Later Telyakovski in [11] has proved the following fact: Theorem T

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∞ n=1

a_nsinnx

with monotone coefficients tending to zero. Young [13] was the first to consider the problem of estimates ofg(x) forx→0 expressed in terms of the coefficients a_n. Then Salem [8], [9], Shogunbekov [10], Aljanˇci´c, Bojani´c and Tomi´c [1]

considered this problem, as well. Later Telyakovski in [11] has proved the following fact:

Theorem T. Assume that an ↓0. Then for x∈ _m+1^π ,_m^π

≡Im, m= 1,2, . . . the following estimate is valid:

g(x) = Xm

n=1

na_nx+O 1 m³

Xm

n=1

n³a_n

! .

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

2a₁sinx

2 ≤g(x), ∀x∈(0, π].

2010Mathematics Subject Classification. 42A20, 42A16.

Key words and phrases. sine series, asymptotic behaviour, multiple-monotone coefficients.

233

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 (1.1)

X∞ n=1

X∞ m=1

am,nsinmxsinny

whose coefficients satisfy conditionsa_m,n →0 for m→ ∞ and all n fixed, and for n→ ∞ and allm fixed.

Fork₁ ≥0, k₂ ≥0 denote 4k1,0a_m,n =

k1

X

i=0

(−1)ⁱ k₁

i

a_m+i,n,

40,k2a_m,n =

k2

X

j=0

(−1)^j k2

j

a_m,n+j,

4k1,k2a_m,n =

k1

X

i=0 k2

X

j=0

(−1)^i+j k₁

i k₂

j

a_m+i,n+j.

Parallel with series (1.1) we consider the series of the form (1.2)

X∞ n=1

X∞ m=1

4k1,k2am,nB^k_m¹(x)B^k_n²(y), where

B¹_r(x)≡De_r(x) = sinx+ sin 2x+· · ·+ sinrx, r ≥1;

B^ν_r(x) = Xr

µ=1

B^ν_µ⁻¹(x), (ν= 2,3, . . .). Let

(1.3)

X∞ ν=1

X∞ µ=1

c_µ,ν

be double numerical series and S_m,n =

Xn

ν=1

Xm

µ=1

c_µ,ν its rectangular partial sums.

If there exists a numberSsuch that for allε >0 there exist natural numbers k and l such that

|Sm,n −S|< ε

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{a_k,l}satisfies conditions (A) and 4k1,k2a_k,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 tog(x, y).

Throughout this paper the O expressions contain positive constants and they may depend only onk₁ 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 B^ν_r(x)≥ −r^ν−1

2 sinx 2.

Proof. For the proof we shall use mathematical induction. Namely, for ν = 1 we have

B¹_r(x)≡De_r(x) = cos^x₂ −cos r+ ¹₂ x

2 sin^x₂ ≥ cos^x₂ −1 2 sin^x₂

=−1 2tan x

4 ≥ −1 2sinx

2. Assuming that

B^ν_r⁻¹(x)≥ −r^ν−2 2 sinx

2 we obtain

B^ν_r(x) = Xr

µ=1

B^ν_µ⁻¹(x)≥ −1 2sinx

2 Xr

µ=1

µ^ν−2 ≥ −r^ν−1 2 sinx

2,

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,k2am,n ≥ 0, fork1 ≥1, k2 ≥1. Then for x∈Ir andy∈I`, (r, `= 1,2, . . .)the following

estimate is valid

g(x, y) = Xr

m=1

X`

n=1

mnxya_m,n (2.1)

+O Xr

m=1

X`

n=1

(rn)²+ (mn)²+ (`m)²

(r`)³ mna_m,n +

k1

X

i=1

Xr

m=1

X`

n=1

(`² +n²)m³n

r⁴⁻ⁱ`³ 4i−1,0a_m,n +

k2

X

j=1

Xr

m=1

X`

n=1

(r² +m²)mn³

r³`^4−j 40,j−1a_m,n +

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

(mn)³

r⁴⁻ⁱ`^4−j4i−1,j−1a_m,n

! .

