Fourier-Bessel Dini Lipschitz functions in the space L2α,n

DOI 10.1007/s13370-017-0509-5

Fourier-Bessel Dini Lipschitz functions in the space L ² _{α, n}

S. El Ouadih

· R. Daher

Received: 26 June 2015 / Accepted: 6 June 2017

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Abstract In this paper, using a generalized translation operator, we obtain an analog of Younis’s Theorem 5.2 in Younis (Int J Math Math Sci 9:301–312, 1986) for the generalized Fourier-Bessel transform for functions satisfying the Fourier-Bessel Dini Lipschitz condition in the space L

.

Keywords Singular differential operator · Generalized Fourier-Bessel transform · Generalized translation operator

Mathematics Subject Classification Primary 46L55 · Secondary 44B20

1 Introduction and preliminaries

Integral transforms and their inverses the Bessel transform are widely used to solve various problems in calculus, mechanics, mathematical physics, and computational mathematics (see [3,7]).

In this article, we obtain an analog of Younis’s Theorem 5.2 in [5] on description of the image under the generalized Fourier-Bessel transform of a class of functions satisfying the Dini Lipschitz condition in weighted function spaces on [ 0 , ∞[ . We now give the exact statement of this theorem.

Suppose that f (x ) is a function in the L

(R) space (all functions below are complex- valued), .

is the norm of L

(R), and δ is an arbitrary number in the interval (0, 1).

B

[email protected] R. Daher

[email protected]

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

Theorem 1.1 [12, Theorem 85] Let f ∈ L

(R) . Then the following conditions are equiva- lents:

(i) f ( x + h ) − f ( x )

= O ( h

) , as h → 0, 0 < δ < 1, (ii)

|λ|≥r

| f (λ)|

dλ = O(r

) as r → ∞, where f stands for the Fourier transform of f .

This theorem has been generalized in the case of noncompact rank 1 Riemannian symmet- ric spaces [10], and was extented in [11] for the generalized Fourier transform in the space L

(R

) using a spherical mean operator. The Titchmarsh’s Theorem has been generalized recently for a class of functions satisfying the Lipschitz condition for the Fourier-Bessel transform in [8,9].

On the other hand, Younis Theorem 5.2 [5] characterized the set of functions in L

(R) satisfying the Dini Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely we have:

Theorem 1.2 [5] Let f ∈ L

(R) . Then the following are equivalents:

(a) f ( x + h ) − f ( x ) = O

h^η (log¹_h)^γ

, as h → 0 , 0 < η < 1 , γ ≥ 0, (b)

|λ|≥r

| f (λ)|

dλ = O

r

(log r)

, as r → ∞, where f stands for the Fourier transform of f .

In this paper, we consider a second-order singular differential operator B on the half line which generalizes the Bessel operator B

. We obtain an analog of Theorem 1.2 for the generalized Fourier-Bessel transform associated to B in L

. For this purpose, we use a generalized translation operator.

We briefly overview the theory of generalized Fourier-Bessel transform and related har- monic analysis (see [1,6])

Consider the second-order singular differential operator on the half line B f (x ) = d

f ( x )

d x

+ ( 2 α + 1 ) x

d f ( x )

d x − 4n (α + n ) x

f (x),

where α >

and n = 0 , 1 , 2 , .... For n = 0, we obtain the classical Bessel operator B

f ( x ) = d

f ( x )

d x

+ ( 2 α + 1 ) x

d f ( x ) d x . Let M be the map defined by

M f (x ) = x

f (x), n = 0, 1, ....

Let L

, 1 ≤ p < ∞ , be the class of measurable functions f on [ 0 , ∞[ for which f

= M

f

< ∞,

where

f

=

0

| f ( x )|

x

d x

.

If p = 2, then we have L

= L

([0, ∞[, x

).

For α ≥

, we introduce the Bessel normalized function of the first kind j

defined by j

( z ) = (α + 1 )

(− 1 )

2

n !( n + α + 1 ) , z ∈ C, (1)

where ( x ) is the gamma-function (see [4]). The function y = j

( x ) satisfies the differential equation

B

y + y = 0,

with the condition initial y(0) = 0 and y

(0) = 0. The function j

(x ) is infinitely differen- tiable, even, and, moreover entire analytic.

