Directory UMM :Journals:Journal_of_mathematics:GMJ:
Georgian Mathematical Journal
1(1994), No. 3, 287-302
ON SOME PROPERTIES OF MULTIPLE CONJUGATE
TRIGONOMETRIC SERIES
D. LELADZE
Abstract. We have obtained the estimate in the terms of partial and
mixed moduli of continuity of deviation of Ces´
aro (C, ) means ( =
(α1 , . . . , αn ), αi ∈ R, αi > −1, i = 1, n) of the sequence of rectangular
partial sums of n-multiple (n > 1) conjugate trigonometric series from
n-multiple truncated conjugate function. This estimate implies the
result on the mλ -convergence (λ ≥ 1) of (C, ) means (αi > 0, i =
1, n), provided that the essential conditions are imposed on the partial
moduli of continuity. Finally, it is shown, that the mλ -convergence
cannot be replaced by ordinary convergence.
1. Let f ∈ L([−π; π]n ), n ∈ N, n > 1, be a function, 2π-periodic in
¯n [f ]
each variable, σn [f ] its n-multiple trigonometric Fourier series, and σ
its conjugate series with respect to n variables (see, e.g., [1]).
We set
m = (m1 , . . . , mn ) (mi ∈ N, i = 1, n);
x = (x1 , . . . , xn ) (xi ∈ R, i = 1, n);
α = (α1 , . . . , αn ) (αi ∈ R, αi > −1, i = 1, n); M = {1, 2, . . . , n}.
By Mj we denote the set of all subsets of M with j elements, by M (A)
(A)
a set M \A (A ⊂ M ), and by Mj the set of all subsets of M (A) with j
elements.
For B = {ii , . . . , ik } ⊂ M we define m(B) = {1/mi1 , . . . , 1/mik } and the
1991 Mathematics Subject Classification. 42B08.
287
288
D. LELADZE
truncated conjugate function with respect to the corresponding variables
=
1
(−2π)k
Z
π
...
π
f¯m(B) (x) =
∆k,sik ∆k−1,sik−1 . . . ∆1,si1 (f ; x) . . .
×
1/mik
1/mi1
×
Z
k
Y
ctg
j=1
sij
dsi1 . . . dsik , f¯m (x) = f¯m(M ) (x),
2
where
∆i,h (f, x) = f (x1 , . . . , xi−1 , xi + h, xi+1 , . . . , xn ) −
− f (x1 , . . . , xi−1 , xi − h, xi+1 , . . . , xn ), i = 1, n.
For f ∈ Lq ([−π; π]n ), 1 ≤ q ≤ +∞ (L∞ = C), we consider its mixed
modulus of continuity
=
sup
ωB m(B); f Lq =
h hk−1 h
1
,
∆ k ∆
k−1 . . . ∆1 (f ; x) . . .
k
q
|h1 | −1, f¯m,n (x, y)
α,β
be the two-dimensional truncated conjugate function and σ
¯mn
(x, y; f ) be
the Ces´aro means of the double conjugate series σ
¯2 [f ].
We have ([1], p.187-188):
Z π/m Z π/n
α,β
α
¯m
¯ β (t) dt ds +
(x, y; f ) = 1/π 2
¯mn
(s)K
σ
ψx,y (s, t; f )K
n
0
2
+ 1/(2π )
0
π/m
Z
0
+ 1/π 2
Z
+ 1/π 2
Z
π/m
0
π/m
0
+ 1/(2π 2 )
+ 1/π 2
Z
π
π/m
+ 1/π
2
Z
π
π/m
2
+ 1/(4π )
Z
Z
t ¯α
ψx,y (s, t; f ) ctg K
m (s) dt ds +
2
π/n
π
π/n
π
π/n
π/m
π
Z
Z
π
Z
Z
0
0
Z
π/n
0
π
π/m
β
α
¯m
(t) dt ds +
(s)Hn,2
ψx,y (s, t; f )K
π/n
π/n
Z
β
α
¯m
(s)Hn,1
(t) dt ds +
ψx,y (s, t; f )K
Z
s ¯β
ψx,y (s, t; f ) ctg K
(t) dt ds +
2 n
α
¯ nβ (t) dt ds +
ψx,y (s, t; f )Hm,1
(s)K
α
¯ β (t) dt ds +
ψx,y (s, t; f )Hm,2
(s)K
n
π
π/n
ψx,y (s, t; f ) ctg
t
s
ctg dt ds +
2
2
290
D. LELADZE
2
+ 1/(2π )
π
Z
π/m
π
2
+ 1/(2π )
Z
π/m
π
+ 1/(2π 2 )
+ 1/π 2
+ 1/π 2
Z
Z
π
π/m
+ 1/(2π )
+ 1/π 2
+ 1/π 2
Z
Z
π
16
X
Z
π/m
Z π
π
s β
(t) dt ds +
ψx,y (s, t; f ) ctg Hn,1
2
π/n
π
s β
ψx,y (s, t; f ) ctg Hn,2
(t) dt ds +
2
π/n
π
t α
(s) dt ds +
ψx,y (s, t; f ) ctg Hm,1
2
π/n
π/n
Z π
π/n
π
β
α
ψx,y (s, t; f )Hm,1
(t) dt ds +
(s)Hn,1
β
α
(t) dt ds +
(s)Hn,2
ψx,y (s, t; f )Hm,1
π
t α
ψx,y (s, t; f ) ctg Hm,2
(s) dt ds +
2
π/n
Z
π/m
Z π
π/m
=
Z
π/m
Z π
π/m
Z π
2
Z
π/n
Z π
π/n
β
α
(t) dt ds +
(s)Hn,1
ψx,y (s, t; f )Hm,2
β
α
ψx,y (s, t; f )Hm,2
(t) dt ds =
(s)Hn,2
(j)
Pmn
(x, y; f ),
(2)
j=1
where
ψx,y (s, t; f ) = f (x + s, y + t) − f (x − s, y + t) −
− f (x + s, y − t) + f (x − s, y − t)
(3)
α
α
¯ α (s)
(s) are the summands of the conjugate Fej´er kernel K
(s), Hm,2
and Hm,1
m
(see [2], pp.157-160)
¯ α (s) = 1 ctg s + H α (s) + H α (s),
K
m
m,1
m,2
2
2
1
α
cos (m + 2 + 2 )s − πα
α
2
.
Hm,1 (s) = −
s 1+α
Aα
m (2 sin 2 )
(4)
(5)
Besides,
α
¯m
|K
(s)| ≤ C(α)m,
α
|Hm,2
(s)| ≤
|s| ≤ π;
C(α)
, π/m ≤ |s| ≤ π, m > 1.
m2 s3
(6)
(7)
The estimate (7) is more precise than the estimate (2.1.14) in [1] and it
can be proved by arguments analogous to those in [2], pp.157-160.
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
291
From (2) we obtain
(8)
α,β
α,β
(x, y; f ) =
(x, y; f ) − Pmn
¯mn
σ
¯mn
(x, y; f ) − f¯mn (x, y) = σ
16
X
′
=
(k)
Pmn
(x, y; f ),
(8)
k=1
where ′ indicates that the eighth member is omitted (the replacement of
1/m (1/n) in f¯mn (x, y) by π/m (π/n) does not matter).
In the sequel we will use the inequality (see [3], p.179)
q o1/q Z d n Z b
o1/q
nZ bZ d
≤
|f (x, y)|q dx
dy,
f (x, y) dy dx
(9)
c
a
c
a
1 ≤ q < +∞.
