Vol 1, No 2 (2010)

CLASSES

m n,

( , , )

p r

ψ

Λ

AND DOUBLE TRIGONOMETRIC FOURIER SERIES

Xhevat Z. Krasniqi

Ave. "Mother Teresa" 5, Prishtinë, 10000, Kosova

University of Prishtina, Department of Mathematics and Computer Sciences

E-mail: [email protected] (Accepted on: 14-10-2010)

ABSTRACT.

I

n this paper we generalize the Lipschitz classes

_Λ

m n,

( , , , )

_{α β}

_{p r}

defined in [6]. Here separately are given, necessary and sufficient conditions, so that a function with double Fourier series belongs the generalized classes

Λ

m n,

( , , )

ψ

p r

.

Key words: Lipschitz classes, double trigonometric Fourier series.

AMS Subject Classification: 42A10, 42B05.

---

1. INTRODUCTION:

Let

f x y

( , )

∈

L

(

T

2

)

,

T

2

≡ −

[

π π

, ] [

× −

π π

, ]

, be

2

π

−

periodic function with respect to each variable and even with respect to both variables

x y

,

; even with respect to

x

and odd with respect to

y

; odd with respect to

x

and even with respect to

y

; odd with respect to both variables

x y

,

, with its Fourier series

_{, ,}

0 0

cos

cos

,

i k i k

i k

a

ix

ky

λ

∞ ∞

= = _1.1

_{, ,}

0 0

cos

sin

,

i k i k

i k

a

ix

ky

λ

∞ ∞

= = _1.2

_{, ,}

0 0

sin

cos

,

i k i k

i k

a

ix

ky

λ

∞ ∞

= = _1.3

_,

1 1

sin

sin

,

i k

i k

a

ix

ky

∞ ∞

= = _1.4

respectively, where

$ %& ' " ()* % + ,%

,

0

1

if

0,

0;

0,

0

2

1

if

4

i

1

f ,

0.

i k

i

k

i

k

i

k

i k

λ

=

=

=

=

>

>

=

>

We say that

f

∈

AW

p

, (1

<

p

< ∞

)

if

2

1,1 , 1 1

( )

,

p p

j s

i k j i s k

ik

a

∞ ∞ ∞ ∞

−

= = = =

∆

< ∞

where

∆

_1,1

a

_{j s,}

=

a

_{j s,}

−

a

_j+1,s

−

a

_{j s,+1}

+

a

_j+1,s+1 (see [2]). Is said to be

f

∈ Λ

m n,

( , , , )

α β

p r

if

1 1 1 2 2 _,

( , )

, , , _{0 0 0 0}

|

( , ; , )|

,

r _r

p _p

m n m n

p r p p

f x y t

dtd

f

dxdy

t

t

π π

α β α β

τ

τ

τ

τ

∆

≡

<∞

where

1

<

p

< ∞ ≤ < ∞

,1

r

, ,

α β

>

0, ,

m n

∈

N

and

[

]

,

0 0

( , ; , )

( 1)

(

2 ) ,

(

2 ) .

m n

i k i k

m n m n

i k

f x y t

τ

+

C C f x m i t y n

k

τ

= =

∆

=

−

+ −

+ −

{

2 2

}

1

( , )

, 0 0 | | | |

( , ; ) supsup

( , ; , )

p p

m n

p m n

t h

h

f

π π

f x y t

dxdy

τ η

η

τ

≤ ≤

Ω

=

∆

is the integral modulus of smoothness of order

( , )

m n

. Let

α

_i

( )

t

_i ,

(

i

=

1, 2)

be positive functions that satisfy the following properties:

(1) The functions

α

_i

( )

t

_i are measurable on

[0, 2 ]

π

and (2) Integrable on

[ , 2 ]

δ

_i

π

for each

δ

_i

∈

(0, 2 )

π

. A function

ψ

( , )

t t

_{1 2}

=

α

₁

( )

t

₁

α

₂

( )

t

₂ satisfies

σ

=

( ,

σ σ

_{1 2}

)

-condition (see [2]) if there exist real numbers

δ

_i

∈

(0, 2 )

π

and

0

i

>

ò

,

(

i

=

1, 2)

so that:

0

( )

and

0

( )

.

i i

i i i

i

t t dt

i i i i

t t

i i

dt

i

δ _σ δ _σ

α

α

−

< ∞

ò

= ∞

We say that

f

∈ Λ

m n,

( , , )

ψ

p r

if

1

1 1 2 2

( , )

, , _{0 0}

( , )

_{0 0}

|

,

( , ; , ) |

,

r _r

p

m n p

p r m n

f

_ψ

≡

ψ

t

τ

π π

∆

f x y t

τ

dxdy

dtd

τ

< ∞

where the function

ψ

( , )

t

τ

satisfies

σ

=

( ,

σ σ

_{1 2}

)

-condition. We notice that if we give the values

α

₁

( )

t

₌

t

−αr−1

and

1 2

( )

r

β

α τ

₌

τ

− −

,

( ,

α β

>

0)

to the function

ψ

( , )

t

τ

, we get the classes

Λ

m n,

( , , , )

α β

p r

defined in [6]. So, the classes

,

( , , )

m n

p r

ψ

Λ

are generalization of the classes

,

_{( , , , )}

m n

p r

α β

Λ

.