From (1) we see that

z

lim

j

( z ) − 1

z

= 0 , (2)

by consequence, there exists c > 0 and η > 0 satisfying

|z| ≤ η ⇒ | j

(z) − 1| ≥ c|z|

. (3) From [2], we have

| j

(x )| ≤ 1, (4)

1 − j

( x ) = O ( x

), 0 ≤ x ≤ 1 . (5) For λ ∈ C , and x ∈ R , put

ϕ

( x ) = x

j

(λ x ). (6)

From [1,6] recall the following properties.

Proposition 1.3

(c) ϕ

satisfies the differential equation

B ϕ

= −λ

ϕ

. (d) For all λ ∈ C, and x ∈ R

|ϕ

(x )| ≤ x

e

.

The generalized Fourier-Bessel transform we call the integral transform (see [1]) F

f (λ) =

0

f (x )ϕ

(x )x

d x, λ ≥ 0, f ∈ L

.

Let f ∈ L

such that F

( f ) ∈ L

= L

([ 0 , ∞[, x

d x ) . Then the inverse generalized Fourier-Bessel transform is given by the formula (see [1])

f (x ) =

0

F

f (λ)ϕ

(x)dμ

(λ), where

d μ

(λ) = a

λ

dλ, a

= 1

4

((α + 1))

.

From [1,6], we have

Proposition 1.4

(e) For every f ∈ L

∩ L

we have the Plancherel formula

0

| f (x )|

x

d x =

0

| F

f (λ)|

dμ

(λ).

(f) The generalized Fourier-Bessel transform F

extends uniquely to an isometric isomor- phism from L

onto L

([ 0 , +∞[, μ

) .

Define the generalized translation operator T

, h ≥ 0 by the relation T

f ( x ) = ( x h )

τ

( M

f )( x ), x ≥ 0 , where τ

are the Bessel translation operators of order α defined by

τ

f ( x ) = c

0

f ( x

+ h

− 2x hcost ) si n

tdt , where

c

=

0

si n

tdt

= (α + 1 )

(π)

α +

. For f ∈ L

, we have (see [1])

F

( T

f )(λ) = ϕ

( h ) F

( f )(λ), (7)

F

( B f )(λ) = −λ

F

( f )(λ). (8)

2 Fourier-Bessel Dini Lipschitz condition

Definition 2.1 Let f ∈ L

, and let

T

f (x) − h

f (x)

≤ C h

log

, γ ≥ 0, i.e.,

T

f (x ) − h

f (x )

= O

h

log

,

for all x in R

and for all sufficiently small h, C being a positive constant . Then we say that f satisfies a Fourier-Bessel Dini Lipschitz of order η, or f belongs to Lip(η, γ ).

Definition 2.2 If however

T

f ( x ) − h

f ( x )

log¹_h

_γ

→ 0 , as h → 0 , γ ≥ 0 ,

i.e.,

T

f (x) − h

f (x)

= o

h

log

,

then f is said to be belong to the little Fourier-Bessel Dini Lipschitz class lip(η, γ ).

Remark It follows immediately from these definitions that lip(η, γ ) ⊂ Lip(η, γ ).

Theorem 2.3 Let η > 1. If f ∈ Lip(η, γ ), then f ∈ lip(1, γ ).

Proof For x ∈ R

and h small, f ∈ Lip(η, γ ) we have T

f (x) − h

f (x)

≤ C h

log

.

Then

log 1 h

T

f ( x ) − h

f ( x )

≤ C h

. Therefore

log

h

T

f (x ) − h

f (x )

≤ C h

, which tends to zero with h → 0. Thus

log

h

T

f ( x ) − h

f ( x )

→ 0 , h → 0 .

Then f ∈ lip ( 1 , γ ) .

Theorem 2.4 If η < ν , then Lip (η, 0 ) ⊃ Lip (ν, 0 ) and lip (η, 0 ) ⊃ lip (ν, 0 ).

Proof We have 0 ≤ h ≤ 1 and η < ν , then h

≤ h

.

Then the proof of this theorem.

3 New results on Fourier-Bessel Dini Lipschitz class Lemma 3.1 For f ∈ L

, we have

h

0

| j

(λ h ) − 1 |

| F

f (λ)|

d μ

(λ)

2

= T

f ( x ) − h

f ( x )

. where m = 0, 1, 2...

Proof We use the formulas (6) and (7) we conclude that

F

( T

f )(λ) = h

j

(λ h ) F

f (λ).

Then

F

((T

− h

I ) f )(λ) = h

( j

(λh) − 1) F

f (λ).

By Proposition 1.4, we have the result.

Theorem 3.2 Let η > 2. If f belong to the Fourier-Bessel Dini Lipschitz class, i.e., f ∈ Lip (η, γ ), η > 2 , γ ≥ 0 .

Then f is equal to the null function in R

.

Proof Assume that f ∈ Lip (η, γ ) . Then

T

f (x ) − h

f (x )

≤ C h

log

, γ ≥ 0.

From Lemma 3.1, we have h

0

| j

(λ h ) − 1 |

| F

f (λ)|

d μ

(λ) ≤ C

h

log

. Therefore

0

| j

(λh) − 1|

| F

f (λ)|

dμ

(λ) ≤ C

h

log

.

Then

0

| j

(λ h ) − 1 |

| F

f (λ)|

d μ

(λ)

h

≤ C

h

log

, Since η > 2 we have

h

lim

h

log

= 0 . Then

h

lim

0

| 1 − j

(λ h )|

λ

h

λ

| F

f (λ)|

d μ

(λ) = 0 . and also from the formula (2) and Fatou Theorem, we obtain

0

λ

| F

f (λ)|

d μ

(λ) = 0 .

Thus λ

F

f (λ) = 0 for all λ ∈ R

, then f is the null function.

Analog of the Theorem 3.2, we obtain this theorem.

Theorem 3.3 Let f ∈ L

. If f belong to lip(2, 0). i.e.,

T

f (x ) − h

f (x)

= O(h

), as h → 0.

Then f is equal to null function in R

.

Now, we give another the main result of this paper analog of Theorem 1.2.

Theorem 3.4 Let f ∈ L

. Then the following conditions are equivalents (i) f ∈ Lip(η, γ ) , 0 < η < 1, γ ≥ 0,

(ii)

r

| F

f (λ)|

dμ

(λ) = O

r

(log r)

, as r → ∞.

Proof ( i ) ⇒ ( ii ) . Suppose that f ∈ Lip (η, γ ) . Then T

f (x ) − h

f (x )

= O

h

log

, h → 0.

From Lemma 3.1, we have T

f ( x ) − h

f ( x )

= h

0

| j

(λ h ) − 1 |

| F

f (λ)|

d μ

(λ).

By formula (3), we get

h η 2h

| j

(λh) − 1|

| F

f (λ)|

dμ

(λ) ≥ c

η

2

h η 2h

| F

f (λ)|

d μ

(λ).

There exists then a positive constant C such that

h η 2h

| F

f (λ)|

dμ

(λ) ≤ C

h η 2h

| j

(λh) − 1|

| F

f (λ)|

dμ

(λ)

≤ C

h

T

f (x ) − h

f (x)

= O

h

log

.

We obtain

r

| F

f (λ)|

d μ

(λ) ≤ C

r

(log r)

, where C

is a positive constant. Now,

r

| F

f (λ)|

d μ

(λ) =

2ⁱr

| F

f (λ)|

d μ

(λ)

≤ C

r

(log r)

+ (2r)

(log 2r)

+ (4r)

(log 4r)

+ · · ·

≤ C

r

( log r )

1 + 2

+ ( 2

)

+ ( 2

)

+ · · ·

≤ K

r

(log r)

, where K

= C

(1 − 2

)

since 2

< 1.

Consequently

r

| F

f (λ)|

d μ

(λ) = O

r

( log r )

, as r → ∞.

(ii ) ⇒ (i). Suppose now that

r

| F

f (λ)|

dμ

(λ) = O

r

( log r )

, as r → ∞,

and write

T

f ( x ) − h

f ( x )

= h

( I

+ I

), where

I

=

0

| j

(λ h ) − 1 |

| F

f (λ)|

d μ

(λ), and

I

=

1/h

| j

(λ h ) − 1 |

| F

f (λ)|

d μ

(λ).

Using the inequality (4), we get I

≤ 4

1/h

| F

f (λ)|

d μ

(λ) = O

h

log

, as h → 0 . Set

φ(λ) =

λ

| F

f (x )|

dμ

(x).

From formula (5) and integration by parts, we have I

= −

0

| j

(λ h ) − 1 |

|φ

(λ) d λ

≤ −h

0

λ

φ

(λ)dλ

≤ −φ 1

h

+ 2h

0

λφ(λ)dλ

≤ 2h

0

λ

(log λ)

dλ

≤ C

h

log

where C

is a positive constant, and this ends the proof.

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