Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
27 (2011), 233–243
www.emis.de/journals
ISSN 1786-0091
CERTAIN ESTIMATES FOR DOUBLE SINE SERIES WITH
MULTIPLE-MONOTONE COEFFICIENTS
XHEVAT Z. KRASNIQI
Abstract. In this paper we obtain estimates of the sum of double sine
series near the origin, with multiple-monotone coefficients tending to zero.
These estimates extend some results of Telyakovski [11] and Popov [7] from
single to multidimensional case.
1. Introduction and Preliminaries
Many authors considered the sine series
g(x) :=
∞
X
an sin nx
n=1
with monotone coefficients tending to zero. Young [13] was the first to consider
the problem of estimates of g(x) for x → 0 expressed in terms of the coefficients
an . Then Salem [8], [9], Shogunbekov [10], Aljančić, Bojanić and Tomić [1]
considered this problem, as well. Later Telyakovski in [11] has proved the
following fact:
π
π
Theorem T. Assume that an ↓ 0. Then for x ∈ m+1
,m
≡ Im , m = 1, 2, . . .
the following estimate is valid:
!
m
m
X
1 X 3
n an .
g(x) =
nan x + O
m3 n=1
n=1
Likewise, among others Popov [7] has proved
Theorem P. For any nonincreasing sequence of positive numbers ak tending
to zero, the following inequality holds:
1
x
− a1 sin ≤ g(x), ∀x ∈ (0, π] .
2
2
2010 Mathematics Subject Classification. 42A20, 42A16.
Key words and phrases. sine series, asymptotic behaviour, multiple-monotone coefficients.
233
234
XHEVAT Z. KRASNIQI
The problem shown above is considered by the present author [4]–[6] as well.
The object of this work is to extend these two theorems, formulated above, from
single to multidimensional case. In fact, we shall investigate the behavior near
the origin of the sum of double sine series with multiple-monotone coefficients.
Let us now consider the following double sine series
∞ X
∞
X
(1.1)
am,n sin mx sin ny
n=1 m=1
whose coefficients satisfy conditions am,n → 0 for m → ∞ and all n fixed, and
for n → ∞ and all m fixed.
For k1 ≥ 0, k2 ≥ 0 denote
k1
X
i k1
4k1 ,0 am,n =
(−1)
am+i,n ,
i
i=0
k2
X
j k2
40,k2 am,n =
(−1)
am,n+j ,
j
j=0
k1 X
k2
X
k2
i+j k1
4k1 ,k2 am,n =
(−1)
am+i,n+j .
i
j
i=0 j=0
Parallel with series (1.1) we consider the series of the form
∞ X
∞
X
(1.2)
k1
k2
4k1 ,k2 am,n B m (x)B n (y),
n=1 m=1
where
e r (x) = sin x + sin 2x + · · · + sin rx, r ≥ 1;
B r (x) ≡ D
r
X
ν
ν−1
B r (x) =
B µ (x), (ν = 2, 3, . . . ) .
1
µ=1
Let
∞ X
∞
X
(1.3)
cµ,ν
ν=1 µ=1
be double numerical series and
Sm,n =
n X
m
X
cµ,ν
ν=1 µ=1
its rectangular partial sums.
If there exists a number S such that for all ε > 0 there exist natural numbers
k and l such that
|Sm,n − S| < ε
CERTAIN ESTIMATES FOR DOUBLE SINE SERIES . . .
235
for all n > k and m > l, then series (1.3) converges in Pringsheim’s sense to
the number S (see [12], page 27).
It’s well-known (see [3]) that if sequence of numbers {ak,l } satisfies conditions
(A) and 4k1 ,k2 ak,l ≥ 0 for all k and l, then:
Lemma 1.1.
(1) Series (1.1) converges almost everywhere in Pringsheim’s
sense, in other words there exists a function g(x, y) such that the sum
of series (1.1) is g(x, y).
(2) Series (1.2) converges almost everywhere in Pringsheim’s sense to g(x, y).