The aim of this paper is to study, apart, necessary conditions and sufficient conditions which guarantee that a function belongs the classes

Λ

m n,

( , , )

ψ

p r

. We recall that for one dimensional case these conditions are given by present author in [7]. We make mention that for the classes

Λ

m n,

( , , , )

α β

p r

these condition are studied in [6]. There are given, as well, at the same time necessary and sufficient conditions so that a function with double Fourier series belongs to the same classes. Due to chronology it is important to point out that some conditions for more simply classes, than those that Tevzadze considered, are given by R. Askey and S. Waigner [5], and M. Izumi and S. Izumi [3], as well.

2. SEVERAL LEMMAS:

In this section we will give some auxiliary statements which we need to prove the main results. Below, some lemmas in two-dimensional case are proved for what we believe that they can be useful to another applications concerning to the double numerical series with positive terms. We start with the following lemma proved in [6].

Lemma: 2.1. Let

f

∈

AW

p,

(1

<

p

< ∞

)

. Then

( , ) ( 1) 2 ( 1) 2

, , 1 1 1,1 ,

{

( , ; )}

{

p

mn p mp np m p n p

p mn p j s

i k _{j i s k}

h

h f

A

h

i

k

a

η

η

η

∞ ∞ + − + −

≤ ≤ _{= =}

Ω

≤

∆

( 1) 2 2

1,1 ,

1 1

p

mp m p p

j s

i k _{j i s k}

h

h

i

k

a

η

∞ ∞

+ − −

≤ > _{= =}

+

∆

2 ( 1) 2

1,1 ,

1 1

p

np p n p

j s

i k _{j i s k}

h

i

k

a

η

η

∞ ∞

− + −

> ≤ _{= =}

+

∆

2

1,1 ,

1 1

( )

}

.

p p

j s

i k _{j i s k}

h

ik

a

η

∞ ∞ −

> > _{= =}

+

∆

Lemma: 2.2. [4] Let

a

_ν,

b

_ν and

β

_n be numbers such that

0

a

_ν

≥

,

b

_ν

≥

0

and _{n n}

, (

1, 2,

)

n

a

_ν

a

n

ν

β

∞

=

=

=

…

. Then:

(1) For

0

<

p

≤

1

we have

(

)

1 1 1

;

p

p p

a

b

p

a

b

ν

ν µ ν ν ν

ν µ ν

β

∞ ∞

= = =

≥

(2) For

1

≤

p

< ∞

we have

(

)

1 1 1

.

p

p p

a

b

p

a

b

ν

ν µ ν ν ν

ν µ ν

β

∞ ∞

= = =

≤

Lemma: 2.3. [4] Let

a

_ν,

b

_ν and

γ

_n be numbers such that

0

a

_ν

≥

,

b

_ν

≥

0

and

1

,

n

n n

a

_ν

a

ν

γ

=

=

(

n

=

1, 2,

…

)

. Then: (1) For

0

<

p

≤

1

, we have

(

)

1 1

;

p

p p

a

_ν

b

_µ

p

a

_ν

b

_{ν ν}

ν µ ν ν

γ

∞ ∞ ∞

= = =

≥

(

)

1 1

.

p

p p

a

_ν

b

_µ

p

a

_ν

b

_{ν ν}

ν µ ν ν

γ

∞ ∞ ∞

= = =

≤

Now, because of their simplicity, we will prove briefly four lemmas for which we spoke at the beginning of this section.

Lemma: 2.4. Let

a

_{µ ν,} ,

b

_{µ ν,} ,

β

_{m n}(1)_, and

β

_{m n}(2)_, be numbers such that

a

_{µ ν,}

≥

0

,

b

_{µ ν,}

≥

0

, _,

(

(2)_,

)

(1)_{, ,}

(

(2)_,

)

p p

n n m n m n m n

m

a

_{µ µ}

a

µ

β

β

β

∞

=

=

,

and m_, m n_, m n(2)_,

, ( ,

1, 2,

)

n

a

_ν

a

m n

ν

β

∞

=

=

=

…

. Then:

(1) For

0

<

p

≤

1

we have

(

)

2 (1) (2)

