aa r X i v : . [ m a t h . F A ] N ov SPHERICAL MEANS IN ODD DIMENSIONSAND EPD EQUATIONS
BORIS RUBIN
Abstract.
The paper contains a simple proof of the Finch-Patch-Rakesh inversion formula for the spherical mean Radon transformin odd dimensions. This transform arises in thermoacoustic to-mography. Applications are given to the Cauchy problem for theEuler-Poisson-Darboux equation with initial data on the cylindri-cal surface. The argument relies on the idea of analytic continua-tion and known properties of Erd´elyi-Kober fractional integrals. Introduction
We consider the spherical mean Radon transform(1.1) (
M f )( θ, t ) = 1 σ n − Z S n − f ( θ − tσ ) dσ,θ ∈ S n − , t ∈ R + = (0 , ∞ ) , where f is a smooth function supported inside the unit ball B = { x ∈ R n : | x | < } , S n − is the unit sphere in R n with the area σ n − = 2 π n/ / Γ( n/ dσ denotes integration against the usualLebesgue measure on S n − . The inversion problem for this transformhas attracted considerable attention in the last decade in view of newdevelopments in thermoacoustic tomography; see [AKK, AKQ, FPR,FHR, KK, Ku, PS1, PS2] and references therein. Explicit inversionformulas for M f in the “closed form” are of particular interest. For n odd, such formulas were obtained by Finch, Patch, and Rakesh in[FPR]. The corresponding formulas for n even were obtained by Finch,Haltmeier, and Rakesh in [FHR]. An interesting inversion formula ofdifferent type, that covers both odd and even cases, was suggested by Mathematics Subject Classification.
Primary 44A12; Secondary 92C55,65R32.
Key words and phrases.
The spherical mean Radon transform, The Euler-Poisson-Darboux equation, Erd´elyi-Kober fractional integrals.The research was supported in part by the NSF grant DMS-0556157 and theLouisiana EPSCoR program, sponsored by NSF and the Board of Regents SupportFund.
Kunyansky [Ku]; see also a survey paper [KK], where diverse inversionalgorithms and related mathematical problems are discussed.In spite of their elegance and ingenuity, most of explicit inversion for-mulas for
M f are still mysterious, and the basic ideas behind them arenot completely understood. We observe, for example, that the deriva-tion for n = 3 in [FPR] relies on implementation of delta functionsand is not completely rigorous. The dimensions n = 5 , , . . . have beentreated using a pretty complicated reduction to the 3-dimensional case.The resulting inversion formula can be written in the form(1.2) f ( x ) = c n ∆ Z S n − [ D n − t n − ϕ θ ] (cid:12)(cid:12)(cid:12) t = | x − θ | dθ, where ϕ θ ( t ) = ( M f )( θ, t ),∆ = n X k =1 ∂ ∂x k , c n = ( − ( n − / π n/ − Γ( n/ , D = 12 t ddt ;cf. [FPR, Theorem 3]. Other formulas in that theorem can be obtainedin the framework of the same method; see Remark 3.1.In the present article (Section 3) we suggest a simple and rigorousproof of (1.2) that handles all odd n simultaneously, so that no reduc-tion to n = 3 is needed. The idea is to treat the spherical mean asa member of a certain analytic family of operators associated to theEuler-Poisson-Darboux equation(1.3) (cid:3) α u ≡ ∆ x u − u tt − n + 2 α − t u t = 0and invoke known facts from fractional calculus [SKM] about Erd´elyi-Kober operators. Section 4 deals with applications. Here we recon-struct the solution u ( x, t ) of the equation (cid:3) α u = λ u , λ ≥
0, from theCauchy data on the cylinder S n − × R + . Section 5 contains commentsand open questions.No claim about the originality of the results presented in this articleis made, but it is felt that the elementary use of operators of fractionalintegration to obtain them might appeal to the applied mathematician.If the reader is not interested in applications in Section 4, he may skipsubsections 2.2 and 2.4. These are not needed for the basic Section 3. Acknowledgements.
I am grateful to professors Mark Agranovsky,Peter Kuchment, and Larry Zalcman for very pleasant discussions andhospitality during my short visits to Bar Ilan and Texas A&M Univer-sities.
PHERICAL MEANS 3 Preliminaries
Erd´elyi-Kober fractional integrals.
