In 1917 Radon [19] introduced a transform f Rf for f a suitable real-valued function on R 2 by. (Rf)(L) = f

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1 J Korean Math Soc 40 (2003), No 4, pp COMPLEX ANALYSIS AND THE FUNK TRANSFORM T N Bailey, M G Eastwood, A R Gover, and L J Mason Abstract The Funk transform is defined by integrating a function on the two-sphere over its great circles We use complex analysis to invert this transform Introduction In 97 Radon [9] introduced a transform f Rf for f a suitable real-valued function on R 2 by (Rf)(L) = f for L a straight line in R 2 Thus, Rf is a function defined on the set of straight lines in R 2 See, for example, [2] for a review There are many variations on this theme in real integral geometry One such variation was already introduced in 93 by Funk [0] Its definition is just like the Radon transform except that R 2 is replaced by the round sphere S 2 and great circles play the role of straight lines Funk proved that a smooth function f on S 2 lies in the kernel of this transformation if and only if f is odd (see [3] for a modern treatment and a discussion of Funk s motivation in constructing Zoll metrics on the sphere) Our aim, in this article, is to show how the Funk transform F acting on smooth functions may be inverted using complex analysis More precisely, we shall show that an inverse transform T arises as a result of solving a certain -problem on CP 2 \ RP 2 (where is the Cauchy- Riemann operator) We obtain the following explicit formula Using Received October, Mathematics Subject Classification: Primary 53C65; Secondary 32W05, 44A2 Key words and phrases: Funk, Penrose, Radon, Zoll This research was supported by the Australian Research Council and the Isaac Newton Institute for Mathematical Sciences L

2 578 T N Bailey, M G Eastwood, A R Gover, and L J Mason standard coördinates (x, y, z) on R 3, () (T φ) 0 = x 2 φ S 2 y x 2, 0 where φ is extended off S 2 R 3 as a function homogeneous of degree in the sense that φ(λu) = λ φ(u), for λ > 0 There are many similar formulae in the literature (see, for example, []) and we are not claiming any original contribution here From our point of view, the pleasing and original aspect of this particular formula is its unexpected derivation from complex analysis The fact that F is invertible on smooth functions (without recourse to a particular inversion formula) is also well-known It may be derived from the theory of spherical harmonics (see [3, Appendix A]) The motivation for this article is a link between the Radon transform in some generality and a transform in complex geometry introduced, independently, around 967 by Andreotti and Norguet [], Penrose [7], and Schmid [20] (A version of the transform was found in 904 by Bateman [2]) See, for example, [8] for a review of this Penrose transform Our approach to the Funk transform has its roots in understanding this link [5] This aspect, however, will be entirely suppressed in this article Given this motivation, it almost goes without saying that this simple case generalises considerably Indeed, our investigations originally concentrated on the X-ray transform (see, for example, [4]), the real anaue of the classical Penrose transform Much of this article applies mutatis mutandis, as discussed at the end The X-ray transform has also been investigated from this point of view by Sparling [2] and Woodhouse [22] A general link between real and complex integral geometry is presented in [3, 4, 9] A link in terms of D-modules is also available [6, 7] Rather than embark on any general discussion, however, we prefer to concentrate on what we believe to be the simplest non-trivial example This article is organized as follows In we define the Funk transform and establish some of its elementary properties In 2 we discuss the transform T and prove that it provides a right inverse: F T = Id In 3 we obtain () by solving a -problem on CP 2 \ RP 2 As a result of this interpretation, we prove that T is also a left inverse: T F = Id We shall use the standard index notation of differential geometry Greek indices will run over the range, 2, 3, so that X α denotes a vector in R 3 or C 3 Naïvely, one may regard X α as listing the components of X, but it is preferable to regard α as an abstract index in the sense of

