Vol 2, No 6 (2011)

(1)

Available online through

www.ijma.info

International Journal of Mathematical Archive- 2 (6), June – 2011 910

SUFFICIENT CONDITIONS FOR A FUNCTION TO BE

AN APPROXIMATE IDENTITY ON

L (

2

B

_R

(0))

M. Akram*

Department of Mathematics, G. C. University, Katchery Road Lahore, Pakistan

••••_E-mail:_{[email protected], [email protected]}

(Received on: 12-05-11; Accepted on: 23-05-11)

---

ABSTRACT

I

n this paper, we investigate some properties of approximate identities and their Fourier coefficients with respect to some appropriate orthonormal bases of

L (

2

B

_R

(0))

. Sufficient conditions for a function in

L (

2

B

_R

(0)

×

B

_R

(0))

to be an approximate identity on

L (

2

B

_R

(0))

are also proved.

Key Words: Complete space, Jacobi Polynomials, orthonormal bases, Fourier coefficient, Approximate Identity,

L

2 -convergence.

AMS(2000) Classification: 41A30, 41A35, 41A63, 44A35, 65N80, 86-08.

---

1. INTRODUCTION:

Since the interior of the Earth looks like a ball, therefore it becomes necessary to develop new methods and tools to approximate functions on the 3-dimensional ball. The study of the Earth's interior, such as the mass density, the speed of propagation of seismic P and S waves and other rheological quantities, is still a field of research. Similarly, if we look at the 3-dimensional image of a human brain, it has similarity to the 3-dimensional ball. Since biologists deal with the similarly simplified model of a living cell and they are very much satisfied by the results obtained from such a model ([10], [15]), therefore approximating tools on the 3-dimensional ball can also be used to study the human brain. A particular example is to study the electromagnetic potential of the Earth and also the electromagnetic potential in the human brain whether it is produced by the local environment of the brain or it is produced by an electrotherapy for the purpose of the treatment of a cancer patient.

In [2] M. Akram and V. Michel studied locally supported approximate identities on the unit ball, they further studied the regularization of the Helmholtz decomposition on 3d-ball in [3]. In this paper, we investigate some properties of approximate identities and their Fourier coefficients with respect to some appropriate orthonormal bases of

2

L (

B

_R

(0))

. Sufficient conditions for a function in

L (

2

B

_R

(0)

×

B

_R

(0))

to be an approximate identity on

L (

2

B

_R

(0))

are also proved.

2. JACOBI POLYNOMIALS:

In this paper

N

denote the set of all positive integer, Where

N

₀

:=

N

∪

{0}

and

R

represent the set of all real numbers. Here, we present definition and some properties of the Jacobi polynomials. For further details and proofs we refer to [11] and [17].

The functions

P

_n( , )α β

,

n

∈

N

₀

,

with

α β

,

> 1

−

fixed, are called Jacobi polynomials if they satisfy the following properties for all

n

∈

N

₀

:

(i)

P

_n( , )α β is a polynomial of degree n, defined on

[ 1,1].

−

(ii)

1

( , ) ( , )

0

(1

x

) (1

x

)

P

n

( )

x P

m

( )

x dx

= 0

α β α β α β

−

+

for all

m

∈

N

₀

\

{ }

n

.

---

(2)

(iii) ( , )

(1) =

(

1)

=

(

)!

,

(

1) (

1)

! !

n

P

n

α β

α

Γ

+

Γ

+ Γ

+

where

Γ

represents the Gamma function.

The Jacobi polynomials

P

_n( , )α β

,

n

∈

N

₀

,

with

α β

,

> 1

−

fixed satisfy the following Rodriguez's formula:

{

}

( , )

( 1)

(1

) (1

)

( ) =

(1

)

(1

)

.

2 !

n n

d

x

P

x

n

dx

α β α β

−

+α +β

−

+

−

+

Note that the Legendre polynomials

P

_n represent the special case of the Jacobi polynomials

P

_n( , )α β for each

n

with

=

= 0.

α

β

The Jacobi polynomials

P

_n( , )α β for

> 1,

₀

=

1 x

β α

α

β

α β

−

+

have the following property

( , )

1

1 1

( , ) 2

1 = max( , )

,

2 (A)

max

( ) =

1 ( )

= max( , ) <

.

