On the range of the Fourier transform connected with Riemann-Liouville operator

We characterize the range of some spaces of functions by the Fourier transform associated with the Riemann-Liouville operator Rα, α > 0 and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a PaleyWiener-Schwartz theorems.


Introduction
In [3], the first author with the others consider the so-called Riemann-Liouville transform R α ; α 0, defined on the space C * (R 2 ) (the space of continuous functions on R 2 , even with respect to the first variable) by The mapping R α generalizes the mean operator R 0 defined by The dual operator t R 0 of R 0 is defined by t R 0 (g)(r, x) = 1 π R g r 2 + (x − y) 2 , y dy.
The mean operator R 0 and its dual t R 0 play an important role and have many applications; for example, in image processing of the so-called synthetic aperture radar (SAR) data [9,10] or in the linearized inverse scattering problem in acoustics [8].The operators R 0 and t R 0 have been studied by many authors and from many points of view [2,13,14].In [3]; the authors associated to the Riemann-Liouville operator the Fourier transform F α defined by where, j α is a modified Bessel function.They have constructed the harmonic analysis related to the Fourier transform F α (inversion formula, Plancherel formula, Paley-Wiener theorem, Plancherel theorem ...).
Our investigation in the present work consists to characterize the range of some spaces of functions by the Fourier transform F α and to establish a real Paley-Wiener theorem and a Paley-Wiener-Schwartz theorem for this transform.More precisely, in the second section of this paper, we characterize the range of some subspace of L2 [0, +∞[×R; r 2α+1 dr ⊗ dx (the space of square integrable functions on [0, +∞[×R with respect to the measure r 2α+1 dr ⊗ dx).In the third section; we give a new characterization of the Schwartz's space S * (R 2 ) (the space of infinitely differentiable functions on R 2 ; even with respect to the first variable, rapidly decreasing together with all their derivatives) [15,16,18].Using this; we give a nice description of the space S * (Γ) (the space of infinitely differentiable functions on Γ = R 2 ∪ it, x ; (t, x) ∈ R 2 , |t| |x| ; even with respect to the first variable, rapidly decreasing with all their derivatives).In the last section, using the idea of [4]; we establish a real Paley-Wiener theorem and a Paley-Wiener-Schwartz theorem.
We recall that in [21]; the author obtains similar results for the Hankel transform and the generalized Hankel transform on the half line.
• dm n (x) the measure defined on R n , by Proposition 2.1.i.For all non negative measurable function f on Γ + (respectively integrable on Γ + with respect to the measure dγ α ), we have ii.For all non negative measurable function g on [0, +∞[×R (respectively integrable on [0, +∞[×R with respect to the measure dν α ), we have Definition 2.2.The Fourier transform associated with the Riemann-Liouville operator is defined on L 1 (dν α ) by where Γ is the set defined by the relation (2.3) and ϕ µ,λ is the eigenfunction given by (2.1).
This section consists to characterize, by the Fourier transform associated with the Riemann-Liouville operator, a space of functions having only some integral conditions at infinity.This permits in the coming section, to give an other description of the Schwartz's space on the set Γ.
We denote by [3,13] • S(R 2 ) the space of infinitely differentiable functions on R 2 , rapidly decreasing together with all their derivatives, and S * (R 2 ) its subset consisting of even functions with respect to the first variable.
• S * (Γ) the space of infinitely differentiable functions on Γ, even with respect to the first variable, rapidly decreasing together with all their derivatives, which means where To prove the main result of this section, we need the following lemma.
be a bounded function such that i.For all (µ, λ) ∈ R 2 ; the function ii.
Proof.• Suppose firstly that f ∈ S(R 2 ).By integration by parts; we have Then, the result follows from the hypothesis ii) and the fact that f and all its derivatives are bounded on R 2 .
and the required result follows from the first case.
Example 3.2.Let a be a positive real number and let From the asymptotic expansion of the function j α [12,22]; it follows that the functions r −→ r α+ 1 2 j α (r) and g(r) = r 0 s α+ 1 2 j α (s)ds are bounded on [0, +∞[.On the other hand, for all (µ, λ) ∈ R 2 ; Thus, Consequently; from lemma 3.1, we deduce that In the following, to give a nice description of rapidly decreasing functions; we need the following notations Then, for all f ∈ S * (R 2 ); we have the following properties • For all k ∈ N; Where B is the mapping given by the relation (2.7).Now, we are able to prove the main result of this section.