We remind some knownfacts; see, e.g., [SKM, Sec. 18.1]. For
Re α > η ≥ − /
2, theErd´elyi-Kober fractional integral of a function ϕ on R + is defined by(2.1) ( I αη ϕ )( t ) = 2 t − α + η ) Γ( α ) Z t ( t − r ) α − r η +1 ϕ ( r ) dr, t > . For our further needs, it suffices to assume that ϕ is infinitely smoothand supported away from the origin. Then I αη ϕ extends as an entirefunction of α and η , so that(2.2) I η ϕ = ϕ, (2.3) ( I αη ϕ )( t ) = t − α + η ) D m t α + m + η ) ( I α + mη ϕ )( t ) , D = 12 t ddt , (2.4) ( I − mη ϕ )( t ) = t − η − m ) D m t η ϕ ( t ) , where m is a nonnegative integer. The property(2.5) D m = t − (cid:18) ddt t (cid:19) m t allows us to write (2.3) and (2.4) in a different equivalent form. Thecomposition formula and the inverse operator are as follows:(2.6) I βη + α I αη ϕ = I α + βη ϕ, ( I αη ) − ϕ = I − αη + α ϕ. The generalized Erd´elyi-Kober fractional integrals.
Let J ν and I ν be the Bessel function and the modified Bessel function of thefirst kind, respectively [E]. The generalized Erd´elyi-Kober operatorsare defined by(2.7) J αη,λ ϕ ( t ) = t − α + η ) J αλ t η ϕ ( t ) , I αη,λ ϕ ( t ) = t − α + η ) I αλ t η ϕ ( t ) , where λ ≥ J αλ ϕ ( t ) = 2 α λ − α Z t ( t − r ) ( α − / J α − ( λ √ t − r ) ϕ ( r ) r dr, (2.8) I αλ ϕ ( t ) = 2 α λ − α Z t ( t − r ) ( α − / I α − ( λ √ t − r ) ϕ ( r ) r dr ;(2.9)see [L1, L2], [SKM, Sec. 37.2]. As above, we assume ϕ to be infinitelysmooth and supported away from the origin. Integrals (2.8) and (2.9) B. RUBIN are absolutely convergent if
Re α > α by the formulas J αλ ϕ = D m J α + mλ ϕ = J α + m D m ϕ, (2.10) I αλ ϕ = D m I α + mλ ϕ = I α + mλ D m ϕ, (2.11) m ∈ N . These follow from the well-known relation (cid:18) τ ddτ (cid:19) m [ τ ν J ν ( τ )] = τ ν − m J ν − m ( τ )(similarly for I ν ). Clearly, J αη, = I αη, = I αη . If Re α >
Re β > I βη + α,λ J αη,λ ϕ = I α + βη ϕ. The latter extends by analyticity to all complex α and β and yields theinversion formula(2.13) ( J αη,λ ) − f = I − αη + α,λ f. By (2.11) and (2.7), this can also be written as(2.14) ( J αη,λ ) − f = t − η D m I m − αλ t η + α ) f = t − η D m t η + m ) I m − αη + α,λ f. The Euler-Poisson-Darboux equation.
Consider the Cauchyproblem for the Euler-Poisson-Darboux equation (1.3):(2.15) (cid:3) α u = 0 , u ( x,
0) = f ( x ) , u t ( x,
0) = 0 , where f belongs to the Schwartz space S ( R n ); see [B1] for details. If α ≥ (1 − n ) /
2, then (2.15) has a unique solution u ( x, t ) = ( M αt f )( x )where the operator M αt is defined in the Fourier terms by[ M αt f ] ∧ ( y ) = m α ( t | y | ) ˆ f ( y ) ,m α ( ρ ) = Γ( α + n/
2) ( ρ/ − α − n/ J n/ α − ( ρ ) . The operator M αt extends meromorphically to all complex α with thepoles − n/ , − n/ − , . . . . For Re α >
0, it is an integral operator(2.16) ( M αt f )( x ) = Γ( α + n/ π n/ Γ( α ) Z | y | < (1 − | y | ) α − f ( x − ty ) dy. In the case α = 0, M αt f is the spherical mean(2.17) ( M t f )( x ) = 1 σ n − Z S n − f ( x − tσ ) dσ ; PHERICAL MEANS 5 cf. (1.1). Passing to polar coordinates, one can obviously represent M αt f as an Erd´elyi-Kober integral of the spherical mean(2.18) ( M αt f )( x ) = Γ( α + n/ n/
2) ( I αη ϕ x )( t ) , ϕ x ( t ) = ( M t f )( x ) , with η = n/ − The generalized Euler-Poisson-Darboux equation.