Proof. By the Lemma 1.1 we have g(x, y) =

X`

n=1

Xr

m=1

4k1,k2a_m,nB^k_m¹(x)B^k_n²(y) (2.2)

+ X∞ n=`+1

Xr

m=1

4k1,k2a_m,nB^k_m¹(x)B^k_n²(y)

+ X`

n=1

X∞ m=r+1

4k1,k2a_m,nB^k_m¹(x)B^k_n²(y)

+ X∞ n=`+1

X∞ m=r+1

4k1,k2a_m,nB^k_m¹(x)B^k_n²(y)

=A^k_{`,r</sub><sup>1,k2</sup>(x, y) +A<sup>k</sup><sub>∞</sub><sup>1,k</sup><sub>,r</sub><sup>2</sup>(x, y) +A<sup>k</sup><sub>`,}¹_∞^,k2(x, y) +A^k_∞^1,k_,∞²(x, y).

The expressionA^k_{`,r</sub><sup>1,k2</sup>(x, y) can be written as follows (2.3) A<sup>k</sup><sub>`,r}^1,k2(x, y)

= X`

n=1

" _r X

m=1

4k1,k2a_m,nB^k_m¹(x)

#

B^k_n²(y) = X`

n=1

H_r,n^k1,k2(x)B^k_n²(y),

where H_r,n^k1,k2(x) = P_r

m=14k1,k2a_m,nB^k_m¹(x).

Since

B^s_m(x)−B^s_m−1(x) = B^s_m⁻¹(x), (s= 2,3, . . . k1),

then

H_r,n^k1,k2(x) = Xr

m=1

[4k1−1,k2am,n− 4k1−1,k2am+1,n]B^k_m¹(x)

= Xr

m=1

4k1−1,k2a_m,n h

B^k_m¹(x)−B^k_m¹₋₁(x)

i− 4k1−1,k2a_r+1,nB^k_r¹(x)

= Xr

m=1

4k1−1,k2am,nB^k_m¹⁻¹(x)− 4k1−1,k2ar+1,nB^k_r¹(x)

= Xr

m=1

4k1−2,k2a_m,nB^k_m¹⁻²(x)−

k1

X

i=k1−1

4i−1,k2a_r+1,nBⁱ_r(x) ...

= Xr

m=1

40,k2am,nsinmx−

k1

X

i=1

4i−1,k2ar+1,nBⁱ_r(x).

Applying the relation sinu=u+O(u³), asu→0,4i−1,k2a_r,n≥ 4i−1,k2a_r+1,n by our assumption, the well-known estimate Bⁱ_r(x) = O((r+ 1)ⁱ) for i ≥ 1, x∈[0, π], and x∈Ir we obtain

(2.4) H_r,n^k1,k2(x)

= Xr

m=1

m40,k2a_m,nx+O 1 r³

Xr

m=1

m³40,k2a_m,n+

k1

X

i=1

rⁱ4i−1,k2a_r,n

! .

In a similar way using the same arguments as for H_r,n^k1,k2(x) we arrive at the following estimates (y∈I_`):

X`

n=1

40,k2a_m,nB^k_n²(y) = X`

n=1

40,0a_m,nsinny−

k2

X

j=1

40,j−1a_{m,`+1</sub>B<sup>j</sup><sub>`}(y)

= X`

n=1

na_m,ny+O 1

`<sup>3</sup> X`

n=1

n³a_m,n+

k2

X

j=1

`<sup>j</sup>40,j−1a<sub>m,`</sub>

! ,

and X`

n=1

4i−1,k2a_r,nB^k_n²(y) = X`

n=1

n4i−1,0a_r,ny

+O 1

`<sup>3</sup> X`

n=1

n³4i−1,0ar,n+

k2

X

j=1

`<sup>j</sup>4i−1,j−1ar,`

! .

Using (2.3), (2.4), and last estimates we get

A^k_`,r^1,k2(x, y) = Xr

m=1

mx X`

n=1

40,k2a_m,nB^k_n²(y) (2.5)

+O 1

r³ Xr

m=1

m³ X`

n=1

40,k2a_m,nB^k_n²(y)

+

k1

X

i=1

rⁱ X`

n=1

4i−1,k2a_r,nB^k_n²(y)

!

= Xr

m=1

X`

n=1

mnxya_m,n+O 1 r`³

Xr

m=1

X`

n=1

mn³a_m,n

+1 r

Xr

m=1 k2

X

j=1

m`<sup>j</sup>40,j−1am,`+ 1 r³`

Xr

m=1

X`

n=1

m³nam,n

+ 1 (r`)³

Xr

m=1

X`

n=1

(mn)³a_m,n+ 1 r³

Xr

m=1 k2

X

j=1

m³`<sup>j</sup>40,j−1a<sub>m,`</sub>

+1

` X`

n=1 k1

X

i=1

nrⁱ4i−1,0a_r,n+ 1

`<sup>3</sup> X`

n=1 k1

X

i=1

n³rⁱ4i−1,0a_r,n

+

k1

X

i=1 k2

X

j=1

rⁱ`<sup>j</sup>4i−1,j−1a<sub>r,`</sub>

! .

