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Inversion Formulas for Riemann-Liouville Transform and Its Dual Associated with Singular Partial Differential Operators

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Inversion Formulas for Riemann-Liouville Transform and Its Dual Associated with Singular Partial Differential Operators INVERSION FORMULAS FOR RIEMANN-LIOUVILLE TRANSFORM AND ITS DUAL ASSOCIATED WITH SINGULAR PARTIAL DIFFERENTIAL OPERATORS C. BACCAR, N. B. HAMADI, AND L. T. RACHDI Received 21 May 2005; Revised 27 September 2005; Accepted 20 October 2005 t We define Riemann-Liouville transform ᏾α and its dual ᏾α associated with two singu- lar partial differential operators. We establish some results of harmonic analysis for the Fourier transform connected with ᏾α. Next, we prove inversion formulas for the opera- t t tors ᏾α, ᏾α and a Plancherel theorem for ᏾α. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction The mean operator is defined for a continuous function f on R2, even with respect to the first variable by 1 2π R0( f )(r,x) = f (r sinθ,x + r cosθ)dθ, (1.1) 2π 0 which means that R0( f )(r,x)isthemeanvalueof f on the circle centered at (0,x)and t radius r. The dual of the mean operator R0 is defined by t 1 2 2 R0( f )(r,x) = f r +(x − y) , y dy. (1.2) π R t ThemeanoperatorR0 and its dual R0 play an important role and have many applica- tions, for example, in image processing of the so-called synthetic aperture radar (SAR) data [11, 12] or in the linearized inverse scattering problem in acoustics [6]. Our purpose in this work is to define and study integral transforms which general- t ize the operators R0 and R0. More precisely, we consider the following singular partial differential operators: ∂ Δ = , 1 ∂x (1.3) ∂2 2α +1 ∂ ∂2 Δ = + − ,(r,x) ∈]0,+∞[×R, α 0. 2 ∂r2 r ∂r ∂x2 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 86238, Pages 1–26 DOI 10.1155/IJMMS/2006/86238 2 Inversion formulas for Riemann-Liouville transform 2 We associate to Δ1 and Δ2 the Riemann-Liouville transform Rα,definedonᏯ∗(R )(the space of continuous functions on R2, even with respect to the first variable) by ⎧ ⎪ α 1 ⎪ − 2 ⎪ f rs 1 t ,x + rt ⎪ π −1 ⎨⎪ − − R = × − 2 α 1/2 − 2 α 1 α( f )(r,x) ⎪ 1 t 1 s dtds,ifα>0, (1.4) ⎪ ⎪ ⎪ 1 ⎪ 1 dt ⎩ f r 1 − t2,x + rt √ ,ifα = 0. π −1 1 − t2 t 2 The dual operator Rα is defined on the space ᏿∗(R ) (the space of infinitely differen- tiable functions on R2, rapidly decreasing together with all their derivatives, even with respect to the first variable) by ⎧ √ ⎪ +∞ u2−r2 ⎪ 2α 2 2 2 α−1 ⎨⎪ √ f (u,x + v) u − v − r ududv,ifα>0, π − 2− 2 tR ( )( , ) = r u r α f r x ⎪ ⎪ 1 ⎩ f r2 +(x − y)2, y dy,ifα = 0. π R (1.5) For more general fractional integrals and fractional differential equations, we can see the works of Debnath [3, 4] and Debnath with Bhatta [5]. t We establish for the operators Rα and Rα the same results given by Helgason, Ludwig, and Solmon for the classical Radon transform on R2 [10, 14, 17]andwefindtheresults given in [15] for the spherical mean operator. Especially (i) we give some harmonic analysis results related to the Fourier transform associ- ated with the Riemann-Liouville transform Rα; t (ii) we define and characterize some spaces of the functions on which Rα and Rα are isomorphisms; t (iii) we give the following inversion formulas for Rα and Rα: = R 1tR = 1tR R f αKα α( f ), f Kα α α( f ), (1.6) = tR 2R = 2 R tR f α Kα α( f ), f Kα α α( f ), 1 2 ff where Kα and Kα are integro-di erential operators; t (iv) we establish a Plancherel theorem for Rα; t (v) we show that Rα and Rα are transmutation operators. This paper is organized as follows. In Section 2, we show that for (μ,λ) ∈ C2,thedif- ferential system Δ1u(r,x) =−iλu(r,x), Δ ( , ) =− 2 ( , ), 2u r x μ u r x (1.7) ∂u u(0,0) = 1, (0,x) = 0, ∀x ∈ R, ∂r C. Baccar et al. 3 admits a unique solution ϕμ,λ given by 2 2 ϕμ,λ(r,x) = jα r μ + λ exp(−iλx), (1.8) where jα is the modified Bessel function defined by J (s) j (s) = 2αΓ(α +1) α , (1.9) α sα and Jα is the Bessel function of first kind and index α. Next, we prove a Mehler integral