Considerthe more general Cauchy problem(2.19) (cid:3) α u = λ u, u ( x,
0) = f ( x ) , u t ( x,
0) = 0 , where f is a Schwartz function and λ ≥
0. If α ≥ (1 − n ) /
2, then (2.19)has a unique solution u ( x, t ) = ( M αt,λ f )( x ), where the operator M αt,λ isdefined as analytic continuation of the integral( M αt,λ f )( x ) = (2 /λ ) α − Γ( α + n/ π n/ t − n − α (2.20) × Z | y | 2) ( J αη,λ ϕ x )( t ) , ϕ x ( t ) = ( M t f )( x ) ,η = n/ − 1, where J αη,λ is the generalized Erd´elyi-Kober operator (2.7).2.5. More preparations. We restrict M αt f to x = θ ∈ S n − and set(2.22) ( N α f )( θ, t ) = t n +2 α − ( M αt f )( θ ) . In particular, for Re α > 0, owing to (2.16), we have(2.23) ( N α f )( θ, t ) = Γ( α + n/ π n/ Γ( α ) Z R n f ( y ) ( t − | y − θ | ) α − dy where ( ... ) α − has a standard meaning, namely, ( a − b ) α − = ( a − b ) α − if a > b and 0 otherwise. Given a function F on the cylinder S n − × R + ,we denote(2.24) ( P F )( x ) = 1 σ n − Z S n − F ( θ, | x − θ | ) dθ, x ∈ R n , which is a modification of the back-projection (or dual) operator; cf.[H, N], where these notions are used for the classical Radon transform.We also invoke Riemann-Liouville integrals [SKM](2.25) ( I α − u )( s ) = 1Γ( α ) Z s − ( s − t ) α − u ( t ) dt, u ∈ C ∞ [ − , . B. RUBIN The integral (2.25) is absolutely convergent when Re α > α ∈ C , so that(2.26) ( I − m − u )( s ) = ( d/ds ) m u ( s ) , m = 0 , , , . . . . Lemma 2.1. Let B = { x ∈ R n : | x | < } . For Re α > and anyintegrable function f supported in B we have (2.27) ( P N α f )( x ) = c α Z B f ( y ) | x − y | − α ( I α − u )( h ) dy, x ∈ B, where c α = Γ( n/ 2) Γ( α + n/ − α π ( n +1) / Γ(( n − / , h = | x | −| y | | x − y | , u ( t ) = (1 − t ) ( n − / . Proof. By (2.23) and (2.24), changing the order of integration, we get( P N α f )( x ) = Γ( α + n/ σ n − π n/ Γ( α ) Z S n − dθ Z B f ( y )( | x − θ | −| y − θ | ) α − dy = Γ( α + n/ σ n − π n/ Γ( α ) Z B f ( y ) k α ( x, y ) dy, where k α ( x, y ) = Z S n − ( | x | − | y | − θ · ( x − y )) α − dθ = σ n − (2 | x − y | ) − α Z h − ( h − t ) α − (1 − t ) ( n − / dt. This gives the result. (cid:3) Inversion of the Spherical Mean for n Odd Let C ∞ ( B ) be the space of C ∞ -functions on R n supported in B ; f ∈ C ∞ ( B ). By (2.22), (2.18), and (2.4), analytic continuation of N α f at α = 3 − n has the form( N − n f )( θ, t ) = Γ(3 − n/ n/ D n − t n − ϕ θ ( t ) , ϕ θ ( t ) = ( M f )( θ, t ) . If n = 2 k + 3 , k = 0 , , . . . , then (2.26) yields( I − n − u )( h ) = ( d/dh ) k (1 − h ) k = ( − k k ! = ( − ( n − / Γ( n − . Hence, analytic continuation of (2.27) at α = 3 − n is(3.1) P [ D n − t n − ϕ θ ]( x ) = c ( I f )( x ) , c = 2( − ( n − / Γ ( n/ /π, where x ∈ B and(3.2) ( I f )( x ) = Γ( n/ − π n/ Z B f ( y ) dy | x − y | n − PHERICAL MEANS 7 is the Riesz potential of order 2 [SKM]. The latter can be inverted bythe Laplacian, and simple calculations yield(3.3) f ( x ) = c n ∆ Z S n − [ D n − t n − ϕ θ ] (cid:12)(cid:12)(cid:12) t = | x − θ | dθ, c n = ( − ( n − / π n/ − Γ( n/ . Remark . Formula (3.3) coincides (up to notation) with the thirdformula in [FPR, Theorem 3]. Other formulas in that theorem can besimilarly obtained from (2.27) if the latter is applied to ∆ f insteadof f . Here we take into account that I ∆ f = − f , because supp f isseparated from the boundary of B .4. An inverse problem for the EPD equation Let α ≥ (1 − n ) / , λ ≥ 0. Suppose we know the trace u ( θ, t ) of thesolution of the Cauchy problem(4.1) (cid:3) α u = λ u, u ( x, 0) = f ( x ) , u t ( x, 0) = 0 , on the cylinder { ( θ, t ) : θ ∈ S n − , t ∈ R + } and want to reconstruct theinitial function f in the space C ∞ ( B ). This can be easily done using(3.1) and the Erd´elyi-Kober operators. Indeed, by (2.21), u ( θ, t ) = ( M αt,λ f )( θ ) = Γ( α + n/ n/ 2) ( J αη,λ ϕ θ )( t ) , where ϕ θ ( t ) = ( M t f )( θ ), η = n/ − 1. Then by (2.13), for u θ ( t ) ≡ u ( θ, t )we have ϕ θ = Γ( n/ α + n/ 2) ( J αη,λ ) − u θ = Γ( n/ α + n/ I − αη + α,λ u θ . Now, (3.1) yieldsΓ( n/ α + n/ P [ D n − t t n − I − αη + α,λ u θ ] = c I f, c = 2( − ( n − / Γ ( n/ /π, and therefore,(4.2) f = ( − ( n − / π n/ 2) Γ( α + n/ 2) ∆ P [ D n − t t n − I − αη + α,λ u θ ] . The operator I − αη + α,λ can be replaced by any expression from (2.14).It remains to note that once f is known, the solution u ( x, t ) of theequation (cid:3) α u = λ u can be reconstructed from the trace u ( θ, t ) by theformula u ( x, t ) = ( M αt,λ f )( x ); see (2.20). B. RUBIN Comments It is a challenging open problem to appropriately adjust ourmethod to the case when n is even and give alternative proof of thecorresponding inversion formulas from [FHR] and [Ku]. Formula (3.1) provokes the following Conjecture. Let n ≥ be odd. A function ϕ θ ( t ) ≡ ϕ ( θ, t ) belongs tothe range of the operator f → ( M f )( θ, t ) , f ∈ C ∞ ( B ) , if and only if P [ D n − t t n − ϕ θ ] belongs to the range I [ C ∞ ( B )] of the potential (3.2). The “only if” part follows immediately from (3.1). The “if” partrequires studying injectivity of the back-projection operator P , whichis of independent interest; cf. [R], where injectivity and inversion ofthe dual Radon transform is studied in the general context of affineGrassmann manifolds. Various descriptions of the range of the spher-ical mean transform and many related results can be found in [AKQ,AK, FR]. It is worth noting that for n = 3, the inversion formula for M f becomes elementary if f is a radial function, i.e. f ( x ) ≡ f ( | x | ). Lemma 5.1. If f ∈ L loc ( R n ) , f ( x ) ≡ f ( | x | ) , then ( M f )( θ, t ) ≡ F ( t ) ,where (5.1) F ( t ) = 2 n − Γ( n/ π / Γ(( n − / Z t | − t | f ( r ) [ a ( r, t )] n − r dra ( r, t ) = [ r − (1 − t ) ] / [(1 + t ) − r ] / / being the area of the trianglewith sides , t, r .Proof. ( M f )( θ, t ) = 1 σ n − Z S n − f ( | θ − tσ | ) dσ = σ n − σ n − Z − f ( √ t − ts ) (1 − s ) ( n − / ds and (5.1) follows. (cid:3) Corollary 5.2. If n = 3 and f is supported in B , then (5.1) yields (5.2) ( a ) F ( t ) = 12 t Z − t f ( r ) r dr if 0 < t ≤ , and (5.3) ( b ) F ( t ) = 12 t Z t − f ( r ) r dr if 1 ≤ t < . PHERICAL MEANS 9 In the case (a), (5.4) f ( r ) = 2 r (cid:20) ddt ( tF ( t )) (cid:21) t =1 − r . In the case (b), (5.5) f ( r ) = − r (cid:20) ddt ( tF ( t )) (cid:21) t =1+ r . Remark . We note that in (5.4) and (5.5) it is not necessary toknow F ( t ) for all t ∈ (0 , 2) as in (3.3). It suffices to know it onlyfor t ∈ (0 , 1) or t ∈ (1 , M f )( θ, t ) in the general case, when ( M f )( θ, t )is known only for ( θ, t ) ∈ S n − × (0 , 1) or ( θ, t ) ∈ S n − × (1 , References [AKQ] Agranovsky M., Kuchment P., Quinto E. T., Range descriptions for thespherical mean Radon transform , J. Funct. 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A., Sushko, D. V., Image restoration in optical-acoustic to-mography , (Russian) Problemy Peredachi Informatsii 40 (2004), no. 3,81–107; translation in Probl. Inf. Transm. 40 (2004), no. 3, 254–278.[PS2] , A parametrix for a problem of optical-acoustic tomography , (Rus-sian) Dokl. Akad. Nauk 382 (2002), no. 2, 162–164.[PBM] Prudnikov A. P., Brychkov Y. A., Marichev O. I., Integrals and series:special functions, Gordon and Breach Sci. Publ., New York - London,1986.[R] Rubin B., Radon transforms on affine Grassmannians , Trans. Amer.Math. Soc. 356 (2004), 5045–5070.[SKM] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Deriva-tives, Theory and Applications, Gordon and Breach Science Publishers,1993. Department of Mathematics, Louisiana State University, Baton Rouge,LA, 70803 USA E-mail address ::