3 Complex analysis and the Funk transform 579 Penrose [8] The Einstein summation convention X α U α := 3 X α U α α= is then interpreted as the natural pairing between vectors and covectors We would like to thank Robin Graham for helpful observations Definitions and preliminaries By smooth we shall always mean infinitely differentiable Suppose f(x) is a smooth function on R 3 \ {0} homogeneous of degree 2, in the sense that f(λx) = λ 2 f(x), for λ > 0 For linearly independent ξ, η R 3, consider the function (2) φ(ξ, η) = f(uξ + vη) (u dv v du), 2π where the integral is taken around any smooth closed curve in the (u, v)- plane of winding number one about the origin The homogeneity of f implies that the form f(uξ+vη) (u dv v du) is closed, so φ is well-defined independent of choice of curve A change of variables shows that φ(aξ + bη, cξ + dη) = φ(ξ, η) ad bc so φ is a well-defined function of ξ η homogeneous in the sense that φ(λξ η) = λ φ(ξ η), for λ R \ {0} Fix a volume form ɛ αβγ on R 3 and let U α = ɛ αβγ ξ β η γ function of U α and φ(µu α ) = µ φ(u α) Then φ is a In other words, φ is homogeneous of degree on (R 3 ) \ {0} and even On the other hand, the transform f φ clearly kills odd functions This is the Funk transform F : smooth even functions on R 3 \ {0}, homogeneous of degree 2 smooth even functions on R 3 \ {0}, homogeneous of degree

4 580 T N Bailey, M G Eastwood, A R Gover, and L J Mason Of course, a homogeneous function on R 3 is equivalent, by restriction, to a function on S 2, the unit sphere in R 3 This gives the more classical form of the Funk transform: {smooth even functions on S 2 } {smooth even functions on S 2 } This formulation is more geometric if ɛ αβγ is the standard volume form on R 3, the point U is on the unit sphere, and ξ and η are also unit vectors so that U, ξ, η forms an oriented orthonormal basis, then (2) may be rewritten as φ(u) = 2π f(cos θ ξ + sin θ η) dθ 2π 0 This is an integral over great circles: U Γ U φ(u) = 2π f ds, Γ U where ds is arclength In this classical form, the Funk transform is clearly SO(3)-invariant Our formulation in terms of homogeneous functions is manifestly invariant under a larger group, namely SL(3, R) This invariance may be expressed infinitesimally if P β α is trace-free, then ( P β α U β F(f) = F P β α X α ) X β f On the other hand, by Euler s relation for homogeneous functions, U α so, in general, F(f) = F(f) X α (Xα f) = 3f X α f = 3f + 2f = f Xα (3) U β F(f) = F ( ) X β (Xα f)

5 Complex analysis and the Funk transform 58 This suggests trying to define a transform G on smooth functions homogeneous of degree by ( ) g U β G(g) := F X β In other words, we are trying to define a smooth function ψ(u) by (4) U β ψ(u) = g (uξ + vη) (u dv v du), 2π Xβ where, as always, U β = ɛ βγδ ξ γ η δ To see that this is a bona fide definition, it suffices to check that the right hand side of this equation is annihilated by contraction with ξ β and η β For fixed ξ and η, the function g(uξ + vη) is homogenous in the (u, v)-coördinates of degree Using Euler s relation, it follows that d(ug) = g (u dv v du) = g ηβ (u dv v du) v Xβ Therefore, η β 2π g (uξ + vη) (u dv v du) = Xβ 2π d(ug) = 0, as required Contracting the right hand side of (4) with ξ β is similar Clearly, (4) characterises an odd function ψ and, if g is even, then ψ vanishes To summarise, (4) defines a transform G : smooth odd functions on R 3 \ {0}, homogeneous of degree smooth odd functions on R 3 \ {0}, homogeneous of degree 2 The infinitesimal invariance of F specified by (3) can now be separated into two equations: ( ) g (5) U β G(g) = F X β and G(X α f) = F(f) There is also a geometric interpretation of G: U Γ U ψ(u) = g 2π Γ U n ds, where g/ n is the indicated normal derivative to Γ U and ds is arclength