2. The Fourier transform F α (f ) of f satisfies the following properties i.The function F α (f ) is infinitely differentiable on Γ, even with respect to the first variable.
ii.For all belongs to the space L 2 (dν α ).Then, for all (l 1 , l 2 ) ∈ N 2 ; the function 2), we deduce that for all k ∈ N and s ∈ R; then, by derivative's theorem, it follows that the function is infinitely differentiable on R 2 , even with respect to the first variable.Hence, from the relation (2.6), the function F α (f ) is infinitely differentiable on Γ, even with respect to the first variable.
ii.For all (k 1 , k 2 ) ∈ N 2 and using the relations (2.6) and (3.2), we get x) belongs to the space L 2 (dν α ); by Plancherel theorem's; the function (the space of continuous functions g on R 2 ; even with respect to the first variable and such that lim On the other hand; for all (µ, λ) ∈ [0, +∞[×R, we have where a > 0 and by (3.6); there exists a > 1 such that Let ψ be the function defined in example 3.2 by Applying the result of example 3.2; we deduce that This shows that lim and consequently; lim (3.7) Combining the relations (2.6), (3.2), (3.5) and (3.7), we get lim and from the derivative's theorem, We have Using the same argument as in iii) and the example 3.2, with and therefore lim Wich means that lim • Conversely; suppose that f ∈ L 2 (dν α ) and F α (f ) satisfies the assertion 2) of theorem.In particular; for every In virtue of the relations (2.5) and (3.2), we deduce that for all (k 1 , k 2 ) ∈ N 2 ; the function Let's denote by Λ n ; n ∈ N * , the usual Fourier transform defined on and F α the Fourier Bessel transform defined on the space then, there exists a null set and By (3.9); the function f k,µ belongs to L 2 (dm 1 ) and However; by integration by parts; we have On the other hand, from the hypothesis iii) and by writing In particular; for all µ ∈ [0, +∞[; Consequently; for all µ ∈ N c 1 ; Combining the relations (3.11) and (3.13), we get and by iteration, we deduce that Using the relation (3.10), we obtain Since the usual Fourier transform Λ 1 is an isometric isomorphism from Integrating over [0, +∞[ with respect to the measure r 2α+1 dr 2 α Γ(α + 1) and using the fact that the Fourier-Bessel transform dr onto itself, we deduce that By the same way, and using the fact that for all k ∈ N; we deduce that there exists a null set N 2 ⊂ R such that for all λ ∈ N c 2 ; dr .Now; integrating by parts; we have On the other hand, from the hypothesis iii) and by the relation (3.12), we deduce that for all k ∈ N; In particular, for all λ ∈ R; However, from the relation (3.8) we have, and by the relation (3.18), we deduce that for all λ ∈ R By the same way, we have using the relation (3.1) and (3.2), we get By the hypothesis iv), it follows that for all λ ∈ R; Combining the relations (3.16), (3.17), (3.18) and (3.19), we deduce that for all λ ∈ N c 2 ; the function dr and By iteration, for all λ ∈ N c 2 , the function dr and we have Integrating over [0, +∞[×R, with respect to the measure dν α (r, λ) and using the Fubini's theorem and Plancherel theorem's, respectively for F α and Λ 1 ; the relation (3.20) leads to belong to L 2 (dν α ).Then by Hölder's inequality, we deduce that for all (l 1 , l 2 ) ∈ N 2 ; the functions belong to L 1 (dν α ), and by derivative's theorem, the relation (3.4) and the inversion formula for the transform F α , that is we deduce that the functions f and F α (f ) are infinitely differentiable on R 2 , even with respect to the first variable.Moreover, for all and lim 2. For all (k 1 , k 2 ) ∈ N 2 and a ∈ R; a > 0, the function is bounded on [0, +∞[×R.