2

q

n x

' n

n

q

n

if

q

n

P

x

P

x

n

if

q

α β

− − ≤ ≤+

+

≥ −

−

:

Here

x

' is one of the two maximum points nearest

x

₀

.

The Jacobi polynomials satisfies the following recurrence relation. For any

α β

,

> 1

−

and for all

x

∈ −

[

1,1

]

( , ) ( , )

0 1

1 ( ) = 1,

( ) =

(

2) ,

2

2 P

α β

x

P

α β

x

α β

−

+

α β

+

x

and for

n

≥

2,

( , )

2 2 ( , )

1

( , ) 2

2 (

)(

2 2)

( )

= (

2 2)(

2 1)(

2 )

(

)

( )

2(

1)(

2 )

( ).

n

P

x

n

n x

P

x

n

n P

x

α β

α β α β

α β

α

β

α

β

α β

−

+

−

+

−

+

−

+

−

+ −

+

3. SPHERICAL HARMONICS:

Spherical harmonics are the functions most commonly used to represent scalar fields on a spherical surface. In this section we present definitions and some well-known facts from the theory of spherical harmonics. For further details we refer to [6, 9, 14] and the references therein.

Let

D

⊂

R

3 be open and connected. A function

F

∈

C ( )

(2)

D

is called harmonic if and only if

2 3

2 T

1 2 3 2

=1

( ) :=

( ) = 0,

= ( ,

,

)

.

x

i i

F

F x

x

for all x

x x x

D

x

∂

∇

∈

∂

A polynomial

P

on

R

3 is called homogeneous of degree

n

if

P

(

λ

x

) =

λ

n

P x

( )

for all

λ

∈

R

, and all

x

∈

R

3

.

The set of all homogeneous harmonic polynomials on

R

3 with degree

n

∈

N

₀ is denoted by Harm_n

(

R

3

)

, i.e.

{

}

3 3 2

0

Harm (

_n

R

) :=

P

∈

Hom (

_n

R

) |

∇

P

= 0 ,

n

∈

N

.

A spherical harmonic of degree

n

is the restriction of a homogeneous harmonic polynomial on

R

3 with degree

0

n

∈

N

to the unit sphere

Ω

. The collection of all spherical harmonics of degree

n

will be denoted by Harm_n

( )

Ω

, i.e.

{

3

}

0

(3)

Page: 901-918

Y

_n

∈

Harm ( )

_n

Ω

,

1 2

2 L

2

1 |

( ) |

||

.

sup

( )

4

n n

n

Y

ξ

π

∈Ω

+

≤

_Ω

In particular,

1 2 ,

2

1 |

( ) |

,

= 1,..., 2

1. sup

4

n j

n

Y

j

n

ξ

π

∈Ω

+

≤

+

Theorem 3.2 If

m

≠

n

then Harm_m

( )

Ω

is orthogonal to Harm_n

( )

Ω

in the sense of

L ( )

2

Ω

, i.e. if

m

≠

n

, then for all

Y

_m

∈

Harm ( )

_m

Ω

and all

Y

_n

∈

Harm ( )

_n

Ω

,

Y Y

_m

,

_n _L2

_{( )}

_Ω

= 0.

Theorem 3.3 The dimension of Harm_n

( ),

Ω

n

∈

N

₀ is equal to

2 n

+

1

, i.e.

(

)

0

dim Harm ( ) = 2

_n

Ω

n

+

1,

n

∈

N

.

By

{ }

_,

0

, =

,...

n j _n

Y

_j

_n

∈N

−

we will always denote a complete 2

L ( )

Ω

-orthonormal system in

Harm

_0,...,_∞

( )

Ω

, such that

Y

_{n j}_,

∈

Harm ( )

_n

Ω

for all

j

=

−

n

,..., .

n

We call

n

the degree of

Y

_{n j}_, and

j

the order of

Y

_{n j}_,

.

The evaluation of sums with spherical harmonics can be essentially simplified by the following theorem.

Theorem 3.4(Addition Theorem for Spherical Harmonics) For all

ξ η

,

∈ Ω

we have

, ,

=

2

1 ( )

( ) =

(

),

4

n

n j n j n

j n

n

Y

ξ

Y

η

P

ξ η

π

−

+

⋅

where

P

_n is the Legendre polynomial of degree

n

.