Taking into account (6) and (9), we obtain
(1)
(x, y; f )kLq ≤ C(α, β) ω1 (1/m; f )Lq + ω2 (1/n; f )Lq .
kPmn
(10)
It is easy to see, that
(2)
(x, y; f )
Pmn
1
=−
π
Z
where
Z
1
f¯n(2) (x, y) = −
2π
0
1
(5)
(x, y; f ) = −
Pmn
π
1
(1)
f¯m
(x, y) = −
2π
π/m
α
¯ (s) ds,
f¯n(2) (x + s, y) − f¯n(2) (x − s, y) K
m
π/n
Z
0
β
(1)
(1)
¯ n (t) dt,
f¯m
(x, y + t) − f¯m
(x, y − t) K
π
π/m
Z π
[f (x + s, y)−f (x − s, y)] ctg
s
ds,
2
(11)
[f (x, y + t)−f (x, y − t)] ctg
t
dt.
2
(12)
π/n
Hence, using again (6) and (9), we obtain:
(2)
(x, y; f )kLq ≤ C(α, β)ω1 (1/m; f¯n(2) )Lq ;
kPmn
kP (5) (x, y; f )kLq ≤ C(α, β)ω2 (1/n; f¯(1) )Lq .
m
mn
(3)
(13)
(14)
Now, it is easy to see that for estimation of Pmn (x, y; f ) it suffices to
estimate the integral
Z π/m Z π
¯ α (s)ωβ (t) cos nt dt ds,
ψx,y (s, t; f )K
I(m, n) = n−β
m
0
π/n
where
ωβ (t) =
cos 1+β
2 t
t 1+β .
(sin 2 )
292
D. LELADZE
We have
−β
2I(m, n) = n
nZ
0
π/mZ π
π/n
¯ α (s)ωβ (t) cos nt dt ds+
×K
m
+
Z
π/m
0
−
Z
π−π/n
π/m
0
π/n
Z
Z
0
π/mZ π
ψx,y (s, t+π/n; f ) ωβ (t)−
π/n
α
¯ (s) cos nt dt ds+
−ωβ (t+π/n) K
m
π
Z
ψx,y (s, t; f )−ψx,y (s, t+π/n; f ) ×
0
¯ α (s)ωβ (t + π/n) cos nt dt ds −
ψx,y (s, t + π/n; f )K
m
o
α
¯m
ψx,y (s, t + π/n; f )K
(s)ωβ (t + π/n) cos nt dt ds .
Now we note that (see [1], p.56, (2.1.18))
|ωβ (t) − ωβ (t + π/n)| ≤ C(β)/(nt2+β ),
π
≤ t ≤ π.
n
(15)
(6) and (15) yield kI(m, n)kLq ≤ C(α, β)λ(n, β)ω2 (1/n; f )Lq and hence
(3)
(x, y; f )kLq ≤ C(α, β)λ(n, β)ω2 (1/n; f )Lq .
kPmn
(16)
Analogously
(6)
kPmn
(x, y; f )kLq ≤ C(α, β)λ(m, α)ω1 (1/m; f )Lq .
Furthermore, using (6),(7) and (9), we obtain
(k)
(x, y; f )kLq ≤ C(α, β) ω1 (1/m; f )Lq + ω2 (1/n; f )Lq
kPmn
(k = 4, 7, 16).
(17)
(18)
Analogously, taking into account that
Z
β
1 π ¯(1)
(9)
(1)
(x, y − t) Hn,1
f (x, y + t) − f¯m
Pmn (x, y; f ) = −
(t) dt,
π π/n m
Z
α
1 π ¯(2)
(11)
fn (x + s, y) − f¯n(2) (x − s, y) Hm,1
(s) ds,
Pmn
(x, y; f ) = −
π π/m
we can prove
(1)
(9)
)Lq ;
kPmn
(x, y; f )kLq ≤ C(α, β)λ(n, β)ω2 (1/n; f¯m
kP (11) (x, y; f )kLq ≤ C(α, β)λ(m, α)ω1 (1/m; f¯(2) )Lq .
mn
n
(19)
(20)
Using the same arguments and applying (6), (7) and (9), we can prove
(1)
(10)
)Lq ;
(x, y; f )kLq ≤ C(α, β)ω2 (1/n; f¯m
kPmn
(21)
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
(14)
kPmn
(x, y; f )kLq ≤ C(α, β)ω1 (1/m; f¯n(2) )Lq .
293
(22)
Now we observe that the following lemma holds true (see [1], p.160,
Lemma 10):
Lemma 1. Let f ∈ Lq ([−π; π]2 ), 1 ≤ q ≤ +∞, and
φx,y (s, t; f ) = f (x + s, y + t) + f (x − s, y + t) + f (x + s, y − t) +
+ f (x − s, y − t) − 4f (x, y).
(23)
Then
Z π Z π
sin
sin
nt dt ds q =
φx,y (s, t; f )ωα (s)ωβ (t)
ms
m−α n−β
cos
cos
L
π/m π/n
= O λ(m, α)λ(n, β)ω(1/m, 1/n; f )Lq + λ(m, α)ω1 (1/m; f )Lq +
+λ(n, β)ω2 (1/n; f )Lq .
There is another lemma in [1] (see p.171, Lemma 11), which can be
corrected by means of (7) as follows:
Lemma 2. For f ∈ Lq ([−π; π]2 ), 1 ≤ q ≤ +∞, we have
Z π Z π
sin
−α
β
(t)
ms dt ds =
φx,y (s, t; f )ωα (s)Hn,2
m
cos
Lq
π/m π/n
= O λ(m, α)ω1 (1/m; f )Lq + ω2 (1/n; f )Lq ;
Z π Z π
sin
−β
α
(s)
nt dt ds =
φx,y (s, t; f )ωβ (t)Hm,2
n
cos
Lq
π/m π/n
= O λ(n, β)ω2 (1/n; f )Lq + ω1 (1/m; f )Lq .
The lemmas and (3), (5), (7) and (23) yield
(12)
(13)
(15)
Pmn (x, y; f ) + Pmn
(x, y; f )Lq ≤
(x, y; f ) + Pmn
≤ C(α, β) λ(m, α)λ(n, β)ω(1/m, 1/n; f )Lq + λ(m, α)ω1 (1/m; f )Lq +
+λ(n, β)ω2 (1/n; f )Lq .
(24)
Finally, (2)–(24) yield
α,β
σ
¯mn (x, y; f ) − f¯mn (x, y)Lq ≤
≤ C(α, β) λ(m, α)λ(n, β)ω(1/m, 1/n; f )Lq + λ(m, α)ω1 (1/m; f )Lq +
+λ(n, β)ω2 (1/n; f )Lq + λ(m, α)ω1 (1/m; f¯n(2) )Lq +
(1)
(25)
+λ(n, β)ω2 (1/n; f¯m
)Lq ,
which is the formula (1) in the case n = 2.
294
D. LELADZE
Corollary. If f ∈ C([−π; π]n ) (n ≥ 2) and
ωi (δ; f )C = o 1/ lnn−1 (1/δ) (i = 1, n)
as δ → 0+, then
(26)
α
(x; f ) − f¯m (x)kC = 0, α = (α1 , . . . , αn ), αi > 0
σm
lim k¯
mλ →∞
(i = 1, n), λ ≥ 1.
Now we will prove that the mλ -summability in the corollary is essential.
Namely, the following result holds true:
Theorem 2. There exists a function f ∈ C([−π; π]n ) which satisfies
(26) and
α
σm
(0; f ) − f¯m (0)| = +∞,
lim |¯
(27)
m→∞
α = (α1 , . . . , αn ), αi > 0 (i = 1, n), 0 = (0, 0, . . . , 0).
Proof. We will prove the theorem for n = 2, this case being quite typical.