In fact; let m ∈ N; m 3 and m 2(α + 1) a .By a simple calculus and using the fact that f and all its derivatives are bounded on [0, +∞[×R; we deduce that for all (k 1 , k 2 ) ∈ N 2 ; there exists and by 1) of this proof, we deduce that the function m is integrable on [0, +∞[×R with respect to the measure dm 2 (r, x) and by (4.1), we have This shows that the function is bounded on [0, +∞[×R and for all (r, x) ∈ [0, +∞[×R; From 2) there exists On the other hand; 4. For all (k 1 , k 2 ) ∈ N 2 ; the function is bounded on [0, +∞[×R.Indeed; for k 1 1, the result follows from 2) Let's prove that for all k ∈ N; k 1; the function is bounded on [0, +∞[×R.From the fact that f and all its derivatives are bounded, we deduce that there exists C k > 0 such that; and by 3) we deduce that the function belongs to L 1 [0, +∞[×R, dm 2 (r, x) , and by (4.1) we have; Consequently, for all (r, x) ∈ [0, +∞[×R; .
By the same method and using the relation (4.2), we prove that for all (k 1 , k 2 ) ∈ N 2 ; the function This achieves the proof of proposition 4.1.
In the sequel; we give a new description of the Schwartz's space S * (R 2 ).Namely, we have Theorem 4.2.Let f be a continuous function on R 2 , even with respect to the first variable.Then, the following properties are equivalent.
i.For all (k 1 , k 2 ) ∈ N 2 ; the functions ii.The function f is infinitely differentiable on R 2 , even with respect to the first variable, bounded together with all its derivatives on [0, +∞[×R and for all (k 1 , k 2 ) ∈ N 2 ; the function iii.The function f belongs to the space S * (R 2 ).
iv.For all (k 1 , k 2 ) ∈ N 2 ; the functions Proof.• From the proof of proposition 4.1, we deduce that ii) holds if i) is satisfied.
• Suppose that f satisfies ii).Then, for all (k 1 , k 2 ) ∈ N 2 ; we have And by hypothesis, we deduce that for all (k 1 , k 2 ) ∈ N 2 ; the function By the same way, for all (k 1 , k 2 ) ∈ N 2 ; the function On the other hand, for all (k 1 , k 2 ) ∈ N 2 ; Consequently, From (4.3), we deduce that for all (k 1 , k 2 ) ∈ N 2 ; the function is bounded on [0, +∞[×R.By the same way, and using (4.4) it follows that the function Thus, the functions ∂f ∂r and ∂f ∂x satisfy the same hypothesis as the function f .By iteration, we deduce that for all (l 1 , l 2 ) ∈ N 2 ; the function belong to L 2 (dν α ), because the transform F α is an isomorphism from S * (R 2 ) onto itself.
• Lastly, if the hypothesis iv) is satisfied, then by proposition 4.1 we deduce that i) holds.
Corollary 4.3.Let f be a continuous function on Γ, even with respect to the first variable.Then the following assertions are equivalent.
ii.The function f is infinitely differentiable on Γ, bounded together with all its derivatives on Γ + , and for all iii.The function f belongs to S * (Γ).
iv.For all (k 1 , k 2 ) ∈ N 2 ; the functions Proof.let f be a continuous function on Γ, even with respect to the first variable.We consider the function g defined on [0, +∞[×R by Then, • For all (µ, λ) ∈ Γ; • For every (r, x) ∈ [0, +∞[×R; So, if the function f satisfies the assertion i) of this corollary; then for all Consequently, the result follows from theorem 4.2 and the fact that for all g ∈ S * (R 2 ); the function f = g • θ belongs to S * (Γ).

Fourier transform of functions with bounded supports.
In this section, we characterize some spaces of functions by their Fourier transforms.More precisely, we establish a real Paley-Wiener theorem and a Paley-Wiener-Schwartz theorem for the Fourier transform connected with the Riemann-Liouville operator.i.If g has a compact support, then f satisfies the assertion 2) of theorem 3.3.Moreover, the sequence A n α F α (g) 2,γα n converges to σ g , where ii. Conversely, let g ∈ L 2 (dν α ) such that F α (g) satisfies the assertion 2) of theorem 3.3 and the sequence A n α F α (g) 2,γα n has a finite limit σ, then g has a compact support and σ = σ g .Proof.i. Suppose that g has a compact support, then for all (k 1 , k 2 ) ∈ N 2 ; the function belongs to L 2 (dν α ).By theorem 3.3, we deduce that the function f = F α (g) satisfies the assertion 2) of theorem 3.3.From the relation (3.3), we have; Applying Plancherel theorem for the transform F α , it follows that for all n ∈ N; 2,γα = (r 2 + x 2 ) n g 1 2n 2,να . ( has a positive measure.Then 2,γα = (r 2 + x 2 ) n g 1 2n 2,να and by hypothesis, we get; σ σ + ε which is impossible.This shows that g has a bounded support and by the proof of i) we can show that σ = σ g .
