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

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DOI 10.1007/s13370-014-0278-3

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

R. Daher · M. El Hamma

Received: 9 November 2013 / Accepted: 20 June 2014

Abstract In this paper, we prove two estimates useful in applications for the Fourier transform in the space L

^p

(

Rⁿ

), 1 < p ≤ 2, as applied to some classes of functions characterized by a generalized modulus of continuity.

Keywords Fourier transform · Generalized modulus of continuity Mathematics Subject Classification 42B12

1 Introduction and preliminaries

The Fourier transform, as well as Fourier series, is widely used in various fields of calculus, mathematical physics, etc.

In [1], Abilov et al. proved two estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized modulus of continuity. In this paper, we prove these estimates in the space L

^p

(

Rⁿ

) of p-power integrable functions. We prove out that similar results have been established in the context of Bessel transform and Jacobi transform [2,3].

Assume that L

^p

(

Rⁿ

), (1 < p ≤ 2), is the space of p-power integrable functions f :

R

−→

C

with the norm

f

p

=

Rⁿ

| f (y)|

^p

d y

_1/p

.

R. Daher·M. El Hamma (

B

⁾

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

e-mail: [email protected] R. Daher

e-mail: [email protected]

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The Fourier transform for the function f ∈ L

^p

(

Rⁿ

) , 1 < p ≤ 2, is defined by f (ξ) = 1

( 2 π)

^n/2

Rⁿ

f ( x ) e

⁻ⁱ^ξ.x

d x The inversion formula of Fourier transform is defined by

f ( x ) = 1 ( 2 π)

^n/2

Rⁿ

f (ξ) e

ⁱ^ξ.x

d ξ

The Fourier transform above extends to a bounded linear map f −→ f from L

^p

(

Rⁿ

) to L

^q

(

Rⁿ

) , for 1 < p ≤ 2 and

¹_p

+

¹_q

= 1 (see [5]), so

f

q

≤ C f

p

, (1)

where C is a positive constant and f ∈ L

^p

(

Rⁿ

).

In this paper, we estimate the integral

|λ|≥N

| f (λ)|

^q

d λ, ∀ N > 0 . in certain classes of functions in the space L

^p

(

Rⁿ

).

Below, to simplify the calculations, we consider only two variable functions. Similar results are also valid for multivariable functions.

In L

^p

(

R²

) , we consider the operator F

h

f (x, y) = 1

4h

²

_x+h

x−h

_y+h

y−h

f (ξ, η)dξd η, h > 0.

which is analogous to Steklov’s function.

We define the differences of first and higher orders as follows

h

f ( x , y ) = F

_h

f ( x , y ) − f ( x , y ) = ( F

_h

− E ) f ( x , y )

^k_h

f ( x , y ) =

h

(

^k−1_h

f ( x , y )) = ( F

_h

− E )

^k

f ( x , y ) =

k i=0

(− 1 )

^k−i

k i

F

ⁱ_h

f ( x , y ), (2) where

F

⁰_h

f (x, y) = f (x , y), F

ⁱ_h

f (x , y) = F

h

(F

ⁱ⁻¹_h

f (x , y)), i = 1, 2, . . . , k; k = 1, 2, . . . and E is the unit operator in the space L

^p

(

R²

).

The quantity

k

( f, δ) = sup

0<h≤δ

^k_h

f (x, y)

p

is called the generalized modulus of continuity of kth order of the function f ∈ L

^p

(

R²

) . Let W

^r,k_p_,φ

(D) denote the class of functions f ∈ L

^p

(

R²

) having the generalized partial derivatives

∂ f

∂x , ∂

²

f

∂x∂y , · · ·

in the sense of Levi (see [4]) that belong to f ∈ L

^p

(

R²

), such that

k

(D

^r

f, δ) = O(φ(δ

^k

)),

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where

D = ∂

²

∂x

²

+ ∂

²

∂y

²

,

D

⁰

f = f , D

ⁱ

f = D(D

ⁱ⁻¹

f ), i = 1, 2, . . . , r , and φ(t ) is an arbitrary function defined on [0, ∞).

Since

F

h

f (x , y) = 1 2π

_∞

−∞

_∞

−∞

sin(hξ) sin(hη)

hξhη f (ξ, η)e

^i(xξ+ηy)

dξ dη.

Lemma 1.1 Let f ∈ W

^r,k_p_,φ

( D ) . Then

_∞

−∞

_∞

−∞

(ξ

²

+ η

²

)

^qr

1 − sin ( h ξ) sin ( h η) h ξ h η

qk

| f (ξ, η)|

^q

d ξ d η ≤ C

^q

^k_h

D

^r

f ( x , y )

^qp

.

Proof We have

(D

^r

f )(ξ, η) = (−1)

^r

(ξ

²

+ η

²

)

^r

f (ξ, η).

Then

(F

ⁱ_h

D

^r

f )(ξ, η) = (−1)

^r

(ξ

²

+ η

²

)

^r

sin(hξ) sin(hη) hξ hη

_i

f (ξ, η).

From formula (2), we conclude that the Fourier transform of

^k_h

D

^r

f ( x , y ) is (− 1 )

^r

(ξ

²

+ η

²

)

^r

1 −

^sin(hξ)_h_ξ_h^sin(hη)_η

k

f (ξ, η). By formula (1), we have the result.

2 Main result

In this section we give the main result of this paper.