First, let α, β ∈ (0; 1). We set
mk = 2
22
m
m k
k
, nk =
(k)
(2)
m k
mk k ,
(k−1)
(k ∈ N);
(28)
x, k ∈ N, k ≥ 3;
ln x = ln ln x, ln x = ln ln
1
π
11π
, 6m
< x ≤ 60m
,
k
k
ln(1/(x−π/(6mk ))) ln(2) (1/(x−π/(6mk )))
11π
π
1
,
gk (x) = ln(1/(π/(5mk )−x)) ln(2) (1/(π/(5mk )−x)) , 60mk ≤ x <
5mk
π
π
0,
x ∈ (0; π)\ 6mk ; 5m
.
k
Furthermore,
h(x) =
∞
X
gk (x);
k=1
p(y) =
(
1
,
ln(2π/y) ln(2) (2π/y)
1
,
ln(2π/(π−y)) ln(2) (2π/(π−y))
(
h(x)p(y),
f (x, y) =
0,
y ∈ (0; π2 ],
y ∈ [ π2 ; π);
(x, y) ∈ (0; π)2 ,
(x, y) ∈ [−π; π]2 \(0; π)2 .
Finally, outside the square [−π; π]2 , we extend the function f by periodicity with the period 2π in each variable. It is easy to see that f satisfies
(26).
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
295
From now on we set m = 2mk , n = nk + 1. We have
Z πZ π
2
¯
¯ β (y) dy dx −
¯ α (x)K
− fmn (0, 0) = 1/π
f (x, y)K
n
m
0
0
Z π Z π
x
y
f (x, y) ctg ctg dy dx =
−1/(4π 2 )
2
2
1/m 1/n
Z 1/m Z 1/n
¯ α (x)K
¯ nβ (y) dy dx +
= 1/π 2
f (x, y)K
m
α,β
(0, 0; f )
σ
¯mn
0
+1/π 2
0
1/m
Z
0
+1/π 2
+1/π 2
Z
π
1/m
π
Z
π
1/m
Z
Z
π
1/n
1/n
0
¯ β (y) dy dx +
¯ α (x)K
f (x, y)K
m
n
¯ α (x)K
¯ β (y) dy dx +
f (x, y)K
m
n
α
¯ β (y) − 1/4 ctg x ctg y dy dx =
¯ (x)K
f (x, y) K
n
m
2
2
1/n
Z
4
X
=
Rj (m, n).
(29)
j=1
Obviously,
R1 (m, n) = o(1) (m, n → ∞).
(30)
Then,
R2 (m, n) = 1/π
2
Z
1/m
0
− 1/π
2
Z
0
1/m
π
¯ α (x) 1 ctg y dy dx −
f (x, y)K
m
2
2
1/n
Z
Z
= R2′ (m, n) +
π
¯ α (x)H β (y) dy dx =
f (x, y)K
m
n
1/n
R2′′ (m, n),
(31)
where
Hnα (t) =
n
1 X α−1 cos(ν + 1/2)t
.
A
Anα ν=0 ν
2 sin(t/2)
(32)
The following estimates hold true (see [2], (5.12)):
¯ nα (t)| ≤ n,
|K
|Hnα (t)|
≤ C(α)n
|t| ≤ π;
−α −(α+1)
t
,
1/n ≤ |t| ≤ π.
(33)
(34)
296
D. LELADZE
Now we have
Z
π
1
y
ctg dy =
2
2
p(y)
1/n
+
Z
π
Z
π/2
p(y)
1/n
1
y
ctg dy +
2
2
y
1
ctg dy = U1 (n) + U2 (n).
2
2
p(y)
π/2
(35)
Now,
U1 (n) = −C1
Z
π/2
1/n
d(2πy)
≤
2πy ln(2πy) ln | ln(2πy)|
≤ C2 ln(3) n ≤ C3 mk ln mk ;
(36)
|U2 (n)| ≤ M.
(37)
(31) and (35)–(37) yield
|R2′ (m, n)| ≤
C
mk ln mk .
ln mk+1 ln(2) mk+1
(38)
As to R2′′ (m, n), we have
|R2′′ (m, n)|
≤
C(β)
(2)
ln mk+1 ln
≤
Z
mk+1 0
C(β)
1/m
Z
ln mk+1 ln(2) mk+1
π
1/n
m
dy dx ≤
nβ y β+1
.
(39)
(31), (38) and (39) yield
|R2 (m, n)| ≤
C(β)
ln mm+1 ln(2) mk+1
mk ln mk .
(40)
Analogously
R3 (m, n) = R3′ (m, n) + R3′′ (m, n).
(41)
Now,
C
ln m,
ln n ln(2) n
C(α)
|R3′′ (m, n)| ≤
.
ln n ln(2) n
|R3′ (m, n)| ≤
(42)
(43)
(41)–(43) yield
|R3 (m, n)| = o(1) (m, n → ∞).
(44)
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
297
Now let us consider R4 (m, n). We break it into 4 parts as follows
Z π Z 1/nτ
Z 1/mτ Z 1/nτ Z 1/mτ Z π
+
+
+
R4 (m, n) =
1/m
+
π
Z
1/mτ
Z
1/nτ
1/nτ
1/m
1/n
π
1/mτ
1/n
α
¯ (x)K
¯ β (y) −
1/π 2 f (x, y) K
m
n
4
X
x
y
1
Ij (m, n),
dy dx =
− ctg ctg
4
2
2
j=1
(45)
where 1/2 ≤ τ < 1.
We have
|I4 (m, n)| ≤
≤ C(α, β)
Z
π
1/mτ
Z
π
1/nτ
1
1
1
dy dx.
+
+
mα xα+1 y nβ y β+1 x mα nβ xα+1 y β+1
It is easy to see that
|I4 (m, n)| = o(1) (m, n → ∞).
(46)
Now we estimate I2 (m, n).
I2 (m, n) =
Z
2
|Q2 (m, n)| = 1/π
3
X
k=1
1/mτ Z π
Qk (m, n).
f (x, y)
1/nτ
1/m
≤ C(β) max |f | ln m
(47)
−Hnβ (y)
dy dx ≤
2 tg(x/2)
nβτ
nβ
and
(48)
|Q2 (m, n)| = o(1) (m, n → ∞);
Z 1/mτ Z π
α
|Q3 (m, n)| = 1/π 2
f (x, y)Hm
(x)Hnβ (y) dy dx = o(1) (49)
1/nτ
1/m
(m, n → ∞).
Next we will show that |Q1 (m, n)| → +∞ as m, n → ∞.
1
π
π
We have m
< 6m
< 5m
< m1τ . Therefore
k
k
Q1 (m, n) = 1/π
2
Z
π/5mk
π/6mk
α
(x)) dx ×
h(x)(−Hm
298
D. LELADZE
×
Z
π
p(y)
1/nτ
1
(2)
(1)
dy = Q1 (m)Q1 (n).
2 tg(y/2)
(50)
Since m = 2mk , we have cos(i + 1/2)x > 0 for i = 0, 1, . . . , m and
π
α
x ∈ [ 6m
; π ]. Hence (−Hm
(x)) < 0 (see (32)). Therefore we have
k 5mk
(1)
|Q1 (m)|
≥
2
Z
π/5mk
α
(x)h(x) dx ≥
Hm
π/6mk
Z 23π/120mk X
m
cos 9π
C
20
h(x) dx ≥
Aα−1
≥ α
i
Am 21π/120mk i=0
x
Z 23π/120mk
C
C α
1
dx
≥
A
m
α
(2)
Am
ln mk ln mk 21π/120mk x
ln mk ln(2)
= 1/π
mk
.
(51)
(2)
As to Q1 (n), we have (analogously to (36))
(2)
Q1 (n) ≥ Cmk ln mk .
(52)
(50)–(52) imply
|Q1 (m, n)| ≥
C
ln mk ln(2) mk
mk ln mk .
(53)
(28), (40) and (53) yield
|Q1 (m, n)| − R2 (m, n) → +∞
(54)
as m, n → ∞ (for m = 2mk , n = nk + 1).