, , , , , ,

1 1 1 1 1 1

;

p

m n _p

p

m n m n m n m n m n

m n m n

a

b

µ ν

p

a

b

µ ν

β β

∞ ∞ ∞ ∞

= = = = = =

≥

(2) For

1

≤

p

< ∞

we have

(

)

2 (1) (2)

, , , , , ,

1 1 1 1 1 1

.

p

m n _p

p

m n m n m n m n m n

m n m n

a

b

µ ν

p

a

b

µ ν

β β

∞ ∞ ∞ ∞

= = = = = =

≤

Proof. Putting _{, ,}

1

m

m

b

_{µ ν}

c

_ν

µ =

=

and using twice Lemma 2.2 for

0

<

p

≤

1

we have:

(

)

( )

( ) (

)

, , , ,

1 1 1 1 1 1 1

(2) , , , 1 1

(2)

, , ,

1 1 1

2 (2) (1)

, , , ,

1 1 2

,

p p

M N m n M N n

m n m n m

m n m n

M N _p

p

m n m n m n m n

p

N M _p m

p

m n m n n

n m

M N _{p p}

p

m n m n m n m n m n

p

m n n

a

b

a

c

p

a

c

p

a

b

p

a

b

p

a

µν ν

µ ν ν

µ µ

β

β

β

β

= = = = = = =

= =

= = =

= =

=

=

≥

=

≥

=

(

_, (1)_, (2)_,

)

1 1

.

M N _p

m n m n m n m

b

β β

=

Passing on limit when

M N

,

→ ∞

to the last inequality we get required inequality.

2. The Lemma 2.4 for

1

≤

p

< ∞

can be prove in the same way.

Lemma: 2.5. Let

a

_{µ ν,} ,

b

_{µ ν,} ,

γ

_{m n}(1)_, and

γ

_{m n}(2)_, be numbers such that

a

_{µ ν,}

≥

0

,

b

_{µ ν,}

≥

0

, _{, ,} (1)_,

1

m

n m n m n

a

_µ

a

µ

γ

=

=

, and

(

(1)

)

(2)

(

(1)

)

, , , , ,

1

n _{p p}

m m m n m n m n

a

_{ν ν}

a

ν

γ

γ

γ

=

=

,

( ,

m n

=

1, 2,

…

)

. Then: (1) For

0

<

p

≤

1

we have

(

)

2 (1) (2)

, , , , , ,

1 1 1 1

;

p

p p

m n m n m n m n m n

m n m n m n

a

b

µ ν

p

a

b

µ ν

γ

γ

∞ ∞ ∞ ∞ ∞ ∞

= = = = = =

≥

(2) For

1

≤

p

< ∞

we have

(

)

2 (1) (2)

, , , , , ,

1 1 1 1

.

p

p p

m n m n m n m n m n

m n m n m n

a

b

µ ν

p

a

b

µ ν

γ

γ

∞ ∞ ∞ ∞ ∞ ∞

= = = = = =

≤

Proof: The proof of this Lemma is very similiar to the proof of the Lemma 2.4. In this case it is enough to apply twice Lemma 2.3; therefore we leave it to the reader.

Lemma: 2.6. Let

a

_{µ ν,} ,

b

_{µ ν,} ,

β

_{m n}(1)_, and

γ

_{m n}(2)_, be numbers such that

a

_{µ ν,}

≥

0

,

b

_{µ ν,}

≥

0

, _{,n m n, m n}(1)_,

m

a

_µ

a

µ

β

∞

=

=

, and

(

(1)

)

(2)

(

(1)

)

, , , , ,

1

n _{p p}

m m m n m n m n

a

_{ν ν}

a

ν

β

γ

β

=

=

,

( ,

m n

=

1, 2,

…

)

. Then: (1) For

0

<

p

≤

1

we have

(

)

2 (1) (2)

, , , , , ,

1 1 1 1 1

;

p

m _p

p

m n m n m n m n m n

m n n m n

a

b

_{µ ν}

p

a

b

µ ν

β γ

∞ ∞ ∞ ∞ ∞

= = = = = =

≥

(2) For

1

≤

p

< ∞

we have

(

)

2 (1) (2)

, , , , , ,

1 1 1 1 1

.

p

m _p

p

m n m n m n m n m n

m n n m n

a

b

_{µ ν}

p

a

b

µ ν

β γ

∞ ∞ ∞ ∞ ∞

= = = = = =

≤

Proof: Let us prove the case

1

≤

p

< ∞

(the case

0

<

p

≤

1

can be prove in a similar way). If we set _{, ,n}

n

b

_{µ ν}

c

_µ

ν

∞

=

(

)

(

)

, , , ,

1 1 1 1 1 1 1

(1) , , , 1 1

(1)

, , ,

1 1

.

p p

M N m N M m

m n m n n

m n n m

N M _p

p

m n m n m n n m

p

M N _p

p

m n m n m

m n n

a

b

a

c

p

a

c

p

a

b

µ ν µ

µ ν µ

ν ν

β

β

∞

= = = = = = =

= =

∞

= = =

=

≤

=

Applying Lemma 2.3 to the last inequality we have

(

) (

)

(

)

2 (1) (2)

, , , , , ,

1 1 1 1 1 1

2 (1) (2)

, , , , 1 1

.

p

M N m M N _{p p}

p

m n m n m n m n m n

m n m n

M N _p

p

m n m n m n m n m n

a

b

p

a

b

p

a

b

µ ν µ ν

β

γ

β γ

∞

= = = = = =

= =

≤

=

Letting

M N

,

→ ∞

to the last inequality then the proof of the lemma completes.