4. COMPLETE ORTHONORMAL SYSTEM IN HILBERT SPACE

L (

2

B

_R

(0))

:

The following systems of orthogonal polynomials on a three-dimensional ball

B

_R

(0) := {

x

∈

R

3

:| |

x

≤

R

}

are known (see , for example [4], [7], [12], [18]).

Theorem: 4.1 Two complete orthonormal systems in the Hilbert space

L (

2

B

R

(0))

are given by

2

I (0, 1/2)

, , 3 2 ,

4

2

3 | |

| |

( ) :=

2

1 ;

| |

n

n j m m n j

m

n

x

G

x

P

Y

R

x

+

−

0

,

; = 1,..., 2

1. n m

∈

N

j

n

+

II (0,2)

, , 3 ,

2

3 | |

( ) :=

2

1 ;

| |

n j m m n j

m

x

G

x

P

Y

R

x

+

−

(0) \ {0};

R

x

∈

B

n m

,

∈

N

0

;

j

= 1,..., 2

n

+

1.

In both cases,

{

}

( , )

0

a b

m _m

P

∈N represents the Jacobi polynomials (see

Section 2 for further details) and

{ }

_,

0

; =1,...,2

1

n j _n

Y

j

n

∈N

+

stands for a spherical harmonics orthonormal basis (see, for

(4)

The kernel

Φ ∈

_δ

L (

2

B

_R

(0)

×

B

_R

(0))

can be expressed in the form

1 2

1 1 1 2 2 2 _{1 1}, , ₁ _{2 2}, , ₂ =0 =0 , =0 =0

1 1 1 2 2 2

2

1

2

1 ( , ) =

( , ,

,

)

( )

,

n j m n j m

n m j n m j

n

x y

n j m n

j m G

x G

y

δ δ

∧

+

∞

Φ

for all

x y

,

∈

B

_R

(0)

, where the series converges in the

L (

2

B

_R

(0)

×

B

_R

(0))

-sense, that is

(

)

1 2 ₂

1 1 1 2 2 2

, =0 =0 , =0 =0

1 1 1 2 2 2

2

1

2

1 ( , ,

,

)

<

.

n m j n m j

n

n j m n

j m

δ

∧

+

∞

Φ

+∞

Here

Φ

_δ∧

( , ,

n j m n

₁ ₁ ₁

,

₂

,

j m

₂

,

₂

)

are the Fourier coefficients for

Φ

_δ defined as

1 1 1 2 2 2 _{1 1}, , ₁ _{2 2}, , ₂ _{L (}2 ₍₀₎ ₍₀₎₎

, , , ,

1 1 1 2 2 2 (0) (0)

( , ,

,

) :=

,

=

( , )

( )

.

n j m n j m

B_R B_R

n j m n j m

B_R B_R

n j m n

j m

G

x y G

x G

y dxdy

δ δ

δ

∧

×

Φ

5. PROPERTIES OF THE FOURIER COEFFICIENTS OF THE APPROXIMATE IDENTITIES ON 2

L (

B

_R

(0))

:

Lemma: 5.1 Let

Φ ∈

_δ

L (

2

B

_R

(0)

×

B

_R

(0))

be an approximate identity in

L (

2

B

_R

(0))

such that

( , )

= 1

(0)

B_R

Φ

δ

x y dy

for all

x

∈

B

R

(0)

and δ

( , ,

n j m n

1 1 1

,

2

,

j m

2

,

2

)

∧

Φ

be its Fourier coefficients as defined above, then

(0,1, 0, 0,1, 0) = 1.

δ

∧

Φ

Proof: Using the definition of

Φ

∧_δ

( , ,

n j m n

₁ ₁ ₁

,

₂

,

j m

₂

,

₂

)

and the addition theorem, we get

0,1,0 0,1,0 (0) (0)

0,1 0,1

3 3

(0) (0)

0 3 ₍₀₎ ₍₀₎

(0,1, 0, 0,1, 0)

=

( , )

( )

3

3 =

( , )

| |

3

1 =

( , )

.

4 | | | |

B_R B_R

x y G

x G

y dxdy

x

y

x y

Y

dxdy

R

x

R

y

x

y

x y

P

dxdy

R

x

y

δ δ

δ

π

∧

Φ

⋅

Applying Fubini's theorem (see [16]) and the fact that ₀

= 1

| | | |

x

y

P

x

⋅

y

, we get 3 ₍₀₎ ₍₀₎

3 ₍₀₎

3 (0,1, 0, 0,1, 0)

=

( , )

4

3 =

= 1

4

B_R B_R

B_R

x y dydx

R

dx

R

δ δ

π

∧

Φ

because

(0)

( , )

= 1

B_R

Φ

δ

x y dy

.

