On estimates for the Fourier transform in the space L2(Rn)

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Harmonic analysis

On estimates for the Fourier transform in the space L ² ( R ⁿ )

Radouan Daher, Mohamed El Hamma

Department of Mathematics, Faculty of Sciences Aïn Chock, Hassan II University, Casablanca, Morocco

a r t i c l e i n f o a b s t r a c t

Article history:

Received 21 September 2013

Accepted after revision 16 December 2013 Available online xxxx

Presented by the Editorial Board

We obtain new inequalities for the Fourier transform in the space L²(Rⁿ), using a generalized spherical mean operator for proving two estimates in certain classes of functions characterized by a generalized continuity modulus.

©2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

1. Introduction and preliminaries

This work is based mainly on Titchmarsh’s theorem ([8], Theorem 84) in the one-dimensional case. In[2], Abilov et al.

proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilov’s for only two variables, using a translation operator.

In this paper, we prove the analog of Abilov’s results[2]in the Fourier transform for multivariable functions on

R

^{n. For} this purpose, we use a spherical mean operator in the place of the translation operator.

Assume thatL²

(R

ⁿ

)

the space of integrable functions f with the norm:

^f

2

=

Rⁿ

f ( x )

²

dx

1/2

.

The Fourier transform for the function f

∈

^L1

( R

ⁿ

)

is defined by:

f (ξ ) = ( 2 π )

^−n/2

Rⁿ

f ( x ) e

^−ix.ξ

dx .

The inverse Fourier transform is defined by the formula:

f ( x ) = ( 2 π )

^−n/2

Rⁿ

_f (ξ ) e

^ix.ξ

d ξ.

The Plancherel theorem provides an extension of the Fourier transform toL²

( R

ⁿ

)

, i.e.:

Rⁿ

f ( x )

²

dx =

Rⁿ

f (ξ )

²

d ξ.

Let jp

(

z

)

be a normalized Bessel function of the first kind, i.e.:

E-mail address:elmohamed77@yahoo.fr(M. El Hamma).

1631-073X/$ – see front matter ©2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.crma.2013.12.016

j

p

( z ) = ²

^p

Γ ( p + ¹ )

z

^p

J

p

( z ), ∀ z ∈ R

⁺

, (1)

where Jp

(

z

)

is a Bessel function of the first kind.

Consider inL²

(R

ⁿ

)

the spherical mean operator (see[4]):

M

h

f ( x ) = ¹ w

n−¹

Sⁿ⁻¹

f ( x + hw ) dw ,

where Sⁿ⁻¹ is the unit sphere in

R

^n,wn−¹ its total surface measure with respect to the usual induced measure dw.

The finite differences of the first and higher orders are defined by:

h

f ( x ) = ^M

h

f ( x ) − ^f ( x ) = ( M

_h

− ^I ) f ( x ),

^k_h

f ( x ) =

h

^k_h⁻¹

f ( x )

= ( M

_h

− ^I )

^k

f ( x ) =

k

i=⁰

(− ¹ )

^k−i

_i

k

M

_hⁱ

f ( x ),

where M⁰_hf

(

x

) =

^f

(

x

)

, Mⁱ_hf

(

x

) =

^Mh

(

M_hⁱ⁻¹f

(

x

))

fori

=

¹

,

2

, . . . ,

kandk

=

¹

,

2

, . . .

, I is the identity operator inL²

( R

ⁿ

)

. Thekth order generalized modulus of continuity of function f

∈

^L2

( R

ⁿ

)

is defined as:

Ω

_k

( f , δ) =

^k_h

f ( x )

2

, 0 < h δ.