Lemma: 2.7. Let

a

_{µ ν,} ,

b

_{µ ν,} ,

β

_{m n}(2)_, and

γ

_{m n}(1)_, be numbers such that

a

_{µ ν,}

≥

0

,

b

_{µ ν,}

≥

0

, _{m, m n, m n}(2)_,

n

a

_ν

a

ν

β

∞

=

=

, and

(

(2)

)

(1)

(

(2)

)

, , , , ,

1

m _{p p}

n n m n m n m n

a

_{µ µ}

a

µ

β

γ

β

=

=

,

( ,

m n

=

1, 2,

…

)

. Then:

(1) For

0

<

p

≤

1

we have

(

)

2 (2) (1)

, , , , , ,

1 1 1 1 1

;

p

n _p

p

m n m n m n m n m n

m n m m n

a

b

_{µ ν}

p

a

b

µ ν

β γ

∞ ∞ ∞ ∞ ∞

= = = = = =

≥

(2)For

1

≤

p

< ∞

we have

(

)

2 (2) (1)

, , , , , ,

1 1 1 1 1

.

p

n _p

p

m n m n m n m n m n

m n m m n

a

b

_{µ ν}

p

a

b

µ ν

β γ

∞ ∞ ∞ ∞ ∞

= = = = = =

≤

The proof of the Lemma 2.7 is similiar to the proof of the Lemma 2.6 therefore we omit it.

Lemma: 2.8. [9] Let

u

₁

≥

0

,

u

₂

≥

0

. Then for each

κ

>

0

the following double inequality holds

(

)

(

)

₍

₎

1

( )

1 2 1 2 2

( )

1 2

.

C

κ

u

κ

+

u

κ

≤

u

+

u

κ

≤

C

κ

u

κ

+

u

κ

Lemma: 2.9. [9] Let

µ

,

τ

and

a

_ν be numbers such that

0

<

µ τ

< < ∞

and

a

_ν

≥

0

. Then

1 1

1 1

.

a

a

τ µ

τ µ

ν ν

ν ν

∞ ∞

= =

≤

In what follows we denote by

C

a positive constant that depends only on on

m n p r

, , ,

and may be different in different relations.

1. 3. MAIN RESULTS:

For

µ ν

,

∈

N

, let us denote

1/ 1 1/( 1) 1/

2 1/( 1)

1/ 1

1 2 ₀ 1 _{1/( 1)} 1

1/ 1

1 2 ₀ 2 _{1/( 1)} 2

, 1,1 ,

( ) :

( )

( ) :

( )

( ) :

( )

( )

( )

( )

( ) :

( )

( )

( )

( )

:

; ,

1, 2,

mr mr

nr nr

i k j s

j i s k

A

t dt

A

d

b

b

b

t t dt

t dt

b

b

b

d

d

B

a

i k

µ µ ν ν

µ

µ ν

ν

µ

α

ν

α τ τ

µ

µ

µ

µ

α

α

ν

ν

ν

ν

α τ τ

τ

α τ τ

+

+

+

+

∞ ∞

= =

=

=

=

+

=

+

=

+

=

+

=

∆

=

…

and

A

( , ) :

µ ν

=

A

( ) ( ),

µ

A

ν

b

( , ) :

µ ν

=

b

( ) ( ).

µ

b

ν

Theorem: 3.1. Let

m n

,

be natural numbers,

,1

p

f

∈

AW

<

p

< ∞

and

1

≤ < ∞

r

. If

1,1 , 1 1

,

j s j s

a

∞ ∞

= =

∆

< ∞

where

a

_{j s,} are coefficients either of the series (1.1), (1.2), (1.3) or (1.4), then:

(1) For

p

≤

r

we have

1 1 2

1 ( , )

, , ,

1 1

( , )

( , )(

)

;

( , )

r _r

r p

p

m n r

p r

b

f

C

b

B

A

ψ µ ν

µ ν

µ ν

µ ν µν

µ ν

−

∞ ∞ −

= =

≤

(2) For

p

>

r

we have

1/ 2

1 ( , )

, , ,

1 1

( , )(

)

.