From (47)–(49) and (54) we obtain
|I2 (m, n)| − R2 (m, n) → +∞ (m, n → ∞).
(55)
Now let us consider I3 (m, n). As in the case of I2 (m, n), we have
I3 (m, n) =
3
X
(56)
Jν (m, n).
ν=1
Then
Z
|J1 (m, n)| = 1/π
2
1/mτ
α
(x)) dx
h(x)(−Hm
Z
1/nτ
1/n
p(y)
dy ≤
2 tg(y/2)
m
C(α)
ln(2) nτ
(ln(3) nτ −ln(3) n) = α(1−τ ) ln (2) ;
α
m
m
ln n
ατ
≤C(α)
π
(57)
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
Z
|J2 (m, n)| = 1/π 2
≤
π
1/mτ
C(β) ln m
(2)
h(x)
dx
2 tg(x/2)
ln n ln n
Z
2
|J3 (m, n)| = 1/π
Z
1/nτ
1/n
π
1/mτ
dy
nβ y β+1
Z
1/nτ
1/n
mατ nβ
≤C(α, β) α β .
m n
;
Z
1/nτ
1/n
p(y)(−Hnβ (y)) dy ≤
α
(x)Hnβ (y) dy dx ≤
h(x)p(y)Hm
299
(58)
(59)
(56)–(59) yield
I3 (m, n) = o(1) (m, n → ∞).
(60)
Now we consider I1 (m, n). As above,
I1 (m, n) =
3
X
Tj (m, n).
(61)
j=1
We have
Z
|T1 (m, n)| = 1/π 2
1/mτ
1/m
C1 (α)
≤
Z
1/nτ
1/n
1
y
α
(x)) ctg dy dx ≤
h(x)p(y)(−Hm
2
2
C2 (α)
(2)
(2)
mα
ln n;
nk m α
ln mk ln mk ln nk ln
Z 1/mτ Z 1/nτ
1
x
h(x)p(y)(−Hnβ (y)) ctg dy dx ≤
|T2 (m, n)| = 1/π 2
2
2
1/m
1/n
nβ
ln m;
ln mk ln(2) mk ln nk ln(2) nk nβ
Z 1/mτ Z 1/nτ
α
|T3 (m, n)| = 1/π 2
(x)Hnβ (y) dy dx ≤
h(x)p(y)Hm
1/m
≤
C2 (β)
C1 (β)
≤
(63)
1/n
mα nβ
.
ln mk ln(2) mk ln nk ln(2) nk mα nβ
C1 (α, β)
(62)
C2 (α, β)
(64)
(61)–(64) yield
I1 (m, n) = o(1)
(m, n → ∞)
(65)
(we remind once more that m = 2mk , n = nk + 1).
Finally, (29), (30), (44)–(46), (55), (60) and (65) prove the Theorem 2 in
the case n = 2 and α, β ∈ (0; 1).
300
D. LELADZE
For α = 1 we have
Hn1 (t) =
sin(n + 1)t
(n + 1)(2 sin 2t )2
(66)
and an estimate analogous to (34) holds true.
Now we consider the case when α > 1. Using the method represented in
[4], p.507, we obtain
d
n ei(n+1/2)t
X
1
− 21 it
−
e
Re
(1 − e−it )−m −
Aα−m
n
t
(1 − e−it )α
Aα
n 2 sin 2
m=1
∞
o
X
Aα−d−1
(67)
−
e−i(m−n−1/2)t (1 − e−it )−d ,
m
Hnα (t) =
m=n+1
where d = [α].
Taking the real part we obtain that the first term of the finite sum is
0. Therefore if [α] = 1 we apply once more the Abel transformation to the
infinite sum in (67) and obtain
cos((n + 21 + α2 )t − πα
(1 − α)α cos(t/2)
2 )
−
−
t 1+α
α
An (2 sin 2 )
8(n + 1)(n + α)(sin 2t )3
∞
sin(m − 1)t
1 X α−3
A
.
− α
An m=1 m+n+1 (2 sin 2t )3
Hnα (t) =
(68)
Then, again, we have an estimate analogous to (34), which enables us to
fulfil the proof. Namely,
|Hnα (t)| ≤
C2 (α)
C1 (α)
+ 2 3 .
nα tα+1
n t
(69)
If [α] = 2 without further transformation we obtain
cos((n + 21 + α2 )t − πα
(α − 1)α cos(t/2)
2 )
+
+
t 1+α
Aα
8(n
)
+
α − 1)(n + α)(sin 2t )3
(2
sin
n
2
∞
1 X α−3 cos(m − 3/2)t
A
+ α
(70)
An m=1 m+n (2 sin 2t )3
Hnα (t) =
and, again, (69) holds true.
Analogous equations may be obtained if [α] ≥ 3.
Now, when α = 1 and β = 1, we use (66). If α = 1 and β < 1 (or vice
versa), we use (34) and (66). If α = 1 and β > 1 (or vice versa), we use
(66) and (67) (for the corresponding d). If α > 1 and β < 1 (or vice versa),
we use again (67) (for the corresponding d) and (34).
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
301
In the n-dimensional case we define f as follows. We set
2k
mk = 22 ,
(k ∈ N);
1
11π
π
, 6m
< x ≤ 60m
,
k
k
lnn−1 (1/(x−π/(6mk ))) ln(n) (1/(x−π/(6mk )))
11π
π
1
,
gk (x) = lnn−1 (1/(π/(5mk )−x)) ln(n) (1/(π/(5mk )−x)) , 60mk ≤ x <
5mk
π
π
x ∈ (0; π)\ 6mk ; 5m
0,
,
k
Again
h(x) =
∞
X
gk (x).
k=1
Then,
=
lnn−1 Qn
i=2
p(x2 , . . . , xn ) =
1
(2π)n−1
ln(n) Qn
(π/2−|π/2−xi |)
for (x2 , . . . , xn ) ∈ (0; π)n−1 .
And, finally
(
h(x1 )p(x2 , . . . , xn ),
f (x1 , . . . , xn ) =
0,
i=2
(2π)n−1
(π/2−|π/2−xi |)
(71)
(x1 , . . . , xn ) ∈ (0; π)n ,
(x1 , . . . , xn ) ∈ [−π; π]n \(0; π)n .
Outside [−π; π]n we extend the function f by periodicity with the period
2π in each variable.
We observe, that functions of the p(x2 , . . . , xn )-type were for the first time
introduced and applied in the works of L.Zhizhiashvili (see [1], [5]).
Remark 1. For the function f (x1 , . . . , xn ) a stronger condition than
(26) holds true, namely,
ωi (δ; f )C ≤
C(f, n)
n−1
ln
(1/δ) ln(n) (1/δ)
, i = 1, n.
Remark 2. Results analogous to Theorem 1, the corollary and Theorem
2 hold true for the n-multiple Abel–Poisson summability method.
References
1. L.V. Zhizhiashvili, Conjugate functions and trigonometric series. (Russian) Tbilisi University Press, Tbilisi, 1969.
302
D. LELADZE
2. A. Zygmund, Trigonometric series, v.1. (Translated into the Russian)
”Mir”, Moscow, 1965; English original: Cambridge, 1959.
3. G.H. Hardy, J.E. Littlewood and G. P´olya, Inequalities. (Translated
into the Russian) ”IL”, Moscow, 1949; English original: Cambridge, 1934.
4. M.M. Lekishvili, On approximation of periodic functions by (C, α)
means. (Russian) Mat.Sb. 121(163)(1983), No.4(8), 499-509.
5. L.V. Zhizhiashvili, On some problems from the theory of simple and
multiple trigonometric and orthogonal series. (Russian) Uspekhi Mat.Nauk
28(1973), No.2, 65-119.
