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ELEMENTARY INVERSION OF RIESZ POTENTIALS AND RADON-JOHN TRANSFORMS 1 B. RUBIN 1 0 2 Abstract. New simple proofs are given to some elementary ap- n proximateandexplicitinversionformulaeforRieszpotentials. The a results are applied to reconstruction of functions from their inte- J grals over Euclidean planes in integral geometry. 6 2 ] A F 1. Introduction . h The Riesz potential of order α of a sufficiently good function f on t a Rn is defined by m [ 1 f(y)dy 2απn/2Γ(α/2) (Iαf)(x) = , γ (α) = , (1.1) 1 γn(α) ZRn |x−y|n−α n Γ((n−α)/2) v 5 Reα > 0, α 6= n,n+2,n+4,... . 0 1 This operator can be regarded (in a certain sense) as a negative power 5 of “minus-Laplacian”, namely, . 1 0 ∂2 ∂2 ∂2 1 Iα = (−∆)−α/2, ∆ = + +...+ . (1.2) ∂x2 ∂x2 ∂x2 1 1 2 n : v If f ∈ Lp(Rn), then the integral (1.1) converges a.e. provided that i X 1 ≤ p < n/Reα. This condition is sharp. Explicit and approxi- r mate inversion formulae for Iα are of great importance. We refer to a [7, 12, 21, 23, 26, 27, 30] for the basic properties and applications of Riesz potentials. In this article we restrict to real α ∈ (0,n). This case reflects basic features and is sufficient for integral geometric applications in Section 4. Our mainworking toolsareasuitably chosen auxiliary functionw(x) (we call it a reconstructing function) and its scaled version w (x) = t t−nw(x/t), t > 0. In the modern literature such a function is called a 2000 Mathematics Subject Classification. Primary 44A12; Secondary 47G10. Keywordsandphrases. Rieszpotentials,Radontransforms,Wavelettransforms. The work was supported by the NSF grants PFUND-137 (Louisiana Board of Regents), DMS-0556157,and the Hebrew University of Jerusalem. 1 2 B. RUBIN wavelet and a convolution (f ∗w )(x) is called the continuous wavelet t transform of f; cf. [2, 3]. We choose w(x) = (−∆)m[(1+|x|2)m−(n+α)/2], w˜(x) = (−∆w)(x), (1.3) m ∈ N, and keep this notation throughout the paper. The main results of the article are presented by Theorem A below and Theorem B in Section 4. The latter gives an example of applica- tion of Theorem A to elementary inversion of the k-plane Radon-John transform in integral geometry and falls into the general scope of the convolution-backprojection method (cf. [15, 22]). Radon transforms and their generalizations are the basic tools in tomography and numer- ous related areas of pure and applied mathematics; see [1, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 18] and references therein. Theorem A. Let 2m > α, f ∈ Lp(Rn), 1 ≤ p < n/α. Then c f = lim(Iαf ∗t−αw ), c = γ (2m−α), (1.4) α,m t α,m n t→0 ∞ Iαf ∗w˜ t d f = dt, d = (2m−α)c , (1.5) α,m Z t1+α α,m α,m 0 ∞ ∞ where (...) = lim (...) and γ (·) has the same meaning as in 0 ε n ε→0 (1.1). TRhe limit in botRh formulae exists in the Lp-norm and in the a.e. sense. If, moreover, f ∈ C (Rn), then it exists in the sup-norm. 0 We recall that C (Rn) denotes the space of continuous functions on 0 Rn vanishing at infinity. Some comments are in order. Statements of this kind are not new. Regarding (1.4), we observe that approximate inversion of operators of thepotentialtypewasinitiatedbyZavolzhenskii, NoginandSamko[32, 17, 24, 25, 26]. Conceptually this approach is close to the convolution- backprojection method in tomography [15]. An elegant formula (1.4) is due to Samko [26, p. 325], [24, 25]; see also [19]. Below we give a new simple proof of it. Regarding (1.5), wavelet-like inversion formulae for Riesz potentials are also well-known; see, e.g., [21, Section 17]. Our aim is to show that elementary reconstructing functions (1.3) work perfectly in the wavelet inversion scheme and the relevant justification is much simpler than in the general theory; cf. [21]. The case α = 0 in (1.5), when Iα is substituted by the identity operator, represents a variant of Calderoń’s reproducing formula [2, 3]. The validity of formulae (1.4) and (1.5) can be formally explained in the language of the Fourier transform. We have lim(Iαf ∗t−αw )∧(ξ) = fˆ(ξ) lim(t|ξ|)−αwˆ(tξ) = c fˆ(ξ) (1.6) t w t→0 t→0 ELEMENTARY INVERSION OF RIESZ POTENTIALS 3 provided that the limit c = lim |ξ|−αwˆ(ξ) exists. Moreover, since w˜ w ξ→0 in (1.3) is a radial function, we easily get ∞ Iαf ∗w˜ ∧ 1 wˆ(y)dy t ˆ dt (ξ) = d f(ξ), d = , (1.7) hZ0 t1+α (cid:21) w w σn−1 ZRn |y|n+α−2 where σ =2πn/2 Γ(n/2) is the surface area of the unit sphere |x|=1. n−1 This simple argu(cid:14)ment does not work for arbitrary Lp-functions. To cover this case, we develop another technique and give the final answer without using the Fourier transform. 2. Preliminaries We recall some known facts. The Fourier transform of a function f ∈ L1 = L1(Rn) is defined by fˆ(ξ) = f(x)eix·ξdx, x·ξ = x ξ +...+x ξ . (2.1) 1 1 n n ZRn Let S = S(Rn) be the Schwartz space of all C∞ functions which, to- gether with derivatives of all orders, vanish at infinity faster than any inverse power of |x| = (x2 + ... + x2)1/2. We endow S(Rn) with a 1 n standard topology generated by the sequence of norms ||ϕ|| = max(1+|x|)m |(∂jϕ)(x)|, m = 0,1,2,... . m x X |j|≤m The following spaces adapted to Riesz potentials were introduced by Semyanistyi [28]; further generalizations due to Lizorkin and Samko can be found in [13, 26]. Let Ψ = Ψ(Rn) be the subspace of S, consisting of functions ψ(ξ) vanishing at ξ = 0 with all derivatives, and let Φ = Φ(Rn) be the Fourier image of Ψ. We equip Φ with the induced topology of S. Then Φ becomes a topological vector space which is isomorphic to Ψ under the action of the Fourier transform. We denote by Φ′ the space of distributions over Φ. The main reason for introducing the spaces Ψ and Φ is that Ψ is invariant under multiplication by |ξ|α for any α ∈ C and therefore, the Riesz potential Iα is an automorphism of Φ and Φ′. Proposition 2.1. [21, p. 21 ] Let f ∈ Lr, 1 ≤ r < ∞, and g ∈ Lp, 1 ≤ p < ∞. If f = g in the Φ′-sense, then f = g almost everywhere. 4 B. RUBIN 3. Proof of Theorem A The following auxiliary propositions form the heart of the proof. Lemma 3.1. Let w∈L1(Rn) be such that h = Iαw has an integrable decreasing radial majorant. If f∈Lp(Rn), 1≤p<n/α, then lim(Iαf ∗t−αw ) = c (α)f, c (α) = h(x)dx, (3.1) t w w t→0 ZRn where the limit exists in the Lp-norm and in the a.e. sense. If, moreover, f ∈ C (Rn), then the limit exists in the sup-norm. 0 Proof. We have t−α(Iαw )(x) = t−nh(x/t) = h (x). Then t t Iαf ∗t−αw = f ∗t−αIαw = f ∗h , t t t whichyieldsanapproximateidentity[30, p. 62]. Theaboveapplication of Fubini’s theorem is permissible because (Iα|f|)∗|w| < ∞ a.e. (cid:3) There are many ways to choose a reconstructing function w. Usually w or Iαw or both are expressed analytically in a pretty complicated way. As we shall see below, anadvantage ofthe choice (1.3) isthat both w and h = Iαw are elementary and a constant c (α) can be different w from zero. Lemma 3.2. Let w(x) = (−∆)m[(1 + |x|2)m−(n+α)/2]. If α > 0, then w∈L1(Rn). If, moreover, α < min(n,2m) and h = Iαw, then Γ((n+2m−α)/2) h(x)=a (1+|x|2)(α−n)/2−m, a =22m−α , α,m α,m Γ((n+α−2m)/2) so that h ∈ L1(Rn) and 22m−απn/2Γ(m−α/2) c ≡ h(x)dx = = γ (2m−α); α,m n ZRn Γ((n+α−2m)/2) cf. the normalizing constant in (1.1). If α is not an integer, then c 6= 0 for any m > α/2. If α ∈ {1,2,...,n−1}, then c 6= 0 α,m α,m provided, e.g., that m = [(α+2)/2]. Proof. The first statement can be easily checked by differentiation. For instance, we can write ∆ in polar coordinates to get d d w(x)=(Lmψ)(|x|2), L=4r1−n/2 rn/2 , ψ(r)=(1+r)m−(n+α)/2. dr dr This gives w(x) = O((1+|x|2)−(α+n)/2) ