x],j | dx dy
j=1
1
2nη
2mh Z
Z
1
+
1
1
+ c(m, n)η n
j=1
1
[y]
X
[y]
X
j=1
1
1
2mh
+
y −2
y
−2
y −2
[y]
X
jn
j=1
1
∞
X
|∆21 ai,j | dx dy.
1
]+1
i=[ 2mh
Next, as is easy to show,
σm,n
1
1
1
]
[ 2nη
[ 2mh
[ 2mh
s
X] X
X]
X
m n
−2
n
≤ c(m, n) h η
im |∆21 ai,j | +
s
j
r=1
s=1
j=1
i=r+1
393
394
T. TEVZADZE
1
]
[ 2nη
1
[ 2mh
]
m n
+h η
r
X
rm−1
X
s−2
m
r=1
s
X
X
s−2
s=1
1
[ 2nη
]
s=1
s
X
j n |∆1 ar,j | +
j=1
1
[ 2nη
]
1
[ 2mh
]
+ η n h−1
s−2
s=1
r=1
+ hm η n
X
X
j n |ar,j | +
j=1
∞
X
jn
j=1
s
X
1
i=[ 2mh
]+1
|∆21 ai,j | .
Therefore, after simple calculations, we find
σm,n ≤ c(m, n)hm η n
1
[X
2mh ]
1
]
[ 2nη
im+1
+
1
[ 2nη
]
X
X
r
m
r=1
j n−1 |∆12 ai,j | +
j=1
i=1
1
[ 2mh
]
X
1
1
] [ 2nη ]
[ 2mh
j
n−1
|∆1 ar,j | +
X X
r=1
j=1
+cm,n η n
∞
X
r
m−1 n−1
j
j=1
|ar,j | +
1
[ 2nη
]
X
i
j n−1 |∆21 ai,j |.
j=1
1
]+1
i=[ 2mh
The analysis of estimates (2.14) and (2.18) gives
Zπ Zπ
|T2 | dx dy ≤
Zπ Zπ
|T3 | dx dy ≤
4
X
Pα (m, n, h, η).
(2.27)
4
X
Pα (m, n, h, η).
(2.28)
α=1
2mh 2nη
Similarly,
2mh 2nη
α=1
Therefore by virtue of (2.11), (2.26), (2.27) and (2.28) we obtain
A4 (m, n; f ) ≤
4
X
Pα (m, n, h, η).
α=1
For A1 (m, n; f ) we have
A1 (m, n; f ) =
2mh
Z Z2nη
|∆m,n f | dx dy =
0
0
(2.29)
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
=
2mh
Z Z2nη X
s
r X
0
0
∞
X
+
+
+
i=r+1 j=1
i=1 j=1
∞
X
∞ X
s
X
r
∞
X
X
+
i=1 j=s+1
(i + 1)(j + 1)∆2 ai,j ×
i=r+1 j=s+1
n
m X
X
m n
Ki [x + (m − 2µ)h] ×
µ
ν
µ=0 ν=0
4
X
(α)
×Kj [y+(n−2ν)η]dx dy =
A1 (m, n, h, η),
×
395
(−1)µ+ν
(2.30)
α=1
1
1
], s = [ 2nη
]. By Lemma 2 we obtain
where r = [ 2mh
(1)
A1
2mh
Z Z2nηX
s
r X
m n
≤ c(m, n)h η
i=1 j=1
0
0
1
[ 2mh
]
1
[ 2nη
]
X X
≤ c(m, n)hm η n
i=1
im+2 j n+2 |∆2 ai,j |dx dy ≤
im+1 j n+1 |∆2 ai,j |.
(2.31)
j=1
Further,
(4)
A1
=
2mh
Z Z2nη
0
0
∞
X
∞
X
i=r+1 j=s+1
X X
+
(i + 1)(j + 1)∆2 ai,j
m>2µ n>2ν
X X
n
µ+ν m
×
(−1)
+
+
+
µ
ν
m>2µ n≤2ν
m≤2µ n>2ν
m≤2µ n≤2ν
4
X
(4)
× Ki [x + (m − 2µ)h]Kj [y + (n − 2ν)η] dx dy =
A1 (α).
X X
X X
α=1
(4)
is sufficient to estimate A1 (4). For
n
µ ([ m
2 ] ≤ µ ≤ m) and ν ([ 2 ] ≤ ν ≤ n)
the term of this sum
Obviously, it
(4)
A1;µ,ν (4), where
are fixed, a = 2µ−m,
b = 2ν − n, we obtain, after passing to the variables t = x − ah, τ = y − bη,
(4)
Ai;µ,ν (4) ≤ c(m, n)
∞
X
∞
X
1
1
i=[ 2mh
]+1 j=[ 2nη
]+1
|∆2 ai,j |
3
3 X
X
r=1 s=1
where
Fr,s =
Z Z
ur vs
(i + 1)(j + 1)Ki (t)Kj (τ ) dt dτ,
Fr,s ,
396
T. TEVZADZE
h 1 1i
h1
h
i
1i
u1 = − ah, − , u2 = − , , u3 = , (2m − a)h ,
i
i i
i
h 1 1i
h1
i
h
1i
v1 = − bη, − , v2 = − , , v3 = , (2n − b)η ,
j
j j
j
3
[
ur × vs .