6 582 T N Bailey, M G Eastwood, A R Gover, and L J Mason 2 An inverse Suppose ρ(u) is a smooth function on R 3 \ {0} homogeneous of degree 3 Let d 2 U denote the differential form ɛ αβγ U α du β du γ on R 3 A calculation shows that ρ(u) d 2 U is a closed 2-form on R 3 \ {0} Let denote a smooth embedded two-sphere in R 3 \ {0} of degree one Then mass ρ := ρ(u) d 2 U is independent of choice of We shall suppose that this mass vanishes Let us further restrict to be everywhere transverse to the Euler field and consider the integral (6) h(x) = X α U α ρ(u) d 2 U, for X R 3 Since 0 t dt converges, Xα U α is an integrable function on and elementary estimates show that, in fact, h(x) is a continuous function of X R 3 \ {0} Also, h(x) is even and homogeneous of degree zero: h(λx) = = = λx α U α ρ(u) d 2 U X α U α ρ(u) d 2 U + λ ρ(u) d 2 U X α U α ρ(u) d 2 U = h(x) The definition of h(x) depends on However, if we consider (7) h(x) h(y ) = X α U α Y α U α ρ(u) d2 U, and notice that the integrand is Lie propagated by the Euler vector field, it follows that the right hand side does not depend on Thus, (6) or (7) defines h(x) up to an additive constant Lemma The function h(x) defined above is smooth on R 3 \ {0} Proof The formula (7) is manifestly SL(3, R)-invariant Infinitesimally, this invariance shows that h is differentiable and (8) X β h X γ (X) Y β h Y γ (Y ) = X α U α Y α U α (U γ ρ) d 2 U U β

7 Complex analysis and the Funk transform 583 (compare (3)) In particular, the derivatives of h are given by integrals of the same form as is h itself Iterating this conclusion, it follows that h is smooth Suppose ψ is a smooth function homogeneous of degree 2 on R 3 \{0} Then, d(ψɛ βγδ U β du γ ) = ψ ɛ βγδ U β du α du γ + ψɛ βγδ du β du γ = ψ U α ɛ βγδ du β du γ ψ d 2 U + ψɛ βγδ du β du γ 2 2 U δ = ψ d 2 U 2 U δ Therefore, by Stokes theorem, ψ/ U δ has zero mass and we may consider h δ (X) h δ (Y ) = X α U α ψ Y α U α d 2 U, U δ an integral of the form (7) (of course, h δ (X) is vector-valued but this does not affect our analysis and, in particular, Lemma is still valid) Equation (8) now reads X β hδ β hδ (X) Y Xγ Y γ (Y ) = X α U α Y α U α (U γ ψ ) d 2 U U β U δ Contracting over γ and δ yields (9) X β g(x) Y β g(y ) = X α U α ψ Y α U α d 2 U, U β where g(x) = 2 hγ / X γ It is clear that (9) defines g uniquely Indeed, if we choose an inner product on R 3, take X and Y to be orthonormal, and contract with X, then g(x) = X α U α S 2 Y α U α X ψ β d 2 U U β Thus, with standard coördinates on R 3, (0) g 0 = x ψ S 2 y x, 0 for example Lemma shows that g is smooth Also notice that g is an odd function and, if ψ is even, then g vanishes Let us write S for the

8 584 T N Bailey, M G Eastwood, A R Gover, and L J Mason transform ψ g Thus, S : smooth odd functions on R 3 \ {0}, homogeneous of degree 2 Invariance under SL(3, R) implies that () X α S(ψ) = S Xβ smooth odd functions on R 3 \ {0}, homogeneous of degree ( (U β ψ) Now suppose φ is a smooth even function homogeneous of degree on R 3 \ {0} Then, φ/ U γ is a smooth odd function, homogeneous of degree 2 and so we may consider S( φ/ U γ ) According to (), X α φ S( Xβ U γ ) = S ( ) ( U β φ U γ and, contracting over β and γ yields ( ) φ X α f(x) = S where f(x) = (S( φ/ U γ ))/ X γ Incorporating the definition of S more explicitly, we have shown that (2) X α X β f(x) Y α Y β f(y ) = X γ U γ 2 φ Y γ U γ d 2 U, U β defines a smooth function f homogeneous of degree 2 Writing T for the transform φ f, we have smooth even functions on smooth even functions on T : R 3 \ {0}, homogeneous of R 3 \ {0}, homogeneous of degree degree 2 )) The definition of T and the invariance of S described by () may be combined into the following two equations ( ) φ (3) X α T (φ) = S and T (U β ψ) = X β S(ψ) (compare (5)) A main result of this article is: Theorem The transforms F and T are mutually inverse as are the transforms G and S Suppose ψ is a smooth odd function on R 3 \ {0} homogeneous of degree 2 Then, using (3) and (5), ( ) (F T )(U β ψ) = F X β S(ψ) = U β (G S)(ψ)