representation of ϕμ,λ and give some properties of Rα. In Section 3, we define the Fourier transform Fα connected with Rα, and we establish some harmonic analysis results (inversion formula, Plancherel theorem, Paley-Wiener t theorem) which lead to new properties of the operator Rα and its dual Rα. 2 t In Section 4, we characterize some subspaces of ᏿∗(R ) on which Rα and Rα are 1 isomorphisms, and we prove the inversion formulas cited below where the operators Kα 2 and Kα are given in terms of Fourier transforms. Next, we introduce fractional powers of the Bessel operator, ∂2 2α +1 ∂ = + , (1.10) α ∂r2 r ∂r and the Laplacian operator, ∂2 ∂2 Δ = + , (1.11) ∂r2 ∂x2 1 2 that we use to simplify Kα and Kα . t Finally, we prove the following Plancherel theorem for Rα: +∞ +∞ 2 2α+1 = 3 tR 2 f (r,x) r dr dx Kα α( f ) (r,x) dr dx, (1.12) R 0 R 0 3 ff where Kα is an integro-di erential operator. t In Section 5, we show that Rα and Rα satisfy the following relations of permutation: ∂2 tR Δ f = tR ( f ), tR Δ f = Δ tR ( f ), α 2 ∂r2 α α 1 1 α (1.13) ∂2 f Δ R ( f ) = R , Δ R ( f ) = R Δ f . 2 α α ∂r2 1 α α 1 2. Riemann-Liouville transform and its dual associated with the operators Δ1 and Δ2 t In this section, we define the Riemann-Liouville transform Rα and its dual Rα,andwe give some properties of these operators. It is well known [21]thatforeveryλ ∈ C,the system 2 αv(r) =−λ v(r); (2.1) v(0) = 1; v(0) = 0, 4 Inversion formulas for Riemann-Liouville transform where α is the Bessel operator, admits a unique solution, that is, the modified Bessel function r → jα(rλ). Thus, for all (μ,λ) ∈ C × C, the system Δ1u(r,x) =−iλu(r,x), Δ ( , ) =− 2 ( , ), 2u r x μ u r x (2.2) ∂u u(0,0) = 1, (0,x) = 0, ∀x ∈ R, ∂r admits the unique solution given by 2 2 ϕμ,λ(r,x) = jα r μ + λ exp(−iλx). (2.3) The modified Bessel function jα has the Mehler integral representation, (we refer to [13, 21]) Γ 1 (α +1) 2 α−1/2 jα(s) = √ 1 − t exp(−ist)dt. (2.4) πΓ(α +1/2) −1 In particular, ∀ ∈ N ∀ ∈ R (k) k , s , jα (s) 1. (2.5) On the other hand, sup jα(rλ) = 1iff λ ∈ R. (2.6) r∈R This involves that sup ϕμ,λ(r,x) = 1iff (μ,λ) ∈ Γ, (2.7) (r,x)∈R2 where Γ is the set defined by Γ = R2 ∪ (iμ,λ); (μ,λ) ∈ R2, |μ| |λ| . (2.8) Proposition 2.1. The eigenfunction ϕμ,λ given by (2.3) has the following Mehler integral representation: ⎧ ⎪ 1 ⎪ α 2 α−1/2 2 α−1 ⎪ cos μrs 1 − t2 exp − iλ(x + rt) 1 − t 1 − s dtds, ⎪ − ⎨⎪ π 1 ( , ) = if α>0, ϕμ,λ r x ⎪ ⎪ ⎪ 1 ⎪ 1 dt ⎩ cos rμ 1 − t2 exp − iλ(x + rt) √ , if α = 0. π −1 1 − t2 (2.9) C. Baccar et al. 5 Proof. From the following expansion of the function jα: ∞ J (s) + (−1)k s 2k j (s) = 2αΓ(α +1) α = Γ(α +1) , (2.10) α α !Γ( + +1) 2 s k=0 k α k we deduce that ∞ + (−1)k rμ 2k j r μ2 + λ2 = Γ(α +1) j (rλ), (2.11) α !Γ( + +1) 2 α+k k=0 k k α and from the equality (2.4), we obtain Γ 1 − 2 2 (α +1) 2 2 α 1/2 jα r μ + λ = √ jα−1/2 rμ 1 − t exp(−irλt) 1 − t dt. π Γ(α +1/2) −1 (2.12) Then, the results follow by using again the relation (2.4)forα>0, and from the fact that j−1/2(s) = coss,forα = 0. (2.13) Definition 2.2. The Riemann-Liouville transform Rα associated with the operators Δ1 2 2 and Δ2 is the mapping defined on Ꮿ∗(R ) by the following. For all (r,x) ∈ R , ⎧ ⎪ α 1 ⎪ − 2 ⎪ f rs 1 t ,x + rt ⎪ π −1 ⎨⎪ − − R = × − 2 α 1/2 − 2 α 1 α( f )(r,x) ⎪ 1 t 1 s dtds,ifα>0, (2.14) ⎪ ⎪ ⎪ 1 ⎪ 1 dt ⎩ f r 1 − t2,x + rt √ ,ifα = 0. π −1 1 − t2 Remark 2.3. (i) From Proposition 2.1 and Definition 2.2,wehave ϕμ,λ(r,x) = Rα cos(μ.)exp(−iλ.) (r,x). (2.15) (ii) We can easily see, as in [2], that the transform Rα is continuous and injective from 2 2 Ᏹ∗(R ) (the space of infinitely differentiable functions on R , even with respect to the first variable) into itself. 6 Inversion formulas for Riemann-Liouville transform 2 2 Lemma 2.4. For f ∈ Ꮿ∗(R ), f bounded, and g ∈ ᏿∗(R ), +∞ +∞ 2α+1 t Rα( f )(r,x)g(r,x)r dr dx = f (r,x) Rα(g)(r,x)dr dx, (2.16) R 0 R 0 t where Rα is the dual transform defined by ⎧ √ ⎪ +∞ u2−r2 ⎪ 2α 2 2 2 α−1 ⎨⎪ √ g(u,x + v) u − v − r ududv, if α>0, π − 2− 2 tR ( )( , ) = r u r α g r x ⎪ ⎪ 1 ⎩ g r2 +(x − y)2, y dy, if α = 0.
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