[−ah, −(2m − a)h] × [−bη, (2n − b)η =
r,s=1
Taking into account the properties of Fejer kernels in the respective intervals,
we have
Z
Z Z
dt dτ
dτ
2 2 2
, F2,1 ≤ i j
,
F1,1 ≤ ij
ij(tτ )2
i
jτ 2
v1
u 1 v1
Z
Z Z
dt
dt dτ
2
,
,
F
≤
ij
F3,1 ≤ ij
1,2
2
ij(tτ )
j
it2
u1
u 3 v1
Z
4
2
dt
F2,2 ≤ i2 j 2 , F3,2 ≤ ij 2
,
ij
j
it2
u3
Z Z
Z
dτ
dt dτ
2 2
F1,3 ≤ ij
,
F
,
≤
i
j
2,3
ij(tτ )2
i
jτ 2
u1 v 3
v3
Z Z
dt dτ
F3,3 ≤ ij
.
ij(tτ )2
u3 v 3
Therefore
∞
X
(4)
A1 (4) ≤ c(m, n)
1
i=[ 2mh
]+1
(2)
∞
X
ij|∆2 ai,j |.
(2.32)
1
j=[ 2nη
]+1
(3)
A1 and A1 are estimated by the same scheme as used in deriving (2.31)
and (2.32). Therefore for A1 we have
A1 ≤
4
X
Ps (m, n, h, η).
(2.33)
s=1
It remains to consider A2 (m, n; f ) (A3 is estimated similarly). We have
1 [ 1 ]
] 2nη
[x
Zπ Z2nη
Zπ Z2nη X
∞
X
X
m,n
A2 =
|∆ f | dx dy =
+
2mh 0
1
]
[x
+
X
2mh 0
∞
X
i=0 j=[ 1 ]+1
2nη
+
∞
X
∞
X
1
1
]+1 j=[ 2nη
i=[ x
]+1
i=0 j=0
1
[ 2nη
]
X
i=[ r1 ]+1 j=0
(i + 1)(j + 1)∆2 ai,j ×
+
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
m X
n
X
m n
Ki [x + (m − 2µ)h] ×
µ
ν
µ=0 ν=0
4
X
(α)
× Kj [y + (n − 2ν)η] dx dy =
A2 (m, n, h, η).
×
397
(−1)µ+ν
(2.34)
α=1
Again, by Lemma 2 we obtain
(1)
A2
≤ c(m, n)hm η n
1 [ 1 ]
] 2nη
[x
Zπ Z2nηX
X
2mh 0
i=1 j=1
1
2mh
≤ c(m, n)hm η n
Z
x−2
1
]
[ 2mh
(2)
(2)
A2
X
im+2
j n+1 |∆2 ai,j |dx ≤
j=1
im+1 j n+1 |∆2 ai,j |.
(2.35)
j=1
we have
1
Zπ Z2nη [ X
2mh ]
+
=
2mh 0
n
m X
X
Further,
1
[ 2nη
]
1
]
[ 2nη
X X
i=1
For A2
[x]
X
i=1
1
≤ c(m, n)hm η n
im+2 j n+2 |∆2 ai,j |dx dy ≤
∞
X
1
i=[ 2mh
]+1
1
]+1
i=[ x
1
[X
2nη ]
(i + 1)(j + 1)|∆2 ai,j | ×
j=0
m n
(−1)µ+ν
×
Ki [x + (m − 2µ)h] ×
µ
ν
µ=0 ν=0
(2)
(2)
× Kj [y + (n − 2ν)η] dx dy = A2 (1) + A2 (2).
1
1
[ 2nη
]
(2)
A2 (1)
≤ c(m, n)hm η n
X
j n+1
j=1
≤ c(m, n)h η
X
j
1
[ 2mh
]
n+1
X
r=1
1
]
[ 2nη
X X
i=1
x−2
j=1
X
im+2 |∆2 ai,j |dx dy ≤
i=[x]+1
1
]
[ 2mh
1
]
[ 2mh
j=1
≤ c(m, n)hm η n
1
[ 2mh
]
1
1
[ 2nη
]
m n
2mh
Z
(2.36)
r
−2
X
im+2 |∆2 ai,j |dx ≤
i=r+1
im+1 j n+1 |∆2 ai,j |.
(2.37)
398
T. TEVZADZE
Similarly (see (2.32)) we find
∞
X
(2)
A2 (2) ≤ c(m, n)η n
1
]+1
i=[ 2mh
1
[ 2nη
]
X
ij n+1 |∆2 ai,j |.
(2.38)
j=1
By (2.35)–(2.37) we obtain
(2)
A2 ≤ P1 (m, n, h, η) + P3 (m, n, h, η).