9 Complex analysis and the Funk transform 585 Thus, if we can show that T is a right inverse for F, then the corresponding statement for G and S will follow Similarly, if f is a smooth even function on R 3 \ {0} homogeneous of degree 2, then ( ) (S G)(X α f) = S F(f) = X α (T F)(f) Thus, if we know that S is a left inverse for G, then the corresponding statement for T and F will follow We shall spend the rest of this section showing that T is indeed a right inverse for F That S is a left inverse for G will be a consequence of the link with complex analysis to be developed in the following section (see Proposition 2) Proposition The transform T is a right inverse to the Funk transform F Proof Suppose φ is a smooth even function homogeneous of degree on R 3 \ {0} Write f = T (φ) as in (2) Fix an inner product on R 3 so that we can view both and φ and f as smooth functions on the round sphere S 2 It suffices to show that (4) F T (φ) 0 0 = φ 0 0 Both F and T are SO(3)-invariant In particular, they are invariant under rotations about the z-axis By averaging, it follows that it suffices to check that (4) holds for φ which are invariant under such rotations Using standard coordinates on R 3, equation (2) gives f 0 = x 2 φ S 2 y x 2, 0 as in () In spherical polar coördinates, φ S 2 is a function of latitude only This allows one to compute explicitly and after integration by parts (twice), conclude that f 0 0 = φ 0 0 The details of this calculation will be omitted By rotational invariance, f has the same value at any point on the equator Since the Funk transform of f at a pole is obtained by averaging f over the equator, (4) follows

10 586 T N Bailey, M G Eastwood, A R Gover, and L J Mason 3 A link with complex analysis Define a mapping C 3 R 3 Z α = ξ α + iη α U α := ɛ αβγ ξ β η γ Notice that if Z is replaced by λz, then U is replaced by λλu Also U = 0 if and only if Z is a (possibly) complex multiple of a real 3-vector Thus, Z α U α defines a mapping π : CP 2 \ RP 2 S 2 where S 2 is the space of rays in R 3 \ {0} (and can be identified with the round sphere in the presence of an inner product) If ψ(u) is a smooth odd function homogeneous of degree 2 on R 3 \ {0}, then we may compose with the mapping above to obtain ψ(z), homogeneous of degree ( 2, 2) in the sense that ψ(λz) = λ 2 λ 2 ψ(z) Thus, ψ(z) may be regarded as a smooth section of a appropriate homogeneous line bundle on CP 2 \ RP 2 Now ω := 4 ψ(z)u β dz β = i 8 ψ(z)ɛ αβγz α Z β dz γ is a (0, )-form on C 3 homogeneous of degree It descends to give a (0, )-form on CP 2 \ RP 2 homogeneous of degree which we shall also denote by ω It is easy to check that this form is -closed and it is natural to ask whether it is -exact Even though H (CP 2 \ RP 2, O( )) 0, this is indeed the case: Lemma 2 The following integral (5) G(Z) = ψ(v ) 8π Z α d 2 V for [Z] CP 2 \ RP 2 V α defines a smooth function G homogeneous of degree satisfying G = ω Here, as in 2, we have chosen an arbitrary smoothly embedded two-sphere in R 3 \ {0} of degree one everywhere transverse to the Euler field Proof As in 2, a short calculation shows that the integrand is Lie propagated by the Euler vector field The integral is therefore independent of choice of The singularity in the integrand is like /(x + iy) in the complex plane and is therefore integrable