Let W^r₂^,_,φ^k

(

D

)

denote the class of functions f

∈

^L2

(R

ⁿ

)

such that Dⁱf

∈

^L2

(R

ⁿ

)

,i

=

¹

,

2

, . . . ,

r(in the sense of Levi, see[6]) and

Ω

k

D

^r

f , δ

= ^O φ

δ

^k

,

where the operator D

=

n i=¹ ∂²

∂x²_i is the Laplace operator andx

= (

x1

,

x2

, . . . ,

xn

)

D⁰f

=

^{f, Dif}

=

^D

(

Dⁱ⁻¹f

)

,i

=

¹

,

2

, . . . ,

r and

φ (

t

)

is an arbitrary function defined on

[

⁰

, ∞)

^i.e.:

W

^r₂^,_,φ^k

( D ) = f ∈ ^L

²

R

ⁿ

, D

ⁱ

f ∈ ^L

²

R

ⁿ

, such that Ω

k

D

^r

f , δ

= ^O φ

δ

^k

, i = ¹ , 2 , . . . , r .

According to[4], we have:

M

_h

f ( x ) = ( 2 π )

^−n/2

Rⁿ

_f (ξ ) j

n−2 2

| ξ | ^h e

^ix.ξ

d ξ

and

f ( x ) = ( 2 π )

^−n/2

Rⁿ

_f (ξ ) e

^ix.ξ

d ξ,

i.e.:

M

_h

f ( x ) − ^f ( x ) = ( 2 π )

^−n/2

Rⁿ

_f (ξ ) j

n−2

2

| ξ | ^h

− ¹ e

^ix.ξ

d ξ.

By Parseval’s identity, we obtain:

M

_h

f ( x ) − ^f ( x )

²

2

=

Rⁿ

f (ξ )

²

j

n−2

2

|ξ | ^h

− ¹

2

d ξ.

It is easy to show that any function f

∈

^Wr2^,,φ^k

(

D

)

implies:

^k_h

D

^r

f ( x )

²

2

=

Rⁿ

|ξ |

^2r

1 − ^j

ⁿ−2

2

|ξ | ^h

2k

f (ξ )

²

d ξ. (2)

From[3], we have:

1 − ^j

ⁿ⁻²

2

( r ) ^min 1 , r

²

, (3)

where the symbol

means that the left-hand side is bounded above and below by a positive constant times the right-hand side.

2. Main result

Taking into account what was said in Section1for some classes of functions characterized by the generalized modulus of continuity, we can prove two estimates for the integral:

|ξ|^N

_f (ξ )

²

_d ξ.

Proposition 2.1.Let r

∈ N ∪ {

⁰

}

^{and k}

∈ N

^{. If f}

∈

^Wr₂^,_,φ^k

(

D

)

for some fixed

φ

defined on

[

⁰

, ∞)

^{, then}

|ξ|N

f (ξ )

²

d ξ = ^O N

^−2r

φ

²

N

^−k

as N

→ ∞

^.

Proof. In the terms of jp

(

z

)

, we have (see[1]):

1 − j

p

( z ) = O ( 1 ), z 1 , (4)

1 − ^j

p

( z ) = ^O z

²

, 0 ^{z 1} , (5)

√ hz J

_p

( hz ) = ^O ( 1 ), hz ⁰ . (6)

Let f

∈

^Wr₂^,_,φ^m

(

D

)

. By Hölder inequality, we have:

|ξ|^N

_f (ξ )

²

_d ξ −

|ξ|^N

_f (ξ )

²

_j

n−2 2

h |ξ | d ξ =

|ξ|^N

1 − j

n−2 2

h |ξ | _f (ξ )

²

_d ξ

=

|ξ|N

1 − ^j

ⁿ−2 2

h |ξ | f (ξ )

^2−1k

f (ξ )

^1k

d ξ

|ξ|N

f (ξ )

²

d ξ

^2k_2k⁻¹

|ξ|N

1 − ^j

ⁿ−2 2

h |ξ |

2k

f (ξ )

²

d ξ

_2k¹

|ξ|N

_f (ξ )

²

_d ξ

^2k_2k⁻¹

|ξ|N

1

|ξ |

^2r

1 − j

n−2 2

h |ξ|

2k

|ξ|

^2r

_f (ξ )

²

_d ξ

_2k¹

N

^−kr

|ξ|^N

_f (ξ )

²

_d ξ

^2k_2k⁻¹

|ξ|^N

1 − j

n−2 2

h |ξ|

2k

|ξ |

^2r

_f (ξ )

²

_d ξ

_2k¹

.