Throughout this paper the O expressions contain positive constants and
they may depend only on k1 and k2 .
2. Main Results
To prove our main results we need first the following lemma.
Lemma 2.1. For ν = 1, 2, 3, . . . and x ∈ (0, π] the following inequality is true
ν
B r (x)
rν−1
x
≥−
sin .
2
2
Proof. For the proof we shall use mathematical induction. Namely, for ν = 1
we have
x
1
cos
−
cos
r
+
x
cos x2 − 1
1
2
2
e r (x) =
B r (x) ≡ D
≥
2 sin x2
2 sin x2
1
x
1
x
= − tan ≥ − sin .
2
4
2
2
Assuming that
ν−1
Br
(x) ≥ −
rν−2
x
sin
2
2
we obtain
ν
B r (x) =
r
X
1
x X ν−2
x
rν−1
ν−1
B µ (x) ≥ − sin
sin ,
µ
≥−
2
2 µ=1
2
2
µ=1
r
which completes the proof of the Lemma 2.1.
We shall prove two theorems. The first theorem gives an estimate of the sum
g(x, y) near the origin from above, while the second one gives the estimate from
below.
Theorem 2.2. Assume that {am,n } satisfies conditions (A) and 4k1 ,k2 am,n ≥
0, for k1 ≥ 1, k2 ≥ 1. Then for x ∈ Ir and y ∈ I` , (r, ` = 1, 2, . . . ) the following
236
XHEVAT Z. KRASNIQI
estimate is valid
(2.1)
r X̀
X
g(x, y) =
mnxyam,n
m=1 n=1
r X̀
X
(rn)2 + (mn)2 + (`m)2
+O
(r`)3
m=1 n=1
+
k1
X
r X̀
X
(`2 + n2 ) m3 n
r4−i `3
i=1 m=1 n=1
+
4i−1,0 am,n
k2 X
r X̀
X
(r2 + m2 ) mn3
r3 `4−j
j=1 m=1 n=1
mnam,n
40,j−1 am,n
!
k1 X
k2 X
r X̀
X
(mn)3
4i−1,j−1 am,n .
+
r4−i `4−j
i=1 j=1 m=1 n=1
Proof. By the Lemma 1.1 we have
(2.2)
g(x, y) =
r
X̀ X
k1
k2
4k1 ,k2 am,n B m (x)B n (y)
n=1 m=1
r
∞ X
X
+
k1
k2
k1
k2
4k1 ,k2 am,n B m (x)B n (y)
n=`+1 m=1
+
+
∞
X̀ X
4k1 ,k2 am,n B m (x)B n (y)
n=1 m=r+1
∞
∞
X
X
k1
k2
4k1 ,k2 am,n B m (x)B n (y)
n=`+1 m=r+1
=
Ak`,r1 ,k2 (x, y)
1 ,k2
1 ,k2
1 ,k2
+ Ak∞,r
(x, y) + Ak`,∞
(x, y) + Ak∞,∞
(x, y).
The expression Ak`,r1 ,k2 (x, y) can be written as follows
(2.3) Ak`,r1 ,k2 (x, y)
=
#
" r
X̀ X
n=1
k1 ,k2
where Hr,n
(x) =
Since
s
k1
4k1 ,k2 am,n B m (x)
m=1
X̀
k1
4k1 ,k2 am,n B m (x).
s
s−1
k2
k1 ,k2
Hr,n
(x)B n (y),
n=1
m=1
Pr
k2
B n (y) =
B m (x) − B m−1 (x) = B m (x), (s = 2, 3, . . . k1 ),
CERTAIN ESTIMATES FOR DOUBLE SINE SERIES . . .
237
then
k1 ,k2
Hr,n
(x) =
=
=
r
X
m=1
r
X
m=1
r
X
k1
[4k1 −1,k2 am,n − 4k1 −1,k2 am+1,n ] B m (x)
h k
i
k1
k1
1
4k1 −1,k2 am,n B m (x) − B m−1 (x) − 4k1 −1,k2 ar+1,n B r (x)
k1 −1
4k1 −1,k2 am,n B m
k1
(x) − 4k1 −1,k2 ar+1,n B r (x)
m=1
=
r
X
k1 −2
4k1 −2,k2 am,n B m (x)
k1
X
−
i
4i−1,k2 ar+1,n B r (x)
i=k1 −1
m=1
..