Theorem: 5.2 Let

Φ ∈

_δ

L (

2

B

_R

(0)

×

B

_R

(0))

be an approximate identity in

L (

2

B

_R

(0))

then

1 1 1 2 2 2 ₁ ₂ ₁ ₂ _{1 2}, 0

( , ,

,

) =

.

lim

δ

n j m n

j m

n

,

n

m

,

m

j j

δ

∧ → +

Φ

(5)

Page: 901-918

(

)

(

)

1 1 1 2 2 2 ₁ ₂ ₁ ₂ _{1 2},

, , , , , , , ,

1 1 1 2 2 2 1 1 1 2 2 2

(0) (0) (0)

, , , , , ,

2 2 2 2 2 2 1 1 1

(0)

( , ,

,

)

_,

=

( , )

( )

=

( )

.

j j

n m

n j m n j m n j m n j m

B_R B_R B_R

n j m n j m n j m

B_R

n j m n

j m

_n

_m

x y G

x G

y dxdy

G

x G

x dx

G

x

G

x G

x dx

δ

∧

Φ

−

Φ

−

Φ

å

−

Taking the absolute values of both hand sides, we have

(

)

(

)

(

)

(

)

1 1 1 2 2 2 _{1 2}, ₁, ₂ _{1 2},

, , , , , ,

2 2 2 2 2 2 1 1 1

(0)

, , , , , ,

2 2 2 2 2 2 1 1 1

(0)

( , ,

,

)

=

( )

.

n n m m j j

n j m n j m n j m

B_R

n j m n j m n j m

B_R

n j m n

j m

G

x

G

x G

x dx

G

x

G

x G

x dx

δ

∧

Φ

−

Φ

−

≤

Φ

−

å

Now using Hölder's inequality (see [5]) we get

(

)

1 1 1 2 2 2 _{1 2}, ₁, ₂ _{1 2},

1 1

2 ₂ ₂

2

, , , , , ,

2 2 2 2 2 2 1 1 1

(0) (0)

, , , , _{L (}2 ₍₀₎₎ , , _{L (}2 ₍₀₎₎

2 2 2 2 2 2 1 1 1

, , , , _{L (}2 2 2 2 2 2 2

( , ,

,

)

( )

|

( ) |

=

n n m m j j

n j m n j m n j m

B_R B_R

n j m n j m _B n j m _B

R R

n j m n j m

n j m n

j m

G

x

G

x

dx

G

x

dx

G

δ

δ δ δ

δ

∧

Φ

−

≤

Φ

−

Φ

−

Φ

−

å

P

(0))

.

B_R

Taking the limit

δ

→ +

0

and using the fact that

Φ

_δ is an approximate identity in

L (

2

B

_R

(0))

we get

1 1 1 2 2 2 _{1 2}, ₁, ₂ _{1 2}, 0

, , , , _{L (}2 ₍₀₎₎ 2 2 2 2 2 2

0

( , ,

,

)

lim

= 0.

n n m m j j

n j m n j m _B

R

n j m n

j m

G

δ δ

δ

∧ → +

→ +

Φ

−

≤

P

Φ

å

−

P

This proves the result.

Lemma: 5.3 Let

Φ ∈

_δ

L (

2

B

_R

(0)

×

B

_R

(0))

be given such that

(0)

|

( , ) |

= 1

B_R

Φ

δ

x y

dy

for all

x

∈

B

R

(0)

and

1 1 1 2 2 2

( , ,

n j m n

,

j m

,

)

δ

∧

Φ

be its Fourier coefficients as defined above. If the system

G

_{n j m}I_{, ,} is used in the definition of the Fourier coefficients then

1 1 1 2 2 2 _{1 1} _{2 2}

( , ,

n j m n

,

j m

,

)

_{n m} _{n m}

,

δ

∧

Φ

≤ Π

Π

(1)

with

1 1

1 1 1

1/2

4

2

3

2

1 =

,

3

n m

m

n

m

n

m

+

Π

and

2 2

2 2 2

1/2

4

2

3

2

1 =

,

3

n m

m

n

m

n

m

+

Π

where

n n m m

₁

,

₂

,

₁

,

₂

∈

N

₀,

j

₁

= 1,..., 2

n

₁

+

1

and

j

₂

= 1,..., 2

n

₂

+

1.