From formula(2), we have the inequality:

|ξ|N

1 − ^j

ⁿ−2 2

h |ξ|

2k

|ξ|

^2r

f (ξ )

²

d ξ

^k_h

D

^r

f ( x )

²

2

.

Therefore

|ξ|^N

_f (ξ )

²

_d ξ

|ξ|^N

_f (ξ )

²

_j

n−2 2

h |ξ|

d ξ + N

^−kr

|ξ|^N

_f (ξ )

²

_d ξ

^2k_2k⁻¹

^k_h

D

^r

f ( x )

^1k

2

.

Then

|ξ|^N

1 − ^j

ⁿ−² 2

h | ξ | _f (ξ )

²

d ξ = ^O

N

^−kr

|ξ|^N

f (ξ )

²

d ξ

^2k_2k⁻¹

^k_h

D

^r

f ( x )

₂^1k

.

Settingh

=

¹_N in the last inequality and from inequality(3), we obtain:

|ξ|^N

_f (ξ )

²

_d ξ = O N

^−kr

|ξ|^N

_f (ξ )

²

_d ξ

^2k_2k⁻¹

φ

^1k

1

N

k

.

Therefore

|ξ|^N

f (ξ )

²

d ξ = ^O N

^−2r

φ

²

N

^−k

.

This completes the proof ofProposition 2.1.

2

Theorem 2.1.Let

φ (

t

) =

^{tν. Then}

|ξ|^N

f (ξ )

²

d ξ = ^O N

^−r−kν

⇐⇒ ^f ∈ ^W

^r₂^,_,φ^k

( D )

where r

=

⁰

,

1

, . . .

;k

=

¹

,

2

, . . .

; 0

< ν <

2.

Proof. We prove sufficiency by usingProposition 2.1let f

∈

^Wr2^,,^kt^ν

(

D

)

, then

|ξ|^N

_f (ξ )

²

_d ξ

¹₂

= O N

^−r−kν

.

To prove necessity, let:

|ξ|N

f (ξ )

²

d ξ = ^O N

^−r−kν

i.e.

|ξ|N

f (ξ )

²

d ξ = ^O

N

^−2r−2kν

.

It is easy to show that there exists a function f

∈

^L2

( R

ⁿ

)

such that D^rf

∈

^L2

( R

ⁿ

)

and

D

^r

f ( x ) = ( − ¹ )

ⁿ

( 2 π )

ⁿ²

Rⁿ

| ξ |

^r

_f (ξ ) e

^ix.ξ

d ξ. (7)

From formula(7)and Parseval’s identity, we have:

^k_h

D

^r

f ( x )

²

2

=

Rⁿ

1 − j

n−2 2

h |ξ |

2k

|ξ|

^2r

f (ξ )

²

d ξ.

This integral is split into two:

Rⁿ

=

|ξ|<N

+

|ξ|N

= ^I

1

+ ^I

2

whereN

= [

^h−1

]

. We estimate them separately from(4); we have:

I

2

=

|ξ|^N

f (ξ )

²

| ξ |

^2r

1 − ^j

ⁿ−2

2

h | ξ |

2k

d ξ

= O

|ξ|^N

_f (ξ )

²

|ξ |

^2r

d ξ

= O

_∞

l=0

N+^l|ξ|^N+^l+¹

|ξ|

^2r

_f (ξ )

²

_d ξ

= ^O

_∞

l=⁰

( N + ^l + ¹ )

^2r

N+^l|ξ|^N+^l+¹

f (ξ )

²

d ξ

= ^O

_∞

l=⁰

( N + ^l + ¹ )

^2r

|ξ|^N+^l

f (ξ )

²

d ξ −

|ξ|^N+^l+¹

f (ξ )

²

d ξ

= ^O

_∞

l=⁰

( N + ^l + ¹ )

^2r

|ξ|N+l

f (ξ )

²

d ξ −

∞ l=⁰

( N + ^l + ¹ )

^2r

|ξ|N+l+1

f (ξ )

²

d ξ

= ^O

N

^2r

|ξ|^N

f (ξ )