.
=
r
X
40,k2 am,n sin mx −
m=1
k1
X
i
4i−1,k2 ar+1,n B r (x).
i=1
Applying the relation sin u = u+O(u3 ), as u → 0, 4i−1,k2 ar,n ≥ 4i−1,k2 ar+1,n
i
by our assumption, the well-known estimate B r (x) = O ((r + 1)i ) for i ≥ 1,
x ∈ [0, π], and x ∈ Ir we obtain
k1 ,k2
(2.4) Hr,n
(x)
=
r
X
m40,k2 am,n x + O
m=1
!
k1
r
X
1 X 3
m 40,k2 am,n +
ri 4i−1,k2 ar,n .
r3 m=1
i=1
k1 ,k2
In a similar way using the same arguments as for Hr,n
(x) we arrive at the
following estimates (y ∈ I` ):
X̀
k2
40,k2 am,n B n (y)
=
X̀
40,0 am,n sin ny −
n=1
n=1
=
X̀
k2
X
j
40,j−1 am,`+1 B ` (y)
j=1
nam,n y + O
n=1
!
k2
X
1 X̀ 3
n am,n +
`j 40,j−1 am,` ,
`3 n=1
j=1
and
X̀
n=1
k2
4i−1,k2 ar,n B n (y) =
X̀
n=1
+O
n4i−1,0 ar,n y
!
k2
X
1 X̀ 3
n 4i−1,0 ar,n +
`j 4i−1,j−1 ar,` .
`3 n=1
j=1
238
XHEVAT Z. KRASNIQI
Using (2.3), (2.4), and last estimates we get
(2.5) Ak`,r1 ,k2 (x, y)
=
r
X
mx
m=1
X̀
k2
40,k2 am,n B n (y)
n=1
r
1 X 3 X̀
k2
m
40,k2 am,n B n (y)
3
r m=1
n=1
!
k
1
X X̀
k2
+
ri
4i−1,k2 ar,n B n (y)
+O
i=1
r
X
n=1
X̀
r
1 X X̀
=
mnxyam,n + O
mn3 am,n
3
r`
m=1 n=1
m=1 n=1
k2
r
r
1 XX
1 X X̀ 3
j
+
m` 40,j−1 am,` + 3
m nam,n
r m=1 j=1
r ` m=1 n=1
k2
r
r
1 X X̀
1 XX
3
+
(mn) am,n + 3
m3 `j 40,j−1 am,`
(r`)3 m=1 n=1
r m=1 j=1
k1
k1
1 X̀ X
1 X̀ X
i
nr 4i−1,0 ar,n + 3
n3 ri 4i−1,0 ar,n
+
` n=1 i=1
` n=1 i=1
!
k1 X
k2
X
+
ri `j 4i−1,j−1 ar,` .
i=1 j=1
But
k2
k2
r
r
1 XX
1 XX
j
m` 40,j−1 am,` =
m`j−1 `40,j−1 am,`
r m=1 j=1
r m=1 j=1
≤
≤
k2
r
1 XX
4 X̀ 3
n 40,j−1 am,`
m`j−1 3
r m=1 j=1
` n=1
k2
X
j=1
r
4 X X̀
mn3 40,j−1 am,n ,
r`4−j m=1 n=1
CERTAIN ESTIMATES FOR DOUBLE SINE SERIES . . .