r r

p

m n r

p r

f

_ψ

C

b

B

_{µ ν}

µ ν

µ ν µν

∞ ∞ −

= =

Proof:. It is obvious that

{

}

(

)

1 1

( , ) ( , )

, , _{0 0} 1 2

1/ 1/ _{( , )}

1 2 1/( 1) 1/( 1) 1 1

( , ) 1 1

( ) ( )

( , ; )

( ) ( )

( , ; )

( ) ( )

1/ ,1/ ;

.

r

r

m n m n

p r p

r m n p r m n p

f

t

t

f

dtd

t

t

f

dtd

A

A

f

ψ

µ ν µ ν µ ν

µ ν

α α τ

τ

τ

α α τ

τ

τ

µ ν

µ ν

∞ ∞ + + = = ∞ ∞ = =

≤

Ω

=

Ω

=

Ω

Using Lemma 2.1 and applying Lemma 2.8 three times we have that

{

( , )

}

( 1) 2 ( 1) 2 /

, , ,

1 1 1 1

/ ( 1) 2 2

,

1 1 1

2 ( 1) 2 ,

1 1 1

( ) ( )

( ) ( )

( ) ( )

{

r p r

mn mr nr m p n p p

p r i k

i k

r p

mr m p p p

i k i k

nr p n p p

i k i k

f

C

A A

i

k

B

A A

i

k B

A A

i k

B

µ ν ψ

µ ν

µ

µ ν ν

ν

µ ν µ

µ ν µ ν

µ ν µ

µ ν ν

∞ ∞ − − + − + − = = = = ∞ ∞ ∞ − + − − = = = = ∞ ∞ ∞ − − + − = = = =

≤

+

+

/ / 4 2 2 ,

1 1 1

( ) ( )

}

.

r p

r p p p p

i k j

i k j

A A

i k B

P

µ ν µ ν

µ ν

∞ ∞ ∞ ∞ − − = = = = =

+

≡

3.1 Let

r

≥

p

. Let us consider first the quantity

P

₁. Then the Lemma 2.4 gives

/ ( 1) 2 ( 1) 2 (1) (2)

1 , , ,

1 1 2 1 _/ (1) (2) , , , 1 1

( ) ( )

( ) ( )(

)

.

r p mr nr m p n p p

r _{r p}

p r

P C

A

A

B

C

A

A

B

µ ν µ ν µ ν µ ν

µ ν µ ν µ ν µ ν

µ

ν µ ν

µ

ν

β β

µ

ν µν

β β

∞ ∞ − − + − + − = = ∞ ∞ − = =

≤

=

It is obvious that

β

_{µ ν}(1)_, depends only on

µ

and

β

_{µ ν}(2)_, depends only on

ν

and for them the following relations hold:

(

)

(1)

1/ ( 1) 1 1/ 1/ 1 0 1

( )

( )

1

1

·

( )

( )

₁

2

( )

( )

( )

2

( )

mr mr i mr mr i mr _i i mr mr mr mr

A i i

A

i

t dt

A

i

_i

t t

dt

A

b

A

µ µ µ µ

µ

β

µ

µ

α

µ

µ

α

µ

µ

µ

∞ − = ∞ ₊ =

+

=

+

≤

=

=

and in the same manner we can show that (2)

2

1

( )

( )

nr

b

A

ν

ν

β

ν

≤

.

Therefore we have

2 1 1 1 1 , 1 1

( ) ( )

( , )(

)

.

( , )

r r p

p r

b

b

P

C

A

B

A

µ ν µ ν

µ

ν

µ ν µν

µ ν

∞ ∞ −

= =

≤

3.2 Now we etimate

P

₂. By the Lemma 2.6 we get

/ ( 1) 2 2 (1) (2)

2 , , ,

1 1 2 1 _/ (1) (2) , , , 1 1

( ) ( )

( ) ( )(

)

.

r p

mr m p p p

r _{r p}

p r

P

C

A

A

B

C

A

A

B

µ ν µ ν µ ν µ ν

µ ν µ ν µ ν µ ν

µ

ν µ

µ

ν

β γ

µ

ν µν

β γ

∞ ∞ − + − − = = ∞ ∞ − = =

≤

=

As we showed (1)

2

1

( )

( )

mr

b

A

µ

µ

β

µ

≤

, and since

1 2 1/ ( 1)

( 2) ( 2) 2

1

( )

_{( )}

( )

( )

,

( )

( )

j

d

_b

A

A j

A

A

ν

ν

ν ν

α τ

τ

_ν

ν γ

γ

ν

ν

+ =

=

=

=

then 2 1 1 2 2 , 1 1

( ) ( )

( , )(

)

.