²

d ξ +

∞ l=⁰

( N + ^l + ¹ )

^2r

− ( N + ^l )

^2r

|ξ|N+l

f (ξ )

²

d ξ

= ^O

N

^2r

|ξ|^N

f (ξ )

²

d ξ +

∞ l=¹

( N + ^l )

^2r−1

|ξ|N+l

f (ξ )

²

d ξ

= ^O

N

^2r

N

^−2r−2kν

+ ^O

_∞

l=¹

( N + ^l )

^2r−1

( N + ^l )

^−2r−2kν

= O N

^−2kν

+ O N

^−2kν

= O h

^2kν

i.e.:

I

₂

= ^O h

^2kν

.

Now, we estimate I1, by virtue of(5):

I

₁

=

|ξ|<N

| ξ |

^2r

1 − ^j

ⁿ−2

2

h | ξ |

2k

f (ξ )

²

d ξ

= ^O h

^4k

|ξ|<^N

|ξ |

^2r

|ξ |

^4k

f (ξ )

²

d ξ

= O h

^4k

|ξ|<N

|ξ |

^2r+4k

_f (ξ )

²

_d ξ

= ^O h

^4k

^N−1

l=⁰

l|ξ|<l+1

|ξ|

^2r+4k

f (ξ )

²

d ξ

= ^O h

^4k

^N−1

l=0

( l + ¹ )

^2r+4k

l|ξ|<l+¹

f (ξ )

²

d ξ

= ^O h

^4k

^N−1

l=0

( l + ¹ )

^2r+4k

|ξ|^l

f (ξ )

²

d ξ −

|ξ|^l+¹

f (ξ )

²

d ξ

= O h

^4k

1 +

N−¹

l=0

( l + 1 )

^2r+4k

− l

^2r+4k

|ξ|^l

_f (ξ )

²

_d ξ

= ^O h

^4k

1 +

N−¹

l=¹

l

^2r+4k−1

|ξ|^l

f (ξ )

²

d ξ

= ^O h

^4k

1 +

N−1

l=¹

l

^2r+4k−1

l

^−2r−2kν

= ^O h

^4k

1 +

N−1

l=⁰

l

^{4k−2kν−1}

= ^O h

^4k

O

N

^4k−2kν

= ^O h

^2kν

i.e.:

I

₁

= ^O h

^2kν

.

Combining the estimates for I₁and I₂ gives:

^k_h

D

^r

f ( x )

2

= ^O h

^kν

.

The necessity is proved.

2

Theorem 2.1in the caser

=

^{0 andk}

=

1 can be found in the works of Platonov[7]and Gioev[5].

Acknowledgements

The authors would like to thank the referee for his valuable comments and suggestions.

References

[1]V.A. Abilov, F.V. Abilova, Approximation of functions by Fourier–Bessel sums, Izv. Vysš. Uˇcebn. Zaved., Mat. 8 (2001) 3–9.

[2]V.A. Abilov, M.V. Abilova, M.K. Kerimov, Some remarks concerning the Fourier transform in the spaceL2(Rⁿ), Zh. Vychisl. Mat. Mat. Fiz. 48 (12) (2008) 2113–2120. Comput. Math. Math. Phys. 48 (2008) 2146–2153.

[3]W.O. Bray, M.A. Pinsky, Growth properties of the Fourier transforms, Filomat 26 (4) (2012) 755–760.

[4]W.O. Bray, M.A. Pinsky, Growth properties of Fourier transforms via moduli of continuity, J. Funct. Anal. 255 (2008) 2265–2285.

[5]D. Gioev, Moduli of continuity and average decay of Fourier transform: Two sided estimates, Contemp. Math. 458 (2008) 377–392.

[6]S.M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems, Nauka, Moscow, 1996 (in Russian).

[7]S.S. Platonov, The Fourier transform of functions satisfying the Lipschitz condition on rank one symmetric spaces, Sib. Math. J. 45 (6) (2005) 1108–1118.

[8]E.C. Titchmarsh, Introduction of the Theory of Fourier Integrals, Oxford University Press, 1937.