239
and in a similar manner we can find
k2
k2
r
r
X
1 XX
4 X X̀
3 j
m ` 40,j−1 am,` ≤
(mn)3 40,j−1 am,n ,
3 `4−j
r3 m=1 j=1
r
m=1 n=1
j=1
k1
k1
r
X
1 X̀ X
4 X X̀ 3
i
nr 4i−1,0 ar,n ≤
m n4i−1,0 am,n ,
` n=1 i=1
r4−i ` m=1 n=1
i=1
k1
k1
r
X
1 X̀ X
4 X X̀
3 i
n r 4i−1,0 ar,n ≤
(mn)3 4i−1,0 am,n ,
4−i `3
`3 n=1 i=1
r
m=1 n=1
i=1
k1 X
k2
X
r ` 4i−1,j−1 ar,` ≤
i j
i=1 j=1
k1 X
k2
X
i=1 j=1
r
16 X X̀
(mn)3 4i−1,j−1 am,n .
4−i
4−j
r `
m=1 n=1
Therefore,
(2.6)
Ak`,r1 ,k2 (x, y)
=
r X̀
X
mnxyam,n
m=1 n=1
r X̀
X
(rn)2 + (mn)2 + (`m)2
+O
(r`)3
m=1 n=1
+
k1
X
r X̀
X
(`2 + n2 ) m3 n
i=1 m=1 n=1
+
r4−i `3
4i−1,0 am,n
k2 X
r X̀
X
(r2 + m2 ) mn3
j=1 m=1 n=1
r3 `4−j
mnam,n
40,j−1 am,n
!
k1 X
k2 X
r X̀
X
(mn)3
+
4i−1,j−1 am,n .
r4−i `4−j
i=1 j=1 m=1 n=1
k2
1 ,k2
Now, we estimate Ak∞,r
(x, y). Indeed, since B n (y) = O y −k2 for y ∈ (0, π],
(2.4), and x ∈ Ir , y ∈ I` , then
r
X
k1
1 ,k2
(2.7) Ak∞,r
(x, y) = O y −k2
4k1 ,k2 −1 am,`+1 B m (x)
m=1
= O `k2
=O
r
X
k1
4k1 ,k2 −1 am,` B m (x)
m=1
r
X
1
m`k2 40,k2 −1 am,`
r m=1
!
k1
r
X
1 X 3 k2
m ` 40,k2 −1 am,` +
+ 3
ri `k2 4i−1,k2 −1 ar,` .
r m=1
i=1
240
XHEVAT Z. KRASNIQI
It is not difficult to prove that
k2
r
r
X
1 X k2
4 X X̀
m` 40,k2 −1 am,` ≤
mn3 40,j−1 am,n .
4−j
r m=1
r`
m=1 n=1
j=1
and
k2
r
r
X
1 X 3 k2
4 X X̀
m
`
4
a
≤
(mn)3 40,j−1 am,n .
0,k2 −1 m,`
3 `4−j
r3 m=1
r
m=1 n=1
j=1
From last two estimates we have
r
r
1 X 3 k2
1 X k2
m` 40,k2 −1 am,` + 3
m ` 40,k2 −1 am,`
r m=1
r m=1
≤
k2 X
r X̀
X
4 (r2 + m2 ) mn3
r3 `4−j
j=1 m=1 n=1
40,j−1 am,n ,
and
k1
X
r ` 4i−1,k2 −1 ar,` =
i k2
i=1
k1
X
ri−1 r`k2 4i−1,k2 −1 ar,`
i=1
≤
k1
X
ri−1 `k2
i=1
≤
k1
X
r
i−1 k2 −1
`
i=1
≤
r
4 X 3
m 4i−1,k2 −1 am,`
r3 m=1
r
16 X X̀
(mn)3 4i−1,k2 −1 am,`
3
(r`) m=1 n=1
k1 X
k2 X
r X̀
X
16(mn)3
i=1 j=1 m=1 n=1
r4−i `4−j
4i−1,j−1 am,n .
Therefore, from these estimates and (2.7) we obtain
1 ,k2
(2.8) Ak∞,r
(x, y) = O
k2 X
r X̀
X
(r2 + m2 ) mn3
j=1 m=1 n=1
r3 `4−j
40,j−1 am,n
!
k1 X
k2 X
r X̀
X
(mn)3
+
4i−1,j−1 am,n .
r4−i `4−j
i=1 j=1 m=1 n=1
CERTAIN ESTIMATES FOR DOUBLE SINE SERIES . . .