11 Complex analysis and the Funk transform 587 The fibres of π may be parameterised by choosing linearly independent ρ, σ R 3 and considering Z = ρ + ζσ for ζ in the upper half-plane The singularity in the integrand is like /(x+ζy) and its derivative with respect to ζ has a singularity like y/(x + ζy) 2 both of which are integrable Hence, we may differentiate under the integral sign to conclude that G is holomorphic on each fibre These fibres are hemispheres of standard CP cycles in CP 2 All other such CP s intersect RP 2 in a single point and may be parameterised as follows Choose ρ, σ, τ R 3 linearly independent and let Z = ρ + iσ ζτ for ζ C If G = ω, then G ζ = G α Z Z α ζ α G = τ Z α = 4 τ α U α ψ(u) Conversely, if this equation holds for all choices of ρ, σ, and τ, then G = ω Invariance under SL(3, R) implies that it suffices to check this equation when ρ, σ, and τ are the standard basis vectors in R 3 and is the round sphere S 2 In this case, writing ζ = p + iq gives U α = ɛ αβγ (ρ β pτ β )(σ γ qτ γ ) = pρ α + qσ α + τ α Thus, it suffices to show that if ψ is a smooth odd function on R 3 \ {0} homogeneous of degree 2, then (6) G(ζ) = ψ(x, y, z) 8π S 2 x + iy ζz is a smooth function satisfying (7) G ζ = 4ψ(p, q, ) We may parameterise a hemisphere of S 2 by setting x = s + s 2 + t 2 y = t + s 2 + t 2 z = + s 2 + t 2 The standard area element is ds dt/( + s 2 + t 2 ) 3/2 in these coördinates Since the integrand of (6) is even we may rewrite G(ζ) = R2 ψ( s, t, +s 2 +t 2 +s 2 +t 2 +s ) ds dt 2 +t2 (s + it ζ)( + s 2 + t 2 ) = ψ(s, t, )ds dt R 2 ζ (s + it)

12 588 T N Bailey, M G Eastwood, A R Gover, and L J Mason The fundamental solution of the Laplacian in R 2 is (ζζ) Since ζ (ζζ) = ζ and equation (7) follows immediately 2 ζ ζ = 4 Laplacian, Lemma 3 The equation G = ω uniquely determines G Proof Since RP 2 is a totally real submanifold of CP 2, any holomorphic function defined on Ω \ RP 2, for Ω an open subset of CP 2, extends uniquely to Ω Since this is a local observation, the same is true for holomorphic sections of a line bundle Thus, and the result follows Γ(CP 2 \ RP 2, O( )) = Γ(CP 2, O( )) = 0 Now suppose that g is a smooth odd function homogeneous of degree on R 3 \ {0} and that ψ = G(g) as characterised by equation (4) Lemma 4 In this case, (8) G(Z) = G(ξ + iη) = g(uξ + vη) (u dv v du) u + iv Proof Differentiating under the integral sign and using the chain rule, G Z := ( G β 2 ξ β + i G ) η β = g 8π X β (uξ + vη) (u dv v du) = 4 U βψ(u) Contracting both sides of this equation with dz β gives G = ω and the result follows from Lemma 3 Proposition 2 The transform S is a left inverse to the transform G Proof Let us denote by E( ), the underlying smooth bundle of the holomorphic line bundle O( ) on CP 2 Recall that g is homogeneous of degree and odd on R 3 \ {0} Thus, g(λx) = λ g(x) for all λ R 3 \ {0} rather than just positive λ We may then use this equation with complex λ to regard g as a smooth section of E( ) RP2 From (8) we shall deduce that g is a suitable limit of G(Z) as Z CP 2 \ RP 2 approaches RP 2 To complete the proof we shall use (5) to obtain a formula for this limit in terms of ψ On RP 2 this formula will reduce to (9)

13 Complex analysis and the Funk transform 589 Choose linearly independent vectors ρ, σ R 3 and consider Z = ρ + ζσ for ζ C \ R As in the proof of Lemma 2, this parameterises the two hemispheres of a standard CP cycle in CP 2 each of which is a fibre of π : CP 2 \ RP 2 S 2 Writing ζ = p + iq and Z = ξ + iη as in (8), we obtain ξ = ρ + pσ and η = iqσ Notice that ξ and η are linearly independent Suppose q > 0 and define u and v by u = q and v = t p for < t < As a limiting case of (8), bearing in mind that g is odd, we obtain G(ρ + ζσ) = 2 = 2π = 2πi g(uξ + vη) (u dv v du) u + iv g(q(ρ + tσ)) q dt i(t ζ) g(ρ + tσ) dt t ζ This is precisely Cauchy s integral formula If q < 0, the same conclusion holds save for a change of sign It follows that G(ρ + ζσ) has smooth limits as ζ approaches the real axis from either side and that g(ρ + tσ) is the sum of these two limits This is the sense in which g is the limit of G We now compute this limit from the formula (5) In terms of standard orthogonal coördinates on R 3, g 0 0 = lim ɛ 0 8π S 2 ψ x + ɛiy + ψ x ɛiy = lim ɛ 0 8π S 2 2xψ x 2 + ɛ 2 y 2 A short computation, bearing in mind that ψ is homogeneous of degree 2, gives [ ( x 2 + ɛ 2 y 2 ) ] d ψ(z dy y dz) y 2 [ 2xψ = x 2 + ɛ 2 y 2 + ( x 2 + ɛ 2 y 2 y 2 ) ] ψ (z dx dy + y dz dx + x dy dz), x

14 590 T N Bailey, M G Eastwood, A R Gover, and L J Mason where y 0 Let us integrate this two-form over the cap S 2 {y > δ} By Stokes theorem, S 2 {y>δ} [ d ( x 2 + ɛ 2 y 2 y 2 ) ψ(z dy y dz)] ( x 2 + ɛ 2 δ 2 ) = δ S 2 {y=δ} δ 2 ψ dz δ( ɛ 2 + δ 2 ) ψ 0 S 2 {y=δ} as δ 0 A similar conclusion applies to the antipodal cap S 2 {y < δ} Noting that the 2-form z dx dy + y dz dx + x dy dz restricts to the standard area form on the unit sphere, we obtain 8π S 2 2xψ x 2 + ɛ 2 y 2 = 8π ( x 2 + ɛ 2 y 2 S 2 y 2 ) ψ x The right hand side has a limit as ɛ 0, namely ( ) x 2 ψ 8π S 2 y 2 x = x ψ S 2 y x This coincides with (0), as required We conclude this article with a brief discussion of the modifications needed to approach the X-ray transform (see [6]) by similar means Suppose f is a smooth even function on R 4 \ {0} homogeneous of degree 2 The definition (2) of φ(ξ, η) = φ(ξ η) for linearly independent ξ, η R 4 is unchanged However, φ is not an arbitrary smooth function, but satisfies the differential equation (9) 2 φ ξ [α η β] = 0, the corresponding equation being vacuous in the Funk case If φ is regarded as an odd function on Gr + 2 (R4 ), the Grassmannian of oriented 2-planes in R 4, then (9) becomes a second order scalar differential equation called the ultrahyperbolic wave equation If g(x) is a smooth odd function on R 4 \ {0}, homogeneous of degree, then ψ β (ξ, η) = g (uξ + vη) (u dv v du) 2π Xβ