(6)

5 5 1 1

2 2 2 2

1 2 1 2

1 1 1 2 2 2

(2

3) (2

1) (2

1)

( , ,

,

)

,

12 m

m

n

n j m n

j m

δ

∧

+

Φ

≤

₍₂₎

where

n n m m

₁

,

₂

,

₁

,

₂

∈

N

₀,

j

₁

= 1,..., 2

n

₁

+

1

and

j

₂

= 1,..., 2

n

₂

+

1

.

Proof: Using the definition of δ

( , ,

n j m n

₁ ₁ ₁

,

₂

,

j m

₂

,

₂

)

∧

Φ

, we get

1 1 1 2 2 2 ₍₀₎ ₍₀₎ _{1 1}, , ₁ _{2 2}, , ₂

|

( , ,

,

) |

_B _B

|

( , )

n j m

( )

n j m

( ) |

.

R R

n j m n

j m

x y G

x G

y

dxdy

δ δ

∧

Φ

≤

Φ

(3) Taking system I of Theorem 4.1 in Equation (3) we have

1 1 2 2

1 1 1 2 2 2 3 3 ₍₀₎ ₍₀₎

2 1 2 2

1 2

,

2 2

1 1 1 2

, 2 2

4

2 3 4

2

3 |

( , ,

,

) |

( , )

| |

(0,

1/2)

(0,

1/2)

(2

1)

2

1 | |

.

| |

B_R B_R

n n

m n j m

n j

m

n

m

n

n j m n

j m

x y

R

x

y

n

P

Y

P

R

x

R

y

Y

dxdy

y

δ δ

∧

+

Φ

≤

Φ

+

×

−

By using equation

(A)

and Theorem 3.1 we get

1 1 1 2 2 2 3 1 1 2 2 ₍₀₎ ₍₀₎

1 1

2 2

1 1 1 2 2 2

1 2

1 |

( , ,

,

) |

4

2 3 4

2

3 ( , )

1/2

2

1 1/2

2

1 .

4

B_R B_R

n j m n

j m

m

n

m

n

x y

R

m

n

m

n

dxdy

m

δ δ

π

∧

Φ

≤

+

Φ

+

×

Applying Tonelli's theorem (see [16],) we get

1 1 1

1 1 1 2 2 2 1 1

1

2 2 2

2 2 3 ₍₀₎ ₍₀₎

2

1/2

2

1 |

( , ,

,

) |

4

2

3

4 1/2

2 1 1

4

2

3 ( , )

.

4

B_R B_R

m

n

n j m n

j m

m

n

m

n

m

n

x y dydx

m

R

δ

π

∧

+

Φ

≤

+

×

+

Φ

After integrating we obtain

1 1 1 2 2 2 ₁ ₁ ₂ ₂

|

Φ

_δ∧

( , ,

n j m n

,

j m

,

) |

≤ Π

_n

_m

Π

_n

_m

.

Taking system II of Theorem 4.1 in Equation ((3)) we have

1 1 1 2 2 2 3 1 2

, ,

1 1 1 2 2 2

(0) (0)

1 |

( , ,

,

) |

2 3 2

3 | |

| |

(0, 2)

( , )

(2

1)

2

1 .

| |

m n j m n j

B_R B_R

n j m n

j m

m

R

x

y

x y P

Y

P

Y

dxdy

R

x

R

y

δ

∧

Φ

≤

+

+ ×

Φ

−

(7)

Page: 901-918

1 1 1 2 2 2 1 2

1 1

2 2

1 1 2 2

3 ₍₀₎ ₍₀₎

1 2

1 1 2 2

1 2

1 3 ₍₀₎ ₍₀₎

|

( , ,

,

) |

2 3 2

3

2

1

2

1

1 ( , )

4

2 1 2

1 (

2)(

1) (

2)(

1)

2 3 2

3

2

4 (2

1 |

( , ) |

B_R B_R

n j m n

j m

m

n

m

n

x y

dydx

m

R

n

m

x y

dydx

R

δ

π

∧

Φ

≤

+

×

Φ

+

≤

+

×

Φ

≤

5 5 1 1

2 2 2 2

2 1 2

3 ₍₀₎

3) (2

1) (2

1)

.