241
In a similar manner we can find the following estimate
(2.9)
1 ,k2
Ak`,∞
(x, y)
=O
k1 X
r X̀
X
(n2 + `2 ) m3 n
r4−i `3
i=1 m=1 n=1
4i−1,0 am,n
!
k1 X
k2 X
r X̀
X
(mn)3
+
4i−1,j−1 am,n .
4−i `4−j
r
i=1 j=1 m=1 n=1
k2
k1
In the end from B m (x) = O x−k1 , B n (y) = O y −k2 for x, y ∈ (0, π], and
x ∈ Ir , y ∈ I` we obtain
1 ,k2
(2.10) Ak∞,∞
(x, y) = O x−k1 y −k2
∞
X
∞
X
n=`+1 m=r+1
!
4k1 ,k2 am,n
= O r ` 4k1 −1,k2 −1 ar,` .
k1 k2
By virtue of the monotonicity of 4k1 ,k2 ar,` , for k1 ≥ 1, k2 ≥ 1, we get
(2.11) rk1 `k2 4k1 −1,k2 −1 ar,` = rk1 −1 `k2 −1 r`4k1 −1,k2 −1 ar,`
r X̀
X
16
≤ 4−k1 4−k2
(mn)3 4k1 −1,k2 −1 ar,`
r
`
m=1 n=1
≤
≤
r X̀
X
16
(mn)3 4k1 −1,k2 −1 am,n
r4−k1 `4−k2 m=1 n=1
k1 X
k2 X
r X̀
X
16 (mn)3
i=1 j=1 m=1 n=1
r4−i `4−j
4i−1,j−1 am,n .
Thus, from (2.10) and (2.11) we get
(2.12)
!
k1 X
k2 X
r X̀
3
X
(mn)
1 ,k2
(x, y) = O
Ak∞,∞
4i−1,j−1 am,n .
4−i `4−j
r
i=1 j=1 m=1 n=1
Inserting (2.6), (2.8), (2.9) and (2.12) into (2.2) we obtain (2.1).
As a direct consequence of the theorem 2.2 is the following corollary:
242
XHEVAT Z. KRASNIQI
Corollary 2.3. [4] Assume that {am,n } satisfies conditions (A) and 41,1 am,n ≥
0. Then for x ∈ Ir and y ∈ I` , (r, ` = 1, 2, . . . ) the following estimate is valid
(2.13) g(x, y) =
r X̀
X
m=1 n=1
mnxyam,n + O
r
1 X X̀
mn3 am,n
r`3 m=1 n=1
!
r X̀
r X̀
X
X
1
1
+ 3
m3 nam,n +
(mn)3 am,n .
3
r ` m=1 n=1
(r`) m=1 n=1
Now we prove the statement that gives the estimate of g(x, y) from below.
Theorem 2.4. Assume that ak,l satisfy conditions (A) and 4k1 ,k2 ak,l ≥ 0.
Then the following estimate is valid
∞ ∞
x
y X X k1 −1 k2 −1
1
sin sin
(2.14)
m
n
4k1 ,k2 am,n ≤ g(x, y).
4
2
2 n=1 m=1
Proof. Based on Lemma 1.1 and Lemma 2.1 we immediately obtain (2.14). Theorem 2.4, for k1 = 1, k2 = 1, implies the following statement proved in
[4].
Corollary 2.5. Assume that ak,l satisfy conditions (A) and 41,1 ak,l ≥ 0. Then
the following estimate is valid
x
y
a1,1
sin sin ≤ g(x, y).
4
2
2
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Received October 13, 2009.
Department of Mathematics and Computer Sciences,
University of Prishtina,
Ave. ”Mother Teresa” 5,
Prishtinë, 10000,
Republic of Kosova
E-mail address: xhevat-z-krasniqi@hotmail.com