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NOTES on RADON TRANSFORMS Abstract These Notes Include the Following Selected Topics: Discussion of Radon's Paper (1917); Tota NOTES ON RADON TRANSFORMS B. RUBIN Abstract These notes include the following selected topics: Discussion of Radon's paper (1917); totally geodesic Radon transforms on the sphere and associated analytic families of intertwining operators; Radon trans- forms on Grassmann manifolds and matrix spaces; the generalized Minkowski-Funk transform for non-central spherical sections, and small divisors for spherical harmonic expansions; the Busemann-Petty prob- lem on sections of convex bodies. Basic classical ideas and some recent results are presented in a systematic form. Contents 1. Introduction. 2. Radon's paper. 3. The Minkowski-Funk transform and related topics. 4. Totally geodesic Radon transforms on the sphere. 5. Radon transforms on Grassmann manifolds. 6. The generalized Minkowski-Funk transform and small divisors. 7. The Busemann-Petty problem. 8. Radon transforms on matrix spaces. 9. References. 2000 Mathematics Subject Classi¯cation. Primary 44A12; Secondary 14M15, 53C65. Key words and phrases. Radon transforms, fractional integrals, Grassmann man- ifolds, the Busemann-Petty problem . 1 2 B. RUBIN 1. Introduction According I.M. Gelfand [Ge], one of the basic problems of integral geometry can be stated as follows. Given a manifold X, let ¥ be a certain family of submanifolds of X. We write x 2 X; » 2 ¥, and consider the mapping Z (1.1) f(x) ! (Rf)(») = f » that assigns to each su±ciently good function f on X a collection of integrals of f over submanifolds » 2 ¥. The problem is to study mapping properties of (1.1) (range, kernel, norm estimates) and ¯nd explicit inversion formulas in appropriate function spaces. It is assumed that ¥ itself is endowed with the structure of a manifold. The mapping (1.1) is usually called the Radon transform of f. Example 1.1. X = Rn; ¥ is the family of all hyperplanes in Rn. Example 1.2. X = Sn is the unit sphere in Rn+1; ¥ is the family of all (n ¡ 1)-dimensional subspheres of Sn of radius 1. Example 1.3. X = Gn;k is the Grassmann manifold of k-dimensional n subspaces of R , 1 · k < n; ¥ = Gn;k0 is the similar manifold with k0 > k. An idea to study manifolds of submanifolds goes back to the 19th century (J. PlÄucker, F. Klein, M.S. Lie).1 The celebrated paper by J. Radon [Rad] contains fundamental ideas related to operators (1.1) in important special cases and paves the way to further developments. In this paper one can also ¯nd information about the history of the problem. Namely, the problem was suggested to Radon by Blaschke. Reconstruction of functions on S2 from their integrals over big circles was studied by Minkowski [Min] (1904). In 1913 P. Funk [Fu1], who was a student of D. Hilbert, reproduced Minkowski's solution and showed that the problem reduces to Abel's integral equation. This manuscript was prepared for our lectures at the University of Tokyo in Summer 2006. It represents an updated and extended version of our previous text [Ru13]. We start by reviewing basic ideas of the original paper by J. Radon [Rad], and then proceed to the Minkowski-Funk transform and its gen- eralizations for lower-dimensional central sections and Grassmannians 1Lie's interest to the group theory was influenced by Klein, who was PlÄucker's student. NOTES ON RADON TRANSFORMS 3 (Sections 2-5). All these transforms and analytic families of intertwin- ing operators generated by them, are of primary importance in the in- tegral geometry of star-shaped/convex bodies. In Section 6 we discuss the old problem of P. Ungar (1954) about injectivity of non-central modi¯cations of the Minkowski-Funk transform. To the best of our knowledge, this problem is still unsolved. It has a number of refor- mulations and leads to the realm of number theory. We present some partial results which give a flavor of how challenging the problem is. Section 7 is devoted to the Busemann-Petty problem (1956) on sec- tions of convex bodies. For the hyperplane sections, it was solved only recently due to the e®orts of a number of people. The lower dimen- sional version of the Busemann-Petty problem when dimension of the section is 2 and 3 is still mysterious. There is a remarkable interplay between Radon transforms of di®erent kinds behind this problem. We present the solution to the original Busemann-Petty problem in a clear and simple form, and connect it with known results from Section 3. Section 8 concludes our notes and deals with Radon transforms on the space of real rectangular matrices. Here we follow our recent works [OR1], [OR2], inspired by pioneering results due to Petrov [Pe1], [Pe2]. Acknowledgements. I am grateful to Professor Toshio Oshima for the invitation to talk at the University of Tokyo and the hospitality during my visit. 