16

B_R

m

n

dx

R

π

+

Finally, we obtain

5 5 1 1

2 2 2 2

1 2 1 2

1 1 1 2 2 2

(2

3) (2

1) (2

1)

|

( , ,

,

) |

.

12 m

m

n

n j m n

j m

δ

∧

+

Φ

≤

Lemma: 5.4 Let

Φ ∈

_δ

L (

2

B

_R

(0)

×

B

_R

(0))

and let there exist an integrable function

Φ

_δ defined on

B

₂R

(0)

such

that for each

F

∈

L (

2

B

_R

(0))

,

Φ

_δ

å

F

=

Φ ∗

_δ

F

and

(0) 2

|

( ) |

= 1

B_R

Φ

δ

y

dy

, then

1 1 1 2 2 2

|

Φ

_δ∧

( , ,

n j m n

,

j m

,

) | 1.

≤

Proof: As we know,

1 1 1 2 2 2 ₍₀₎ ₍₀₎ _{1 1}, , ₁ _{2 2}, , ₂

( , ,

,

) =

( , )

_n _{j m}

( )

_n _j _m

( )

.

B_R B_R

n j m n

j m

x y G

x G

y dxdy

δ δ

∧

Φ

Applying Fubini's theorem, we get

(

)

1 1 1 2 2 2 ₍₀₎ ₍₀₎ _{2 2}, , ₂ _{1 1}, , ₁

, , , ,

2 2 2 1 1 1

(0)

( , ,

,

)

=

( , )

( )

=

( )

.

n j m n j m

B_R B_R

n j m n j m

B_R

n j m n

j m

x y G

y dy G

x dx

G

x G

x dx

δ δ

δ

∧

Φ

å

This implies

(

)

1 1 1 2 2 2 ₍₀₎ _{2 2}, , ₂ _{1 1}, , ₁

|

( , ,

,

) |

_n _j _m

( )

_n _{j m}

( )

B_R

n j m n

j m

G

x G

x dx

δ δ

∧

Φ

≤

Φ

å

Using Hölder's inequality, we have

(

)

(

)

1 1 1 2 2 2

1 1

2 2 2 2

, , , ,

2 2 2 1 1 1

(0) (0)

1 1

2 2 2 2

, , , ,

2 2 2 1 1 1

(0) (0)

, , _{L (}2 ₍₀₎₎ , , _{L (}2 ₍₀₎₎

2 2 2 1 1 1

( , ,

,

)

( )

=

.

n j m n j m

B_R B_R

n j m n j m

B_R B_R

n j m _B n j m _B

R R

n j m n

j m

G

x

dx

G

x

dx

G

x

dx

G

x

dx

G

δ

∧

Φ

≤

Φ

≤

Φ ∗

å

P

After applying Young's inequality, we obtain

1 1 1 2 2 2 _{L (}1 ₍₀₎₎ _{2 2}, , ₂ _{L (}2 ₍₀₎₎ _{1 1}, , ₁ _{L (}2 ₍₀₎₎ 2

( , ,

,

)

_n _j _m _n _{j m}

.

B _R B_R B_R

n j m n

j m

G

δ δ

∧

Φ

≤ Φ

P

Since _, _,

1 1 1

n j m

G

and _, _,

2 2 2

n j m

G

belongs to an orthonormal system in

L (

2

B

R

(0))

, therefore their

2

L

-norms are equal to 1. Also by hypothesis, ₁

L (B₂_R(0))

= 1

δ

Φ

(8)

1 1 1 2 2 2

|

Φ

_δ∧

( , ,

n j m n

,

j m

,

) | 1.

≤

Next theorem gives the sufficient conditions for a function in

L (

2

B

_R

(0)

×

B

_R

(0))

to be an approximate identity in

2

L (

B

_R

(0))

.