( , )

r r p

p r

b

b

P

C

A

B

A

µ ν µ ν

µ

ν

µ ν µν

µ ν

∞ ∞ −

= =

≤

3.3 Applying the Lemma 2.7 and Lemma 2.5, respectively, to the quantites

P

₃ and

P

₄, in the analogous way as we did for quanties

1

P

and

P

₂, we arrive to these estimates:

2 1 2 1 3 , 1 1

( ) ( )

( , )(

)

( , )

r r p

p r

b

b

P

C

A

B

A

µ ν µ ν

µ

ν

µ ν µν

µ ν

∞ ∞ − = =

≤

3.4 and 2 1 2 2 4 , 1 1

( ) ( )

( , )(

)

.

( , )

r r p

p r

b

b

P

C

A

B

A

µ ν µ ν

µ

ν

µ ν µν

µ ν

∞ ∞ −

= =

≤

3.5 Inserting estimates (3.2)-(3.5) to (3.1) we have

{

}

12

( , ) 1 1

, , ,

1 1

1 2 2 1 2 2

( ) ( )

( , )(

)

( , )

( ) ( )

( ) ( )

( ) ( )

.

( , )

( , )

( , )

{

}

r

r _{r p}

p

m n r

p r

r r r

p p p

b

b

f

C

A

B

A

b

b

b

b

b

b

A

A

A

ψ µν

µ ν

µ ν

µ ν µν

µ ν

µ

ν

µ ν

µ

ν

µ ν

µ ν

µ ν

∞ ∞ −

= =

≤

Owing to the Lemma 2.8, relation (3.6) takes the following form

{

}

1 2 1

( , )

, , ,

1 1

( , )

( , )(

)

.

( , )

r

r _{r p}

p

m n r

p r

b

f

C

b

B

A

ψ µ ν

µ ν

µ ν

µ ν µν

µ ν

−

∞ ∞ −

= =

≤

Let us consider the case when

r

<

p

. We start with estimate of

,

1, 2,3, 4

i

P i

=

, alternately.

Estimating of

P

₁: Using two dimensional case of Lemma 2.9, which can be prove simply applying it twice, and changing the order of summation we have

2 2

( 1) ( 1)

1 ,

1 1 1 1

2 2

( 1) ( 1) , 1 1

( ) ( )

( ) ( )

.

r r

m r n r

mr nr p p r

i k i k

r r

m r n r

r mr nr

p p

i k

i k i k

P

C

A

A

i

k

B

C

i

k

B

A

A

µ ν µ ν

µ ν

µ

ν µ ν

µ

ν µ ν

∞ ∞ + − + −

− −

= = = =

∞ ∞ + − + − ∞ ∞

− −

= = = =

≤

=

Is not dificult to prove estimates: mr

( )

2

mr ₁

( )

mr i

A

i

b i

µ

µ

µ

∞

=

≤

and

1

( )

2

( )

nr nr

nr k

A

k

b k

ν

ν

ν

∞

=

≤

, therefore

2 1

1 , 1 1

1 1

( )

( ) ( ).

r p r

i k i k

P

C

ik

B b i b k

∞ ∞ −

= =

≤

3.7

Estimating of

P

₂: Again, applying two dimensional case of Lemma 2.9 and changing the order of summation we get

2 2 ( 1)

2 ,

1 1 1

2 2 ( 1)

,

1 1 1

2 1

, 1 2 1 1

( ) ( )

( ) ( )

( )

( )

( ),

r r

m r r

mr p p r

i k i k

r r _k

m r r

r mr

p p

i k

i k i

r

p r i k i k

P

C

A

A

i

k

B

C

i

k

B

A

A

C

ik

B b i b k

µ

µ ν ν

µ ν

µ

ν µ

µ

ν µ

∞ ∞ ∞ + − −

−

= = = =

∞ ∞ + − − ∞

−

= = = =

∞ ∞ −

= =

≤

=

≤

3.8 because mr

( )

2

mr ₁

( )

mr i

A

i

b i

µ

µ

µ

∞

=

≤

and ₂

1

( )

( )

k

A

b k

ν

ν

=

=

.

Estimating of

P

₃ and

P

₄: In the same manner we get these estimates:

2 1

3 , 2 1

1 1

( )

( ) ( ).

r p r

i k i k

P

C

ik

B b i b k

∞ ∞ −

= =

≤

3.9 and

2 1

4 , 2 2

1 1

( )

( ) ( ).

r p r

i k i k

P

C

ik

B b i b k

∞ ∞ −

= =

≤

3.10 Estimating of (3.1) together with estimates (3.7), (3.8), (3.9) and (3.10) give

{

}

1 2

( , )

, , ,

1 1

( )

( , ),

r _r

p

m n r

p r i k

i k

f

_ψ

C

ik

B b i k

∞ ∞ −

= =

≤

which fully demonstrates Theorem 3.1.