2. Radon's paper Radon begins his paper with the 2-dimensional case of lines in the plane, and then proceeds to generalizations and other settings. We start by reviewing basic ideas of Radon in the context of Example 1.1 for n dimensions. Each hyperplane » in Rn is de¯ned by » = fx 2 Rn : x ¢ θ = tg; θ 2 Sn¡1; t 2 R; and the Radon transform (1.1) can be represented as Z Z (2.1) (Rf)(») = f(x)dm(x) = f(x)±(x ¢ θ ¡ t)dx Rn xZ¢θ=t = f(tθ + u)du ´ (Rf)(θ; t): θ? Here ±(¢) is the usual delta function of one variable, θ? is the (n ¡ 1)- dimensional subspace orthogonal to θ, dm(x) and du stand for the 4 B. RUBIN relevant induced Lebesque measures. For simplicity, we suppose that f belongs to the Schwartz space S(Rn). Clearly, (2.2) (Rf)(θ; t) = (Rf)(¡θ; ¡t) (the symmetry property). The following statement follows immediately from (1.1). Proposition 2.1. The operator (2.1) commutes with rigid motions of Rn. Namely, if ¿ : x ! γx + y; γ 2 O(n); y 2 Rn, then R : f(¿x) ! (Rf)(¿»): This is the basic property of Rf. It means that in order to solve the equation Rf = ', it su±ces to recover f only at one point, say x = 0, and restrict the consideration to radial functions f(x) ´ f0(jxj). Let us pass to details. If f(x) ´ f0(r); r = jxj, then by rotation (set u = γθv, where γθ 2 SO(n); γθen = θ; en = (0;:::; 0; 1)); (2.1) yields 1 Z Z p 2 2 n¡2 (Rf)(θ; t) = f0(jten + vj)dv = σn¡2 f0( t + s )s ds; Rn¡1 0 2¼(n¡1)=2 σ = jSn¡2j = : n¡2 ¡((n ¡ 1)=2) This gives the following important statement. Proposition 2.2. If f(x) ´ f0(jxj), then (Rf)(θ; t) ´ '0(t), where '0(t) is an even function de¯ned by Z1 2 2 (n¡3)=2 (2.3) '0(t) = σn¡2 f0(r)(r ¡ t ) rdr: jtj The integral (2.3) is of Abel type. To be more precise, we introduce Riemann-Liouville (or Weyl) fractional integrals of the form Z1 1 (2.4) v(t) = (I®u)(t) = u(r)(r ¡ t)®¡1dr; ® > 0: ¡ ¡(®) t The inverse of (2.4) is called a fractional derivative. For su±ciently good v(t), it can be written as µ ¶ d m (2.5) u(t) = (D® v) = ¡ (Im¡®v)(t); 8m 2 N; m ¸ ®: ¡ dt ¡ ¡ ¢ a d m If ® 2 N one can set m = ® and get D¡v = ¡ dt v; see [SKM] for more details. NOTES ON RADON TRANSFORMS 5 By changing variables, we write (2.3) as p (n¡1)=2 (n¡1)=2 p '0( t) = ¼ (I¡ f0( ¢))(t); t > 0; so that (1¡n)=2 (n¡1)=2 p 2 (2.6) f0(r) = ¼ (D¡ '0( ¢))(r ): This is the inversion formula for the Radon transform in the radial case. If n is odd, then µ ¶ 1 d (n¡1)=2 (2.7) f (r) = ¼(1¡n)=2 ¡ ' (r): 0 2r dr 0 Now let us recover f(x) from (Rf)(θ; t) = '(θ; t) in the general case. Fix x, and denote fx(y) = f(x + y). By (2.1), (2.8) (Rfx)(θ; t) = '(θ; t + x ¢ θ); and therefore Z Z 1 (2.9) (Rfx)(γθ; t)dγ = '(θ; t + x ¢ θ)dθ: σn¡1 SO(n) Sn¡1 The right hand side of (2.9) is the mean value of ' over all hyperplanes at distance jtj from x. We denote Z ¤ 1 (2.10) (Mt ')(x) = '(θ; t + x ¢ θ)dθ: σn¡1 Sn¡1 Since R commutes with rotations, the left handR side of (2.9) is the Radon transform of the radial function f0(jyj) = SO(n) fx(γy)dγ. The latter can be written as the spherical mean Z 1 def f0(r) = f(x + rθ)dθ = (Mrf)(x); σn¡1 Sn¡1 and Proposition 2.2 yields the following Theorem 2.3. For t 2 R, Z1 ¤ 2 2 (n¡3)=2 (2.11) (Mt Rf)(x) = σn¡2 (Mrf)(x)(r ¡ t ) rdr jtj (n¡1)=2 (n¡1)=2 p 2 = ¼ (I¡ f0( ¢))(t ); and therefore (1¡n)=2 (n¡1)=2 p¤ (2.12) f(x) = ¼ lim(D¡ [(M Rf)(x)])(r): r!0 ¢ 6 B. RUBIN Now we introduce another important operator. Set t = 0 in (2.10), and denote Z 1 (2.13) (R¤')(x) = '(θ; x ¢ θ)dθ; x 2 Rn; σn¡1 Sn¡1 where '(θ; t) is a function on the manifold of all hyperplanes in Rn. The operator (2.13) is called the dual Radon transform (this motivates \*" in (2.10)).
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