(Received 25.02.1993,
revized version 13.05.1993)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., 380043 Tbilisi
Republic of Georgia
1(1994), No. 3, 287-302
ON SOME PROPERTIES OF MULTIPLE CONJUGATE
TRIGONOMETRIC SERIES
D. LELADZE
Abstract. We have obtained the estimate in the terms of partial and
mixed moduli of continuity of deviation of Ces´
aro (C, ) means ( =
(α1 , . . . , αn ), αi ∈ R, αi > −1, i = 1, n) of the sequence of rectangular
partial sums of n-multiple (n > 1) conjugate trigonometric series from
n-multiple truncated conjugate function. This estimate implies the
result on the mλ -convergence (λ ≥ 1) of (C, ) means (αi > 0, i =
1, n), provided that the essential conditions are imposed on the partial
moduli of continuity. Finally, it is shown, that the mλ -convergence
cannot be replaced by ordinary convergence.
1. Let f ∈ L([−π; π]n ), n ∈ N, n > 1, be a function, 2π-periodic in
¯n [f ]
each variable, σn [f ] its n-multiple trigonometric Fourier series, and σ
its conjugate series with respect to n variables (see, e.g., [1]).
We set
m = (m1 , . . . , mn ) (mi ∈ N, i = 1, n);
x = (x1 , . . . , xn ) (xi ∈ R, i = 1, n);
α = (α1 , . . . , αn ) (αi ∈ R, αi > −1, i = 1, n); M = {1, 2, . . . , n}.
By Mj we denote the set of all subsets of M with j elements, by M (A)
(A)
a set M \A (A ⊂ M ), and by Mj the set of all subsets of M (A) with j
elements.
For B = {ii , . . . , ik } ⊂ M we define m(B) = {1/mi1 , . . . , 1/mik } and the
1991 Mathematics Subject Classification. 42B08.
287
288
D. LELADZE
truncated conjugate function with respect to the corresponding variables
=
1
(−2π)k
Z
π
...
π
f¯m(B) (x) =
∆k,sik ∆k−1,sik−1 . . . ∆1,si1 (f ; x) . . .
×
1/mik
1/mi1
×
Z
k
Y
ctg
j=1
sij
dsi1 . . . dsik , f¯m (x) = f¯m(M ) (x),
2
where
∆i,h (f, x) = f (x1 , . . . , xi−1 , xi + h, xi+1 , . . . , xn ) −
− f (x1 , . . . , xi−1 , xi − h, xi+1 , . . . , xn ), i = 1, n.
For f ∈ Lq ([−π; π]n ), 1 ≤ q ≤ +∞ (L∞ = C), we consider its mixed
modulus of continuity
=
sup
ωB m(B); f Lq =
h hk−1 h
1
,
∆ k ∆
k−1 . . . ∆1 (f ; x) . . .
k
q
|h1 | −1, f¯m,n (x, y)
α,β
be the two-dimensional truncated conjugate function and σ
¯mn
(x, y; f ) be
the Ces´aro means of the double conjugate series σ
¯2 [f ].
We have ([1], p.187-188):
Z π/m Z π/n
α,β
α
¯m
¯ β (t) dt ds +
(x, y; f ) = 1/π 2
¯mn
(s)K
σ
ψx,y (s, t; f )K
n
0
2
+ 1/(2π )
0
π/m
Z
0
+ 1/π 2
Z
+ 1/π 2
Z
π/m
0
π/m
0
+ 1/(2π 2 )
+ 1/π 2
Z
π
π/m
+ 1/π
2
Z
π
π/m
2
+ 1/(4π )
Z
Z
t ¯α
ψx,y (s, t; f ) ctg K
m (s) dt ds +
2
π/n
π
π/n
π
π/n
π/m
π
Z
Z
π
Z
Z
0
0
Z
π/n
0
π
π/m
β
α
¯m
(t) dt ds +
(s)Hn,2
ψx,y (s, t; f )K
π/n
π/n
Z
β
α
¯m
(s)Hn,1
(t) dt ds +
ψx,y (s, t; f )K
Z
s ¯β
ψx,y (s, t; f ) ctg K
(t) dt ds +
2 n
α
¯ nβ (t) dt ds +
ψx,y (s, t; f )Hm,1
(s)K
α
¯ β (t) dt ds +
ψx,y (s, t; f )Hm,2
(s)K
n
π
π/n
ψx,y (s, t; f ) ctg
t
s
ctg dt ds +
2
2
290
D. LELADZE
2
+ 1/(2π )
π
Z
π/m
π
2
+ 1/(2π )
Z
π/m
π
+ 1/(2π 2 )
+ 1/π 2
+ 1/π 2
Z
Z
π
π/m
+ 1/(2π )
+ 1/π 2
+ 1/π 2
Z
Z
π
16
X
Z
π/m
Z π
π
s β
(t) dt ds +
ψx,y (s, t; f ) ctg Hn,1
2
π/n
π
s β
ψx,y (s, t; f ) ctg Hn,2
(t) dt ds +
2
π/n
π
t α
(s) dt ds +
ψx,y (s, t; f ) ctg Hm,1
2
π/n
π/n
Z π
π/n
π
β
α
ψx,y (s, t; f )Hm,1
(t) dt ds +
(s)Hn,1
β
α
(t) dt ds +
(s)Hn,2
ψx,y (s, t; f )Hm,1
π
t α
ψx,y (s, t; f ) ctg Hm,2
(s) dt ds +
2
π/n
Z
π/m
Z π
π/m
=
Z
π/m
Z π
π/m
Z π
2
Z
π/n
Z π
π/n
β
α
(t) dt ds +
(s)Hn,1
ψx,y (s, t; f )Hm,2
β
α
ψx,y (s, t; f )Hm,2
(t) dt ds =
(s)Hn,2
(j)
Pmn
(x, y; f ),
(2)
j=1
where
ψx,y (s, t; f ) = f (x + s, y + t) − f (x − s, y + t) −
− f (x + s, y − t) + f (x − s, y − t)
(3)
α
α
¯ α (s)
(s) are the summands of the conjugate Fej´er kernel K
(s), Hm,2
and Hm,1
m
(see [2], pp.157-160)
¯ α (s) = 1 ctg s + H α (s) + H α (s),
K
m
m,1
m,2
2
2
1
α
cos (m + 2 + 2 )s − πα
α
2
.
Hm,1 (s) = −
s 1+α
Aα
m (2 sin 2 )
(4)
(5)
Besides,
α
¯m
|K
(s)| ≤ C(α)m,
α
|Hm,2
(s)| ≤
|s| ≤ π;
C(α)
, π/m ≤ |s| ≤ π, m > 1.
m2 s3
(6)
(7)
The estimate (7) is more precise than the estimate (2.1.14) in [1] and it
can be proved by arguments analogous to those in [2], pp.157-160.
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
291
From (2) we obtain
(8)
α,β
α,β
(x, y; f ) =
(x, y; f ) − Pmn
¯mn
σ
¯mn
(x, y; f ) − f¯mn (x, y) = σ
16
X
′
=
(k)
Pmn
(x, y; f ),
(8)
k=1
where ′ indicates that the eighth member is omitted (the replacement of
1/m (1/n) in f¯mn (x, y) by π/m (π/n) does not matter).
In the sequel we will use the inequality (see [3], p.179)
q o1/q Z d n Z b
o1/q
nZ bZ d
≤
|f (x, y)|q dx
dy,
f (x, y) dy dx
(9)
c
a
c
a
1 ≤ q < +∞.