Note that these inequalities in our result and all forthcoming ones are understood so that the finiteness of the right-hand side implies the finiteness of the left-hand side. The following theorem gives sufficient conditions in terms of Fourier coefficients so that a function belongs to the classes

Λ

m n,

( , , )

ψ

p r

.

Theorem: 3.2. Let m n, be natural numbers,

1

<

p

≤

2

,

1≤r< ∞ and

1 /

p

+

1 /

q

=

1

. If

a

_{j s,} are coefficients either of the series (1.1), (1.2), (1.3) or (1.4), then:

(1) For

r

≤

q

we have

1 1

( , ) 1 1

1 1 , , ,

1 1

( ) ( )

( ) ( ) |

|

;

( , )

r _r q

r m n

p r

b

b

C

b

b

a

f

A

µ ν ψ

µ ν

µ

ν

µ

ν

µ ν

− ∞ ∞

= =

≤

(2) For r>q we have

1

( , )

1 1 , , ,

1 1

( ) ( ) |

|

.

r

r m n

p r

C

b

b

a

_{µ ν}

f

_ψ

µ ν

µ

ν

∞ ∞

= =

≤

Proof: Let

f x y

(

,

)

be even function with respect to both variables (three other cases can be consider in a similar manner). Then is not difficult to show that Fourier series of

,

( , ; , )

m n

f x y t

τ

∆

is as follows

( )

( )

( )

( )

2

, 1 1 1 2

, 1 1 1

2

, 1 1 2

2

,

1

2

cos cos sin

sin

if , are even

1

2

sin cos sin

sin

if

even and odd

1

2

cos sin sin

sin

if

odd and even

1

2

m n

m n m n

j s j s m n

m n m n

j s j s m n

m n m n

j s j s m n

m n j s s

a

jx

sy

jt

s

m n

a

jx

sy

jt

s

m

n

a

jx

sy

jt

s

m

n

a

τ

τ

τ

∞ ∞

+ +

= =

∞ ∞ + −

+

= =

∞ ∞ + −

+

= =

+ − +

=

−

−

−

−

1 1

sin sin

sin

if , are o

sin j

x

m n

dd

.

j

sy

jt

s

τ

m n

∞ ∞

By the well-known Hausdorf-Young's theorem we have

(

2 2

)

, ,

0 0

1 1

( , ; , )

sin

sin

.

r r

q q

p p mq nq

m n j s

j s

C

π π

f x y t

τ

dxdy

a

jt

s

τ

∞ ∞

= =

∆

≥

Therefore,

{ }

_{( , )} 1/ 1/

, , _{1/( 1) 1/( 1)} 1 2 ,

1 1 1 1

( ) ( )

sin

sin

.

r

r _q

q mq nq

mn

pr j s

j s

f

C

t

a

jt

s

dtd

µ ν µ ν

ψ _{µ ν}

µ ν

α α τ

τ

τ

∞ ∞

+ +

= = = =

≥

Applying twice the inequality

sin

u

2

u

π

≥

for

0

≤

u

≤

π

/ 2

, then the following estimate holds

{

( , )

}

, , ,

1 1 1 1

( ) ( )

.

r

r _q

q

m n mr nr mq nq

p r j s

j s

f

C

A

A

a

j s

µ ν ψ

µ ν

µ ν

µ ν

∞ ∞ − −

= = = =

≥

Let

r

≤

q

, then by the Lemma 2,4 we have

{

( , )

}

(

(1) (2)

)

, , ,

1 1

( ) ( )

.

r r

q _q

m n mr nr mq nq

p r

f

_ψ

C

A

A

a

_{µν µ ν}

µ ν

µ ν

µ ν

µ ν β β

∞ ∞ − −

= =

≥

It is easy to prove the following estimates

β

_µ(1)

≥

b

₁

( ) / ( )

µ

A

µ

and

β

_ν(2)

≥

b

₁

( ) / ( )

ν

A

ν

, therefore the last estimate takes the form

{

( , )

}

1 1 1

, , 1 1 ,

1 1

( ) ( )

( ) ( )

.

( , )

r

r _q

r m n

p r

b

b

f

C

b

b

a

A

ψ µ ν

µ ν

µ

ν

µ

ν

µ ν

− ∞ ∞

= =

≥

3.11 Let r>q, then using the two dimensional case of the Lemma 2.9 and changing the order of the summation we have

{

( , )

}

, , ,

1 1 1 1

, 1 1

, 1 1 1 1

( ) ( )

( ) ( )

( ) ( ).

r

r

m n mr nr mr nr

p r j s

j s r _{mr nr mr nr} j s

j s j s

r j s j s

f

C

A

A

a

j s

C

a

j s

A

A

C

a

b j b s

µ ν ψ

µ ν

µ ν

µ ν

µ

ν

µ ν

µ

ν

∞ ∞

− −

= = = =

∞ ∞ ∞ ∞

− −

= = = =

∞ ∞

= =

≥

=

≥

3.12 Estimates (3.11) and (3.12) prove Theorem 2.2.

4. COROLLARIES:

Here, in the following, we are going to deduce some corollaries from Theorem 3.1 and Theorem 3.2. We noticed in the section 3 that there existed two conditions, separately, in each proved theorem which provide that a function in two variables belongs new classes considered in this paper. Therefore arises the

question: Does exist only a single condition in Theorem 3.1 that includes both its conditions? Under some additional conditions the answer is positive for both theorems.