15 Complex analysis and the Funk transform 59 is a function of ξ η satisfying ξ β ψ β = 0 = η β ψ β and also the differential equations (20) ξ [α ψ β] = 0 = η [α ψ β] Again, ψ β may be interpreted as a field on Gr + 2 (R4 ), satisfying an intrinsically defined system of differential equations (an ultrahyperbolic version of the Dirac equation) The mapping induces C 4 2 R 4 Z α = ξ α + iη α ξ η π : CP 3 \ RP 3 Gr + 2 (R4 ) and ω = 4 ψ α(z)dz α defines a -closed (0, )-form on CP 3 \RP 3 (closure being equivalent to the differential equations (20) on Gr + 2 (R4 )) The formula (8) defines a smooth homogeneous function G of degree on CP 3 \RP 3 satisfying G = ω (in fact, from [5], H (CP 3 \RP 3, O( )) = 0) The same argument as given in the proof of Proposition 2 shows that, in the sense explained in this proof, lim G = g along the fibres of π References [] A Andreotti and F Norguet, La convexité holomorphe dans l espace analytique des cycles d une variété algébrique, Ann Scuola Norm Sup Pisa 2 (967), 3 82 [2] H Bateman, The solution of partial differential equations by means of definite integrals, Proc Lond Math Soc (2) (904), [3] T N Bailey and M G Eastwood, Zero-energy fields on real projective space, Geom Dedicata 67 (997), [4] T N Bailey and M G Eastwood, Twistor results for integral transforms, Radon Transforms and Tomography, Contemp Math Vol 278, Amer Math Soc 200, pp [5] T N Bailey, M G Eastwood, A R Gover, and L J Mason, The Funk transform as a Penrose transform, Math Proc Cambridge Philos Soc 25 (999), 67 8 [6] A D Agnolo and C Marastoni, Real forms of the Radon-Penrose transform, Publ Res Inst Math Sci 36 (2000), [7] A D Agnolo and P Schapira, Radon-Penrose transform for D-modules, J Funct Anal 39 (996), [8] M G Eastwood, Introduction to Penrose transform, The Penrose Transform and Analytic Cohomoy in Representation Theory, Contemp Math Vol 54, Amer Math Soc 993, pp 7 75 [9], Complex methods in real integral geometry, The Sixteenth Winter School on Geometry and Physics, Srní, Suppl Rendi Circ Mat Palermo 46 (997), 55 7

16 592 T N Bailey, M G Eastwood, A R Gover, and L J Mason [0] P Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math Ann 74 (93), [] I M Gelfand, S G Gindikin, and M I Graev, Integral geometry in affine and projective spaces, Itogi Nauki Tekh, Ser Sovrem Probl Mat 6 (980), , translated into English in J Sov Math 8 (980), [2] S G Gindikin, Some notes on the Radon transform and integral geometry, Monatsh Math 3 (992), [3] V Guillemin, The Radon transform on Zoll surfaces, Adv Math 22 (976), 85 9 [4] V Guillemin and S Sternberg, An ultrahyperbolic anaue of the Robinson-Kerr theorem, Lett Math Phys 2 (986), 6 [5] F R Harvey, The theory of hyperfunctions on totally real subsets of a complex manifold with applications to extension problems, Amer J Math 9 (969), [6] F John, The ultrahyperbolic differential equation with four independent variables, Duke Math J 4 (938), [7] R Penrose, Solutions of the zero rest-mass equations, J Math Phys 0 (969), [8] R Penrose and W Rindler, Spinors and Space-time, Vol, Cambridge University Press, 984 [9] J Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Sächs Akad Wiss Leipzig, Math-Nat Kl 69 (97), [20] W Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, PhD dissertation, University of California, Berkeley 967, Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Math Surveys and Monographs Vol 3, Amer Math Soc 989, pp [2] G A J Sparling, Inversion for the Radon line transform in higher dimensions, Trans Roy Soc Lond A356 (998), [22] N M J Woodhouse, Contour integrals for the ultrahyperbolic wave equation, Proc Roy Soc Lond A438 (992), T N Bailey Department of Mathematics University of Edinburgh James Clerk Maxwell Building The King s Buildings Mayfield Road Edinburgh EH9 3JZ, Scotland tnb@mathsedacuk

17 Complex analysis and the Funk transform 593 M G Eastwood Department of Pure Mathematics University of Adelaide South Australia meastwoo@mathsadelaideeduau A R Gover Department of Mathematics University of Auckland Private Bag 9209 Auckland, New Zealand gover@mathaucklandacnz L J Mason Mathematical Institute Saint Giles Oxford OX 3LB, England lmason@mathsoxacuk