Taking into account (6) and (9), we obtain
(1)
(x, y; f )kLq ≤ C(α, β) ω1 (1/m; f )Lq + ω2 (1/n; f )Lq .
kPmn
(10)
It is easy to see, that
(2)
(x, y; f )
Pmn
1
=−
π
Z
where
Z
1
f¯n(2) (x, y) = −
2π
0
1
(5)
(x, y; f ) = −
Pmn
π
1
(1)
f¯m
(x, y) = −
2π
π/m
α
¯ (s) ds,
f¯n(2) (x + s, y) − f¯n(2) (x − s, y) K
m
π/n
Z
0
β
(1)
(1)
¯ n (t) dt,
f¯m
(x, y + t) − f¯m
(x, y − t) K
π
π/m
Z π
[f (x + s, y)−f (x − s, y)] ctg
s
ds,
2
(11)
[f (x, y + t)−f (x, y − t)] ctg
t
dt.
2
(12)
π/n
Hence, using again (6) and (9), we obtain:
(2)
(x, y; f )kLq ≤ C(α, β)ω1 (1/m; f¯n(2) )Lq ;
kPmn
kP (5) (x, y; f )kLq ≤ C(α, β)ω2 (1/n; f¯(1) )Lq .
m
mn
(3)
(13)
(14)
Now, it is easy to see that for estimation of Pmn (x, y; f ) it suffices to
estimate the integral
Z π/m Z π
¯ α (s)ωβ (t) cos nt dt ds,
ψx,y (s, t; f )K
I(m, n) = n−β
m
0
π/n
where
ωβ (t) =
cos 1+β
2 t
t 1+β .
(sin 2 )
292
D. LELADZE
We have
−β
2I(m, n) = n
nZ
0
π/mZ π
π/n
¯ α (s)ωβ (t) cos nt dt ds+
×K
m
+
Z
π/m
0
−
Z
π−π/n
π/m
0
π/n
Z
Z
0
π/mZ π
ψx,y (s, t+π/n; f ) ωβ (t)−
π/n
α
¯ (s) cos nt dt ds+
−ωβ (t+π/n) K
m
π
Z
ψx,y (s, t; f )−ψx,y (s, t+π/n; f ) ×
0
¯ α (s)ωβ (t + π/n) cos nt dt ds −
ψx,y (s, t + π/n; f )K
m
o
α
¯m
ψx,y (s, t + π/n; f )K
(s)ωβ (t + π/n) cos nt dt ds .
Now we note that (see [1], p.56, (2.1.18))
|ωβ (t) − ωβ (t + π/n)| ≤ C(β)/(nt2+β ),
π
≤ t ≤ π.
n
(15)
(6) and (15) yield kI(m, n)kLq ≤ C(α, β)λ(n, β)ω2 (1/n; f )Lq and hence
(3)
(x, y; f )kLq ≤ C(α, β)λ(n, β)ω2 (1/n; f )Lq .
kPmn
(16)
Analogously
(6)
kPmn
(x, y; f )kLq ≤ C(α, β)λ(m, α)ω1 (1/m; f )Lq .
Furthermore, using (6),(7) and (9), we obtain
(k)
(x, y; f )kLq ≤ C(α, β) ω1 (1/m; f )Lq + ω2 (1/n; f )Lq
kPmn
(k = 4, 7, 16).
(17)
(18)
Analogously, taking into account that
Z
β
1 π ¯(1)
(9)
(1)
(x, y − t) Hn,1
f (x, y + t) − f¯m
Pmn (x, y; f ) = −
(t) dt,
π π/n m
Z
α
1 π ¯(2)
(11)
fn (x + s, y) − f¯n(2) (x − s, y) Hm,1
(s) ds,
Pmn
(x, y; f ) = −
π π/m
we can prove
(1)
(9)
)Lq ;
kPmn
(x, y; f )kLq ≤ C(α, β)λ(n, β)ω2 (1/n; f¯m
kP (11) (x, y; f )kLq ≤ C(α, β)λ(m, α)ω1 (1/m; f¯(2) )Lq .
mn
n
(19)
(20)
Using the same arguments and applying (6), (7) and (9), we can prove
(1)
(10)
)Lq ;
(x, y; f )kLq ≤ C(α, β)ω2 (1/n; f¯m
kPmn
(21)
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
(14)
kPmn
(x, y; f )kLq ≤ C(α, β)ω1 (1/m; f¯n(2) )Lq .
293
(22)
Now we observe that the following lemma holds true (see [1], p.160,
Lemma 10):
Lemma 1. Let f ∈ Lq ([−π; π]2 ), 1 ≤ q ≤ +∞, and
φx,y (s, t; f ) = f (x + s, y + t) + f (x − s, y + t) + f (x + s, y − t) +
+ f (x − s, y − t) − 4f (x, y).
(23)
Then
Z π Z π
sin
sin
nt dt ds q =
φx,y (s, t; f )ωα (s)ωβ (t)
ms
m−α n−β
cos
cos
L
π/m π/n
= O λ(m, α)λ(n, β)ω(1/m, 1/n; f )Lq + λ(m, α)ω1 (1/m; f )Lq +
+λ(n, β)ω2 (1/n; f )Lq .
There is another lemma in [1] (see p.171, Lemma 11), which can be
corrected by means of (7) as follows:
Lemma 2. For f ∈ Lq ([−π; π]2 ), 1 ≤ q ≤ +∞, we have
Z π Z π
sin
−α
β
(t)
ms dt ds =
φx,y (s, t; f )ωα (s)Hn,2
m
cos
Lq
π/m π/n
= O λ(m, α)ω1 (1/m; f )Lq + ω2 (1/n; f )Lq ;
Z π Z π
sin
−β
α
(s)
nt dt ds =
φx,y (s, t; f )ωβ (t)Hm,2
n
cos
Lq
π/m π/n
= O λ(n, β)ω2 (1/n; f )Lq + ω1 (1/m; f )Lq .
The lemmas and (3), (5), (7) and (23) yield
(12)
(13)
(15)
Pmn (x, y; f ) + Pmn
(x, y; f )Lq ≤
(x, y; f ) + Pmn
≤ C(α, β) λ(m, α)λ(n, β)ω(1/m, 1/n; f )Lq + λ(m, α)ω1 (1/m; f )Lq +
+λ(n, β)ω2 (1/n; f )Lq .
(24)
Finally, (2)–(24) yield
α,β
σ
¯mn (x, y; f ) − f¯mn (x, y)Lq ≤
≤ C(α, β) λ(m, α)λ(n, β)ω(1/m, 1/n; f )Lq + λ(m, α)ω1 (1/m; f )Lq +
+λ(n, β)ω2 (1/n; f )Lq + λ(m, α)ω1 (1/m; f¯n(2) )Lq +
(1)
(25)
+λ(n, β)ω2 (1/n; f¯m
)Lq ,
which is the formula (1) in the case n = 2.
294
D. LELADZE
Corollary. If f ∈ C([−π; π]n ) (n ≥ 2) and
ωi (δ; f )C = o 1/ lnn−1 (1/δ) (i = 1, n)
as δ → 0+, then
(26)
α
(x; f ) − f¯m (x)kC = 0, α = (α1 , . . . , αn ), αi > 0
σm
lim k¯
mλ →∞
(i = 1, n), λ ≥ 1.
Now we will prove that the mλ -summability in the corollary is essential.
Namely, the following result holds true:
Theorem 2. There exists a function f ∈ C([−π; π]n ) which satisfies
(26) and
α
σm
(0; f ) − f¯m (0)| = +∞,
lim |¯
(27)
m→∞
α = (α1 , . . . , αn ), αi > 0 (i = 1, n), 0 = (0, 0, . . . , 0).
Proof. We will prove the theorem for n = 2, this case being quite typical.