But

1 r

Xr

m=1 k2

X

j=1

m`<sup>j</sup>40,j−1a<sub>m,`</sub> = 1 r

Xr

m=1 k2

X

j=1

m`<sup>j−1</sup>`40,j−1a_m,`

≤ 1 r

Xr

m=1 k2

X

j=1

m`^j−14

`<sup>3</sup> X`

n=1

n³40,j−1a_m,`

≤

k2

X

j=1

4 r`^4−j

Xr

m=1

X`

n=1

mn³40,j−1am,n,

and in a similar manner we can find 1

r³ Xr

m=1 k2

X

j=1

m³`<sup>j</sup>40,j−1a<sub>m,`</sub> ≤

k2

X

j=1

4 r³`^4−j

Xr

m=1

X`

n=1

(mn)³40,j−1a_m,n, 1

` X`

n=1 k1

X

i=1

nrⁱ4i−1,0a_r,n≤

k1

X

i=1

4 r⁴⁻ⁱ`

Xr

m=1

X`

n=1

m³n4i−1,0a_m,n, 1

`<sup>3</sup> X`

n=1 k1

X

i=1

n³rⁱ4i−1,0a_r,n≤

k1

X

i=1

4 r⁴⁻ⁱ`³

Xr

m=1

X`

n=1

(mn)³4i−1,0a_m,n,

k1

X

i=1 k2

X

j=1

rⁱ`<sup>j</sup>4i−1,j−1a<sub>r,`</sub> ≤

k1

X

i=1 k2

X

j=1

16 r⁴⁻ⁱ`^4−j

Xr

m=1

X`

n=1

(mn)³4i−1,j−1a_m,n.

Therefore,

A^k_`,r^1,k2(x, y) = Xr

m=1

X`

n=1

mnxya_m,n (2.6)

+O Xr

m=1

X`

n=1

(rn)²+ (mn)²+ (`m)²

(r`)³ mnam,n

+

k1

X

i=1

Xr

m=1

X`

n=1

(`² +n²)m³n

r⁴⁻ⁱ`³ 4i−1,0a_m,n +

k2

X

j=1

Xr

m=1

X`

n=1

(r²+m²)mn³

r³`^4−j 40,j−1a_m,n +

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

(mn)³

r⁴⁻ⁱ`^4−j4i−1,j−1a_m,n

! .

Now, we estimateA^k_∞^1,k_,r²(x, y). Indeed, sinceB^k_n²(y) =O y^−k2

fory∈(0, π], (2.4), and x∈I_r, y ∈I_`, then

A^k_∞^1,k_,r²(x, y) = O y^−k2X^r

m=1

4k1,k2−1a_m,`+1B^k_m¹(x) (2.7)

=O `^k2X^r

m=1

4k1,k2−1a_m,`B^k_m¹(x)

=O 1 r

Xr

m=1

m`<sup>k2</sup>40,k2−1a<sub>m,`</sub>

+1 r³

Xr

m=1

m³`<sup>k2</sup>40,k2−1a<sub>m,`</sub>+

k1

X

i=1

rⁱ`<sup>k2</sup>4i−1,k2−1a<sub>r,`</sub>

! .

It is not difficult to prove that 1

r Xr

m=1

m`<sup>k2</sup>40,k2−1a<sub>m,`</sub>≤

k2

X

j=1

4 r`^4−j

Xr

m=1

X`

n=1

mn³40,j−1a_m,n.

and 1 r³

Xr

m=1

m³`<sup>k2</sup>40,k2−1a<sub>m,`</sub>≤

k2

X

j=1

4 r³`^4−j

Xr

m=1

X`

n=1

(mn)³40,j−1a_m,n.