In the following, we shall give a new characterization of infinitely differentiable functions with bounded supports, by means of their Fourier transforms.For this, let (σ 1 , σ 2 ) ∈ (R * + ) 2 ; we denote by ; the space of entire functions g on C 2 , slowly increasing of exponential type, i.e, there exists an integer k such that ; the space of entire functions f on C 2 , rapidly decreasing of exponential type, i.e for all k ∈ N; and , its subset consisting of even functions with respect to the first variable.
• E (R 2 ), the space of infinitely differentiable functions on R 2 .
• S (R 2 ), the space of tempered distributions on R 2 .
• D ), the space of infinitely differentiable functions, even with respect to the first variable and with support in and T f the tempered distribution given by the function f .The following result is a consequence of Bernstein's inequality and the theorem of Kolmogoroff [1,5,17].ii.
belongs to L p dm 2 .Then, for all n ∈ N * and k ∈ N; 0 < k < n, we have Proof.
• In the case p = +∞, the proof can be found in [17].
• Suppose that p ∈ [1, +∞[ and let where p is the conjugate exponent of p. Then and Applying lemma 8 of [17] and using the hypothesis, we deduce that the function F is infinitely differentiable on R, and we have Then, by Hölder's inequality, we get and by (5.4), we deduce that for all k ∈ N; On the other hand, using the relation (5.5) we have (5.7)However, applying the theorem of Kolmogoroff to F [11,17], we obtain (5.8) Combining the relations (5.6), (5.7) and (5.8) we obtain • We obtain the result by the same way and using the function where Theorem 5.4.Let p ∈ [1, +∞] and let f be a function satisfying the hypothesis of proposition 5.3.

If there exist
converge respectively to σ f,0 and σ f,1 .
We put; ϕ n (r, x) = n 2 ϕ(nr, nx); n ∈ N * and (5.13) By applying lemma 8 of [17] and using the hypothesis, we deduce that for all n ∈ N * ; the function F n is infinitely differentiable on R 2 and for all k ∈ N; we have By Hölder's inequality, we get where p is the conjugate exponent of p, then, (5.15) >From the relation (5.13), we deduce that the function F n can be written in the form where * is the usual convolution product in R 2 .So, Using the case p = +∞ and the relation (5.15), we deduce that ∀n ∈ N * ; σ Fn,0 σ 0 σ f,0 . ( Consequently, lim inf We assume that r 0 (the same proof holds if r < 0).Let ε > 0 such that a < r − 3ε.There exists a subsequence (5.17) Now, since the sequence (ϕ n ) n is an approximate identity and using the relation (5.13), we deduce that . However, by (5.17) for all n ∈ N; , ψ >= 0 and by (5.18) Using, the relation (5.16), we deduce that p,m 2 = σ f,0 .By the same way, we prove that 2. Suppose that there exists • The case p = +∞.
Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 ; we have This shows that the function f is entire on C 2 , slowly increasing of exponential type and by Paley-Wiener theorem's for the distributions, we deduce that In particular, σ f,0 + σ f,1 is finite and from the first assumption of this theorem, the sequences converge respectively to σ f,0 and σ f,1 .
• The case p ∈ [1, +∞[.Let F n n be the sequence defined by By the relation (5.14); for all (k 1 , k 2 ) ∈ N 2 ; ∂ ∂r . >From the case p = +∞; we deduce that for all n ∈ N, the function F n is entire on C 2 , and for all (z 1 , z 2 ) ∈ C 2 ; . This achieves the proof.
• Conversely, suppose that there exists (M 1 , M 2 ) ∈ (R * + ) 2 such that Again, From the second assertion of theorem 5.4, we deduce that the distribution Λ −1 2 (T f ) has a bounded support.Since, the mapping Λ 2 is a topological isomorphism from S * (R 2 ) onto itself, then Λ −1 2 (f ) lies in D * (R 2 ).Now, from the relation and by lemma 5.5, it follows that F −1 α (f ) belongs to D * (R 2 ).
Also; the Fourier-Bessel transform F α is a topological isomorphism from S * (R 2 ) onto itself.Then, from the relation (2.6), we deduce that the mapping B defined by the relation (2.7) is an isomorphism from S * (R 2 ) onto S * (Γ).