Theorem: 5.5 Let

Φ ∈

_δ

L (

2

B

_R

(0)

×

B

_R

(0))

and the Fourier coefficients

Φ

_δ∧

( , ,

n j m n

₁ ₁ ₁

,

₂

,

j m

₂

,

₂

)

of

Φ

_δ have the properties

1 1 1 2 2 2

1 1 1 2 2 2 _{1 2}, ₁, ₂ _{1 2}, 0

1 1 1 2 2 2 1 2 1 2 1 2

( ) |

( , ,

,

) | 1,

( )

lim

( , ,

,

) =

,

( )

( , ,

,

) = 0

,

n n m m j j

i

n j m n

j m

ii

n j m n

j m

iii

n j m n

j m

for n

n m

m

j

δ δ δ

δ

∧

∧ →

∧

Φ

≤

Φ

≠

then,

Φ

_δ is an approximate identity in

L (

2

B

_R

(0))

.

Proof: For any

F

in

L (

2

B

_R

(0))

we have

(

)

_{L (}2 ₍₀₎₎

1 2

1 1 1 2 2 2 _{1 1}, , ₁ _{2 2}, , ₂ , =0 =1 , =0 =1

1 1 1 2 2 2 _{L (}2 ₍₀₎₎

1 2

1 1 1 2 2 2 _{1 1}, , ₁ _{2 2}, , , =0 =1 , =0 =1

1 1 1 2 2 2

( ) =

( ,.),

2

1

2

1 =

( , ,

,

)

( )

,

2

1

2

1 =

( , ,

,

)

( )

B_R

n j m n j m

n m j n m j

B_R

n j m n j m

n m j n m j

F

x

F

n

n j m n

j m G

x G

F

n

n j m n

j m G

x G

δ δ

δ

∧

Φ

+

∞

Φ

+

∞

Φ

å

2 2 L ( (0))

1

1 1 1 1 1, , 1 1 1, , 1 L (2 (0)) , =0 =1

1 1 1

,

2

1 =

( , ,

)

( )

,

B_R

n j m n j m

B_R

n m j

F

n

n j m G

x G

F

δ

∧

+

∞

Φ

where

Φ

_δ∧

( , ,

n j m

₁ ₁ ₁

) :=

Φ

_δ∧

( , ,

n j m n j m

₁ ₁ ₁

, , ,

₁ ₁ ₁

)

. Let us consider

(

)

(

)

1

1 1 1 _{1 1}, , ₁ _{1 1}, , ₁ _{L (}2 ₍₀₎₎ , =0 =1

1 1 1

1

, , , , 2

1 1 1 1 1 1 L ( (0)) , =0 =1

1 1 1

1

, , 1 1 1 , , 2

1 1 1 1 1 1 L ( (0))

, =0 =1

1 1 1

2

1 =

( , ,

)

,

2

1 ,

2

1 =

( , ,

) 1

,

.

n j m n j m

B_R

n m j

n j m n j m

B_R

n m j

n j m n j m

B_R

n m j

n

F

n j m G

G

F

n

G

F G

n

G

n j m

G

F

δ δ

δ

∧

+

∞

Φ

−

Φ

+

∞

−

+

∞

Φ

−

å

By Parseval's relation (see [19]), we have

(

)

2 2

2

2 1 1 1 , , 2

L ( (0)) _, ₌₀ ₁=1 2 2 2 L ( (0))

1 1

2

1 =

( , ,

) 1

_n _j _m

,

.

BR _{n m} j B_R

n

F

n j m

G

F

δ δ

∧

∞

₊

Φ

å

−

Φ

−

P

Taking the limit

δ

→ +

0

on both hand sides, we get

(

)

1 ₂ 2

2

2 1 1 1 , , ₂

L ( (0)) 2 2 2 L ( (0))

0 0 _, ₌₀ ₌₁

1 1 1

2

1 =

( , ,

) 1

,

.

lim

_B

lim

n j m

B

R n m j R

n

F

n j m

G

F

δ δ

∧

→ + → +

+

∞

Φ

å

−

Φ

−

P

(9)

Page: 901-918

sum. After simplifying, we obtain

(

)

1 2 2

2 1 1 1 , , ₂

L ( (0)) 2 2 2 L ( (0))

0 _, ₌₀ ₌₁ 0

1 1 1

2

1

2 ₌

_{( , ,}

_{) 1}

_,

_{= 0.}

lim

_B

lim

n j m

B

R n m j R

n

F

n j m

G

F

δ δ

∧

→ + → +

+

∞

Φ

å

−

Φ

−

P

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[2] M. Akram, V. Michel: Locally supported approximate identities on the unit ball, Rev Mat Complut 23, 2010, 233 - 249.

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e

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′

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