Theorem 2.1 For functions f (x , y) ∈ L

^p

(

R²

), 1 < p ≤ 2, in the class W

^r_p^,_,φ^k

(D),

sup

W^r,k_p,φ(D) ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η

1/q

= O

N

^−2r

φ( c N )

^k

,

where

¹_p

+

¹_q

= 1, c > 0 is a fixed constant and φ(t) is an arbitrary function defined on the interval [ 0 , ∞) .

Proof Let f ∈ W

^r,k_p,φ

(D). Taking into account the Hölder inequality yields

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ξ²+η²≥N²

| f (ξ, η)|

^q

dξd η −

ξ²+η²≥N²

sin(hξ) sin(hη)

hξ hη | f (ξ, η)|

^q

d ξdη

=

ξ²+η²≥N²

1 − sin ( h ξ) sin ( h η) h ξ h η

| f (ξ, η)|

^q

d ξ d η

=

ξ²+η²≥N²

1 − sin(hξ) sin(hη) hξ hη

| f (ξ, η)|

^q⁻¹^/^k

| f (ξ, η)|

¹^/^k

dξ dη

≤

ξ²+η²≥N²

| f (ξ, η)|

^q

dξ dη

^qk−1

qk

ξ²+η²≥N²

1 − sin(hξ) sin(hη) hξhη

_qk

| f (ξ, η)|

^q

dξ dη

¹

qk

=

ξ²+η²≥N²

| f (ξ, η)|

^q

dξ dη

^qk−1

qk

×

ξ²+η²≥N²

1 (ξ

²

+ η

²

)

^qr

(ξ

²

+ η

²

)

^qr

1 − sin(hξ) sin(hη) hξ hη

_qk

| f (ξ, η)|

^q

d ξdη

¹

qk

≤ N

^−2r/k

ξ²+η²≥N²

| f (ξ, η)|

^q

dξ dη

^qk−1

qk

×

ξ²+η²≥N²

(ξ

²

+ η

²

)

^qr

1 − sin ( h ξ) sin ( h η) h ξ h η

qk

| f (ξ, η)|

^q

d ξ d η

¹

qk

From Lemma 1.1, we have the inequality

ξ²+η²≥N²

(ξ

²

+ η

²

)

^qr

1 − sin(hξ) sin(hη) hξhη

qk

| f (ξ, η)|

^q

d ξ d η ≤ C

^q

^k_h

D

^r

f ( x , y )

^qp

. Therefore

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξdη ≤

ξ²+η²≥N²

sin(hξ) sin(hη)

hξ hη | f (ξ, η)|

^q

dξ dη + C

^1/k

N

⁻^2r/k

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η

^qk−_qk¹

^k_h

D

^r

f ( x , y )

^1/kp

Let

I =

ξ²+η²≥N²

sin(hξ) sin(hη)

hξhη | f (ξ, η)|

^q

dξ dη From [1], we have

|I| ≤

√ 2 N h

ξ²+η²≥N²

| f (ξ, η)|

^q

dξdη Consequently

ξ²+η²≥N²

sin(hξ) sin(hη)

hξhη | f (ξ, η)|

^q

dξ dη ≤ 4 √

2 N h

ξ²+η²≥N²

| f (ξ, η)|

^q

dξd η.

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Then

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η ≤ 4 √ 2 N h

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η + C

^1/k

N

^−2r/k

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η

^qk−1

qk

^k_h

D

^r

f ( x , y )

¹p^/^k

Setting h =

_N^c

in the last inequality and choosing c > 0 such that 1 −

⁴^√_c²

≥

¹₂

, we obtain 1 − 4 √

2 c

mat hop

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η

≤ C

^1/k

N

^−2r/k

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η

^qk−1

qk

^k_h

D

^r

f ( x , y )

^1/p ^k

Hence

ξ²+η²≥N²

| f (ξ, η)|

^q

dξd η = O( N

⁻^{2r q}

^k_c_/_N

D

^r

f (x, y)

^qp

) Since

^k_c/N

D

^r

f (x , y)

p

= O

φ c N

_k

, we obtain

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η

_1/q

= O

N

^−2r

φ c N

k

which proves Theorem 2.1.

Corollary 2.2 Let f ∈ W

^r_p^,_,^k_tν

(D), (ν > 0), then

ξ²+η²≥N²

| f (ξ, η)|

^q

dξ dη = O(N

^{−2qr−qkν}

) where r = 0 , 1 , 2 , . . . ; k = 1 , 2 , . . .

Proof Let f ∈ W

^r_p^,_,^k_tν

(D) and φ(t) = t

^ν

. Then From Theorem 2.1, we have

ξ²+η²≥N²

| f (ξ, η)|

^q

d ξ d η = O ( N

^{−2qr−qkν}

) Thus, the proof is finished.

References

1. Abilov, V.A., Abilova, F.V., Kerimov, M.K.: Some Remarks Concerning the Fourier Transform in the Space L₂(Rⁿ). Comput. Math. Math. Physics 48(12), 2146–2153 (2008)

2. R. Daher and M. El Hamma, On Estimates for the Bessel Transform in the Space Lp,α(R+)Thai Journal of Mathematics Vol (11) (2013) No. 3. pp. 697–702.

3. Daher, R., El Hamma, M.: On Estimates for the Jacobi transform in the Space L²(R⁺, (α,β)(t)dt)Inter.

J. of Applied Mathematics. 25(1), 13–23 (2012)

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4. Nikol’skii, S.M.: Approximation of Functions of Several Variables and Embedding Theorems. Nauka, Moscow (1969). [in Russian]

5. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Prince- ton N. J (1971)

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