From last two estimates we have 1

r Xr

m=1

m`<sup>k2</sup>40,k2−1a<sub>m,`</sub>+ 1 r³

Xr

m=1

m³`<sup>k2</sup>40,k2−1a<sub>m,`</sub>

≤

k2

X

j=1

Xr

m=1

X`

n=1

4 (r²+m²)mn³

r³`^4−j 40,j−1a_m,n,

and

k1

X

i=1

rⁱ`<sup>k2</sup>4i−1,k2−1a<sub>r,`</sub>=

k1

X

i=1

rⁱ⁻¹r`<sup>k2</sup>4i−1,k2−1a<sub>r,`</sub>

≤

k1

X

i=1

rⁱ⁻¹`^k2 4 r³

Xr

m=1

m³4i−1,k2−1a_m,`

≤

k1

X

i=1

rⁱ⁻¹`<sup>k2−1</sup> 16 (r`)³

Xr

m=1

X`

n=1

(mn)³4i−1,k2−1a_m,`

≤

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

16(mn)³

r⁴⁻ⁱ`^4−j 4i−1,j−1a_m,n. Therefore, from these estimates and (2.7) we obtain

(2.8) A^k_∞^1,k_,r²(x, y) =O

k2

X

j=1

Xr

m=1

X`

n=1

(r²+m²)mn³

r³`^4−j 40,j−1a_m,n +

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

(mn)³

r⁴⁻ⁱ`^4−j4i−1,j−1am,n

! .

In a similar manner we can find the following estimate

(2.9) A^k_`,¹_∞^,k2(x, y) =O

k1

X

i=1

Xr

m=1

X`

n=1

(n²+`²)m³n

r⁴⁻ⁱ`³ 4i−1,0am,n

+

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

(mn)³

r⁴⁻ⁱ`^4−j4i−1,j−1a_m,n

! .

In the end fromB^k_m¹(x) =O x^−k1

, B^k_n²(y) =O y^−k2

forx, y ∈(0, π], and x∈Ir, y ∈I` we obtain

(2.10) A^k_∞¹_,^,k_∞²(x, y) =O x^−k1y^−k2 X∞ n=`+1

X∞ m=r+1

4k1,k2am,n

!

=O

r^k1`<sup>k2</sup>4k1−1,k2−1a<sub>r,`</sub>

.

By virtue of the monotonicity of4k1,k2a_r,`, for k₁ ≥1, k₂ ≥1, we get

r^k1`<sup>k2</sup>4k1−1,k2−1ar,`=r^k1−1`<sup>k2−1</sup>r`4k1−1,k2−1ar,`

(2.11)

≤ 16 r^4−k1`^4−k2

Xr

m=1

X`

n=1

(mn)³4k1−1,k2−1a_r,`

≤ 16 r^4−k1`^4−k2

Xr

m=1

X`

n=1

(mn)³4k1−1,k2−1a_m,n

≤

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

16 (mn)³

r⁴⁻ⁱ`^4−j 4i−1,j−1a_m,n.

Thus, from (2.10) and (2.11) we get

A^k_∞^1,k_,∞²(x, y) = O

k1

X

i=1 k2

X

j=1

Xr

m=1

X`

n=1

(mn)³

r⁴⁻ⁱ`^4−j4i−1,j−1a_m,n

! . (2.12)

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:

Corollary 2.3. [4]Assume that{a_m,n}satisfies conditions (A) and41,1a_m,n ≥ 0. Then for x∈I_r and y∈I<sub>`</sub>, (r, `= 1,2, . . .) the following estimate is valid (2.13) g(x, y) =

Xr

m=1

X`

n=1

mnxya_m,n+O 1 r`³

Xr

m=1

X`

n=1

mn³a_m,n

+ 1 r³`

Xr

m=1

X`

n=1

m³na_m,n+ 1 (r`)³

Xr

m=1

X`

n=1

(mn)³a_m,n

! . Now we prove the statement that gives the estimate ofg(x, y) from below.

Theorem 2.4. Assume that ak,l satisfy conditions (A) and 4k1,k2ak,l ≥ 0.

Then the following estimate is valid

(2.14) 1

4sinx 2siny

2 X∞ n=1

X∞ m=1

m^k1−1n^k2−14k1,k2a_m,n ≤g(x, y).

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 thata_k,l satisfy conditions (A) and41,1a_k,l ≥0. Then the following estimate is valid

a_1,1 4 sinx

2siny

2 ≤g(x, y).

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Received October 13, 2009.

Department of Mathematics and Computer Sciences, University of Prishtina,

Ave. ”Mother Teresa” 5, Prishtin¨e, 10000,

Republic of Kosova

E-mail address: [email protected]