(2.39)
It is likewise easy to show that
(3)
(4)
A2 ≤ P2 (m, n, h, η) + P4 (m, n, h, η), A2 ≤ P4 (m, n, h, η). (2.40)
Keeping in mind (2.33), (2.34), (2.38)–(2.40), we find
A2 ≤
4
X
Ps (m, n, h, η).
(2.41)
s=1
With regard to (2.29), (2.33), and (2.41) we conclude that
A≤
4
X
Ps (m, n, h, η).
s=1
3. Let now Rk be a k-dimensional Euclidean space of points x = (x1 ,. . . ,xk )
with ordinary linear operations and the Euclidean norm kxk. The product
of two vectors x, y ∈ Rk is understood componentwise and so are the
inequalities x < y, x ≤ y. It is assumed that T k = [0, 2π]k .
If n = (n1 , . . . , nk ) is a multi-index with non-negative integral components, 0 = (0, . . . , 0), 1 = (1, . . . , 1), and (an )n≥0 is a k-multiple numerical
sequence, then
∆i an = an − an+(δi1 ,...,δik ) ,
where δij is the Kronecker symbol, i, j ∈ M = {1, . . . , k}.
We introduce the notation
∆M an ≡ ∆1···k an = ∆1 (∆2 (· · · (∆k an ) · · · )),
for each B, ∅ 6= B ⊂ M , ∆B an is defined similarly, and assume that λ(n)
is the number of zero coordinates of the vector n (see [7]). We will also use
the notation B ′ = M \ B, where B ⊂ M .
Definition 2. A sequence (an )n≥0 will be called quasiconvex if the series
X
Y
X
(ni + 1)|∆B (∆M an )|
n≥0 B⊂M, B6=∅ i∈B
converges.
ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS
399
If the sequence (an )n≥0 is quasiconvex and
lim an = 0,
knk→∞
then the series
X
2−λ(n) an
k
Y
(∗∗)
cos ni xi
i=1
n≥0
converges on (0, 2π)k to some function f : Rk → R summable on T k and is
its Fourier series.
Let m, i, δ, h ∈ Rk , where m and i have non-negative integral components, while δ has positive components. Consider
Z X
Pk
k
Y
mj
ij
j=1
(−1)
ωm (δ; f )1 = sup
f [x + (m − 2i)h] dx
ij
−δ≤h≤δ
j=1
0≤i≤m
Tk
as an integral modulus of smoothness of order m of the function f ∈ L(T k ).
The validity of the following analogue of Lemmas 1 and 2 is obvious.
Lemma 4. Let p ∈ Rk have natural components, h > 0 and 2mh ≤
x ≤ π1. Then
X
Pk
k h
i
Y
mj
i
Kpj [xj + (mj − 2ij )hj ] ≤
|Tpm | =
(−1) j=1 j
ij
j=1
0≤i≤m
≤ c(m)
k
Y
m
mj −1
j
x−2
j hj Pj
,
|Tpm | ≤ c(m)
j=1
k
Y
m
mj +1
hj j Pj
.
j=1
We have
Theorem 2. Let a sequence (an )n≥0 be quasiconvex and
Then for the sum f of the series (∗∗) we have
X X
(m)
ω1 (δ; f ) ≤ c(m)
B:B⊂M
×
Y
ν∈B
where Nν = [ δ1ν ], ν ∈ M .
X
lim an = 0.
knk→∞
×
1≤nν ≤Nν Nµ +1≤nµ ≤∞
ν∈B
µ∈B ′
ν +1
Nν−mν im
ν
Y
µ∈B ′
iµ |∆B (∆M an )|
400
T. TEVZADZE
(As usual, the empty product is assumed to be zero.)
To prove the theorem note that the Hardy transformation is defined in Rk
for k > 2 too and its structure becomes more complicated as the dimension
increases. Nevertheless, the estimates from the proof of Theorem 1 can hold
for the case k > 2 as well.
References
1. S. Aljanˆciˆc and M. Tomiˆc, Sur le module de continuite integral des
s´eries de Fourier a coefficients convexes. C. R. Acad. Sci. Paris 259(1964),
No. 9, 1609–1611.
¨
2. S. Aljanˆciˆc and M. Tomiˆc, Uber
den Stetigkeitsmodul von FourierReihen mit monotonen Koeffizienten. Math. Z. 88(1965), No. 3, 274–284.
3. M. and S. Izumi, Modulus of continuity of functions defined by trigonometric series. J. Math. Anal. and Appl. 24(1968), 564–581.
4. S. A. Telyakovski, The integrability of trigonometric series. Estimation of the integral modulus of continuity. (Russian) Mat. Sbornik
91(133)(1973), No. 4(8), 537–553.
5. T. Sh. Tevzadze, Some classes of functions and trigonometric Fourier
series. (Russian) Some questions of function theory (Russian), v. II, 31–92,
Tbilisi University Press, 1981.
6. E. H. Hardy, On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters.
Quart. J. Math. 37(1906), 53–79.
7. L. V. Zhizhiashvili, Some questions of the theory of trigonometric
Fourier series and their conjugates. (Russian) Tbilisi University Press, 1993.
(Received 29.12.1993; revised version 15.06.1995)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043
Republic of Georgia