Corollary: 4.1. Under the conditions of Theorem 3.1 and with

( , )

( , )

b

µ ν

≤

CA

µ ν

we have:

1 2

1 ( , )

, , ,

1 1

( , )(

)

.

r r

p

m n r

p r

f

_ψ

C

b

B

_{µ ν}

µ ν

µ ν

µν

∞ ∞ −

= =

≤

Corollary: 4.2. Under the conditions of Theorem 3.2 and with

1

( ) ) )

1

( , )

b

µ

b

ν

≥

CA

µ ν

we have:

1

( , )

1 1 , , ,

1 1

( )

( ) |

|

.

r

r m n

p r

C

b

b

a

_{µ ν}

f

_ψ

µ ν

µ

ν

∞ ∞

= =

≤

Corollary: 4.3. Under the conditions of Corollary 4.1 and if Fourier coefficients

a

_{j s,} are monotone in the sense of Hardy, see [1], then:

1 2

1 ( , )

, , ,

1 1

( , )(

)

.

r r

p

m n r

p r

f

_ψ

C

b

a

_{µ ν}

µ ν

µ ν µν

∞ ∞ −

= =

≤

As a special case, for

α

₁

( )

t

₌

t

−αr−1

and

α τ

₂

( )

₌

τ

−βr−1

,

( ,

α β

>

0)

, it is easy to prove the following estimates:

1 1

( , )

r r

and

( , )

r r

.

A

µ ν

C

µ

α −

ν

β −

b

µ ν

C

µ ν

α β

≤

≤

Therefore, from last estimates and the Corollaries 4.1 and 4.2 respectively, the following corollaries hold.

Corollary: 4.4.[6] Let

0

<

α

≤

m

, 0

<

β

≤

n

,1

<

p

< ∞ ≤ < ∞

,1

r

,1 /

p

+

1 /

q

=

1

and

a

_{µ ν,} are the coefficients of the series (1.1). If

1,1 , 1 1

,

j s j s

a

∞ ∞

= =

∆

< ∞

then

1

1 1

1 1

( , )

, , , 1,1 ,

1 1

.

r _r

r r

q q

m n

p r j s

j s

f

C

a

α β α β

µ ν µ ν

µ

ν

∞ ∞ + − + − ∞ ∞

= = = =

≤

∆

Corollary: 4.5. [6] Let

0

<

α

≤

m

, 0

<

β

≤

n

,1

<

p

< ∞ ≤ < ∞

,1

r

,1 /

p

+

1 /

q

=

1

1

1 1

1 1

, 1 1

,

r

r r

q q r

a

α β

µ ν µ ν

µ

ν

∞ ∞ + − + −

= =

< ∞

then

f

∈Λ

m n,

( , , , )

α β

p r

.

REFERENCES:

[1] A. P. Antonov, Classes $Lip(\alpha, p)$ for Double Trigonometric Series with Monotone Coefficients, Moscow Univ. Math. Bull., 63 (2008), No.1,12-16.

[2] B. Lakovich, Imbedding theorems for a class of functions, (Russian) Publ. Inst. Math. (Beograd) (N.S.) 39 (1986), No.53 , 153-160.

[3] M. Izumi and S. Izumi, Lipschitz classes and Fourier coefficients, J. Math. Mech. 18 (1968), No.69, 857-870.

[4] M. K. Potapov and M. Berisha, Moduli of smoothness and the Fourier coefficients of periodic functions of one variable,(Russian) Publ. Inst. Math. (Beograd) (N.S.) 26 (1979), No.40, 215-228.

[5] R. Askey and S. Waigner, Integrability theorems for Fourier series, Duke Math. J. 33 (1966), 223-228.

[6] T. Sh. Tevzadze, Some classes of functions and trigonometric Fourier series. Some problems in the theory of functions, Tbilisi. Gos. Univ., Tbilisi, (in Russian), II (1981), 31-92.

[7] Xh. Z. Krasniqi, On a generalization of Lipschitz's classes, J. Inequal. Pure and Appl. Math., 9 (3) (2008), Art. 73, 7 pp. [8] B. Hardy, E. Litlewood and G. Polya, Inequalities, GIIL Moscow, 1948, 1-456 (in Russian).