First, let α, β ∈ (0; 1). We set
mk = 2
22
m
m k
k
, nk =
(k)
(2)
m k
mk k ,
(k−1)
(k ∈ N);
(28)
x, k ∈ N, k ≥ 3;
ln x = ln ln x, ln x = ln ln
1
π
11π
, 6m
< x ≤ 60m
,
k
k
ln(1/(x−π/(6mk ))) ln(2) (1/(x−π/(6mk )))
11π
π
1
,
gk (x) = ln(1/(π/(5mk )−x)) ln(2) (1/(π/(5mk )−x)) , 60mk ≤ x <
5mk
π
π
0,
x ∈ (0; π)\ 6mk ; 5m
.
k
Furthermore,
h(x) =
∞
X
gk (x);
k=1
p(y) =
(
1
,
ln(2π/y) ln(2) (2π/y)
1
,
ln(2π/(π−y)) ln(2) (2π/(π−y))
(
h(x)p(y),
f (x, y) =
0,
y ∈ (0; π2 ],
y ∈ [ π2 ; π);
(x, y) ∈ (0; π)2 ,
(x, y) ∈ [−π; π]2 \(0; π)2 .
Finally, outside the square [−π; π]2 , we extend the function f by periodicity with the period 2π in each variable. It is easy to see that f satisfies
(26).
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
295
From now on we set m = 2mk , n = nk + 1. We have
Z πZ π
2
¯
¯ β (y) dy dx −
¯ α (x)K
− fmn (0, 0) = 1/π
f (x, y)K
n
m
0
0
Z π Z π
x
y
f (x, y) ctg ctg dy dx =
−1/(4π 2 )
2
2
1/m 1/n
Z 1/m Z 1/n
¯ α (x)K
¯ nβ (y) dy dx +
= 1/π 2
f (x, y)K
m
α,β
(0, 0; f )
σ
¯mn
0
+1/π 2
0
1/m
Z
0
+1/π 2
+1/π 2
Z
π
1/m
π
Z
π
1/m
Z
Z
π
1/n
1/n
0
¯ β (y) dy dx +
¯ α (x)K
f (x, y)K
m
n
¯ α (x)K
¯ β (y) dy dx +
f (x, y)K
m
n
α
¯ β (y) − 1/4 ctg x ctg y dy dx =
¯ (x)K
f (x, y) K
n
m
2
2
1/n
Z
4
X
=
Rj (m, n).
(29)
j=1
Obviously,
R1 (m, n) = o(1) (m, n → ∞).
(30)
Then,
R2 (m, n) = 1/π
2
Z
1/m
0
− 1/π
2
Z
0
1/m
π
¯ α (x) 1 ctg y dy dx −
f (x, y)K
m
2
2
1/n
Z
Z
= R2′ (m, n) +
π
¯ α (x)H β (y) dy dx =
f (x, y)K
m
n
1/n
R2′′ (m, n),
(31)
where
Hnα (t) =
n
1 X α−1 cos(ν + 1/2)t
.
A
Anα ν=0 ν
2 sin(t/2)
(32)
The following estimates hold true (see [2], (5.12)):
¯ nα (t)| ≤ n,
|K
|Hnα (t)|
≤ C(α)n
|t| ≤ π;
−α −(α+1)
t
,
1/n ≤ |t| ≤ π.
(33)
(34)
296
D. LELADZE
Now we have
Z
π
1
y
ctg dy =
2
2
p(y)
1/n
+
Z
π
Z
π/2
p(y)
1/n
1
y
ctg dy +
2
2
y
1
ctg dy = U1 (n) + U2 (n).
2
2
p(y)
π/2
(35)
Now,
U1 (n) = −C1
Z
π/2
1/n
d(2πy)
≤
2πy ln(2πy) ln | ln(2πy)|
≤ C2 ln(3) n ≤ C3 mk ln mk ;
(36)
|U2 (n)| ≤ M.
(37)
(31) and (35)–(37) yield
|R2′ (m, n)| ≤
C
mk ln mk .
ln mk+1 ln(2) mk+1
(38)
As to R2′′ (m, n), we have
|R2′′ (m, n)|
≤
C(β)
(2)
ln mk+1 ln
≤
Z
mk+1 0
C(β)
1/m
Z
ln mk+1 ln(2) mk+1
π
1/n
m
dy dx ≤
nβ y β+1
.
(39)
(31), (38) and (39) yield
|R2 (m, n)| ≤
C(β)
ln mm+1 ln(2) mk+1
mk ln mk .
(40)
Analogously
R3 (m, n) = R3′ (m, n) + R3′′ (m, n).
(41)
Now,
C
ln m,
ln n ln(2) n
C(α)
|R3′′ (m, n)| ≤
.
ln n ln(2) n
|R3′ (m, n)| ≤
(42)
(43)
(41)–(43) yield
|R3 (m, n)| = o(1) (m, n → ∞).
(44)
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
297
Now let us consider R4 (m, n). We break it into 4 parts as follows
Z π Z 1/nτ
Z 1/mτ Z 1/nτ Z 1/mτ Z π
+
+
+
R4 (m, n) =
1/m
+
π
Z
1/mτ
Z
1/nτ
1/nτ
1/m
1/n
π
1/mτ
1/n
α
¯ (x)K
¯ β (y) −
1/π 2 f (x, y) K
m
n
4
X
x
y
1
Ij (m, n),
dy dx =
− ctg ctg
4
2
2
j=1
(45)
where 1/2 ≤ τ < 1.
We have
|I4 (m, n)| ≤
≤ C(α, β)
Z
π
1/mτ
Z
π
1/nτ
1
1
1
dy dx.
+
+
mα xα+1 y nβ y β+1 x mα nβ xα+1 y β+1
It is easy to see that
|I4 (m, n)| = o(1) (m, n → ∞).
(46)
Now we estimate I2 (m, n).
I2 (m, n) =
Z
2
|Q2 (m, n)| = 1/π
3
X
k=1
1/mτ Z π
Qk (m, n).
f (x, y)
1/nτ
1/m
≤ C(β) max |f | ln m
(47)
−Hnβ (y)
dy dx ≤
2 tg(x/2)
nβτ
nβ
and
(48)
|Q2 (m, n)| = o(1) (m, n → ∞);
Z 1/mτ Z π
α
|Q3 (m, n)| = 1/π 2
f (x, y)Hm
(x)Hnβ (y) dy dx = o(1) (49)
1/nτ
1/m
(m, n → ∞).
Next we will show that |Q1 (m, n)| → +∞ as m, n → ∞.
1
π
π
We have m
< 6m
< 5m
< m1τ . Therefore
k
k
Q1 (m, n) = 1/π
2
Z
π/5mk
π/6mk
α
(x)) dx ×
h(x)(−Hm
298
D. LELADZE
×
Z
π
p(y)
1/nτ
1
(2)
(1)
dy = Q1 (m)Q1 (n).
2 tg(y/2)
(50)
Since m = 2mk , we have cos(i + 1/2)x > 0 for i = 0, 1, . . . , m and
π
α
x ∈ [ 6m
; π ]. Hence (−Hm
(x)) < 0 (see (32)). Therefore we have
k 5mk
(1)
|Q1 (m)|
≥
2
Z
π/5mk
α
(x)h(x) dx ≥
Hm
π/6mk
Z 23π/120mk X
m
cos 9π
C
20
h(x) dx ≥
Aα−1
≥ α
i
Am 21π/120mk i=0
x
Z 23π/120mk
C
C α
1
dx
≥
A
m
α
(2)
Am
ln mk ln mk 21π/120mk x
ln mk ln(2)
= 1/π
mk
.
(51)
(2)
As to Q1 (n), we have (analogously to (36))
(2)
Q1 (n) ≥ Cmk ln mk .
(52)
(50)–(52) imply
|Q1 (m, n)| ≥
C
ln mk ln(2) mk
mk ln mk .
(53)
(28), (40) and (53) yield
|Q1 (m, n)| − R2 (m, n) → +∞
(54)
as m, n → ∞ (for m = 2mk , n = nk + 1).
From (47)–(49) and (54) we obtain
|I2 (m, n)| − R2 (m, n) → +∞ (m, n → ∞).
(55)
Now let us consider I3 (m, n). As in the case of I2 (m, n), we have
I3 (m, n) =
3
X
(56)
Jν (m, n).
ν=1
Then
Z
|J1 (m, n)| = 1/π
2
1/mτ
α
(x)) dx
h(x)(−Hm
Z
1/nτ
1/n
p(y)
dy ≤
2 tg(y/2)
m
C(α)
ln(2) nτ
(ln(3) nτ −ln(3) n) = α(1−τ ) ln (2) ;
α
m
m
ln n
ατ
≤C(α)
π
(57)
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
Z
|J2 (m, n)| = 1/π 2
≤
π
1/mτ
C(β) ln m
(2)
h(x)
dx
2 tg(x/2)
ln n ln n
Z
2
|J3 (m, n)| = 1/π
Z
1/nτ
1/n
π
1/mτ
dy
nβ y β+1
Z
1/nτ
1/n
mατ nβ
≤C(α, β) α β .
m n
;
Z
1/nτ
1/n
p(y)(−Hnβ (y)) dy ≤
α
(x)Hnβ (y) dy dx ≤
h(x)p(y)Hm
299
(58)
(59)
(56)–(59) yield
I3 (m, n) = o(1) (m, n → ∞).
(60)
Now we consider I1 (m, n). As above,
I1 (m, n) =
3
X
Tj (m, n).
(61)
j=1
We have
Z
|T1 (m, n)| = 1/π 2
1/mτ
1/m
C1 (α)
≤
Z
1/nτ
1/n
1
y
α
(x)) ctg dy dx ≤
h(x)p(y)(−Hm
2
2
C2 (α)
(2)
(2)
mα
ln n;
nk m α
ln mk ln mk ln nk ln
Z 1/mτ Z 1/nτ
1
x
h(x)p(y)(−Hnβ (y)) ctg dy dx ≤
|T2 (m, n)| = 1/π 2
2
2
1/m
1/n
nβ
ln m;
ln mk ln(2) mk ln nk ln(2) nk nβ
Z 1/mτ Z 1/nτ
α
|T3 (m, n)| = 1/π 2
(x)Hnβ (y) dy dx ≤
h(x)p(y)Hm
1/m
≤
C2 (β)
C1 (β)
≤
(63)
1/n
mα nβ
.
ln mk ln(2) mk ln nk ln(2) nk mα nβ
C1 (α, β)
(62)
C2 (α, β)
(64)
(61)–(64) yield
I1 (m, n) = o(1)
(m, n → ∞)
(65)
(we remind once more that m = 2mk , n = nk + 1).
Finally, (29), (30), (44)–(46), (55), (60) and (65) prove the Theorem 2 in
the case n = 2 and α, β ∈ (0; 1).
300
D. LELADZE
For α = 1 we have
Hn1 (t) =
sin(n + 1)t
(n + 1)(2 sin 2t )2
(66)
and an estimate analogous to (34) holds true.
Now we consider the case when α > 1. Using the method represented in
[4], p.507, we obtain
d
n ei(n+1/2)t
X
1
− 21 it
−
e
Re
(1 − e−it )−m −
Aα−m
n
t
(1 − e−it )α
Aα
n 2 sin 2
m=1
∞
o
X
Aα−d−1
(67)
−
e−i(m−n−1/2)t (1 − e−it )−d ,
m
Hnα (t) =
m=n+1
where d = [α].
Taking the real part we obtain that the first term of the finite sum is
0. Therefore if [α] = 1 we apply once more the Abel transformation to the
infinite sum in (67) and obtain
cos((n + 21 + α2 )t − πα
(1 − α)α cos(t/2)
2 )
−
−
t 1+α
α
An (2 sin 2 )
8(n + 1)(n + α)(sin 2t )3
∞
sin(m − 1)t
1 X α−3
A
.
− α
An m=1 m+n+1 (2 sin 2t )3
Hnα (t) =
(68)
Then, again, we have an estimate analogous to (34), which enables us to
fulfil the proof. Namely,
|Hnα (t)| ≤
C2 (α)
C1 (α)
+ 2 3 .
nα tα+1
n t
(69)
If [α] = 2 without further transformation we obtain
cos((n + 21 + α2 )t − πα
(α − 1)α cos(t/2)
2 )
+
+
t 1+α
Aα
8(n
)
+
α − 1)(n + α)(sin 2t )3
(2
sin
n
2
∞
1 X α−3 cos(m − 3/2)t
A
+ α
(70)
An m=1 m+n (2 sin 2t )3
Hnα (t) =
and, again, (69) holds true.
Analogous equations may be obtained if [α] ≥ 3.
Now, when α = 1 and β = 1, we use (66). If α = 1 and β < 1 (or vice
versa), we use (34) and (66). If α = 1 and β > 1 (or vice versa), we use
(66) and (67) (for the corresponding d). If α > 1 and β < 1 (or vice versa),
we use again (67) (for the corresponding d) and (34).
ON SOME PROPERTIES OF TRIGONOMETRIC SERIES
301
In the n-dimensional case we define f as follows. We set
2k
mk = 22 ,
(k ∈ N);
1
11π
π
, 6m
< x ≤ 60m
,
k
k
lnn−1 (1/(x−π/(6mk ))) ln(n) (1/(x−π/(6mk )))
11π
π
1
,
gk (x) = lnn−1 (1/(π/(5mk )−x)) ln(n) (1/(π/(5mk )−x)) , 60mk ≤ x <
5mk
π
π
x ∈ (0; π)\ 6mk ; 5m
0,
,
k
Again
h(x) =
∞
X
gk (x).
k=1
Then,
=
lnn−1 Qn
i=2
p(x2 , . . . , xn ) =
1
(2π)n−1
ln(n) Qn
(π/2−|π/2−xi |)
for (x2 , . . . , xn ) ∈ (0; π)n−1 .
And, finally
(
h(x1 )p(x2 , . . . , xn ),
f (x1 , . . . , xn ) =
0,
i=2
(2π)n−1
(π/2−|π/2−xi |)
(71)
(x1 , . . . , xn ) ∈ (0; π)n ,
(x1 , . . . , xn ) ∈ [−π; π]n \(0; π)n .
Outside [−π; π]n we extend the function f by periodicity with the period
2π in each variable.
We observe, that functions of the p(x2 , . . . , xn )-type were for the first time
introduced and applied in the works of L.Zhizhiashvili (see [1], [5]).
Remark 1. For the function f (x1 , . . . , xn ) a stronger condition than
(26) holds true, namely,
ωi (δ; f )C ≤
C(f, n)
n−1
ln
(1/δ) ln(n) (1/δ)
, i = 1, n.
Remark 2. Results analogous to Theorem 1, the corollary and Theorem
2 hold true for the n-multiple Abel–Poisson summability method.
References
1. L.V. Zhizhiashvili, Conjugate functions and trigonometric series. (Russian) Tbilisi University Press, Tbilisi, 1969.
302
D. LELADZE
2. A. Zygmund, Trigonometric series, v.1. (Translated into the Russian)
”Mir”, Moscow, 1965; English original: Cambridge, 1959.
3. G.H. Hardy, J.E. Littlewood and G. P´olya, Inequalities. (Translated
into the Russian) ”IL”, Moscow, 1949; English original: Cambridge, 1934.
4. M.M. Lekishvili, On approximation of periodic functions by (C, α)
means. (Russian) Mat.Sb. 121(163)(1983), No.4(8), 499-509.
5. L.V. Zhizhiashvili, On some problems from the theory of simple and
multiple trigonometric and orthogonal series. (Russian) Uspekhi Mat.Nauk
28(1973), No.2, 65-119.
(Received 25.02.1993,
revized version 13.05.1993)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., 380043 Tbilisi
Republic of Georgia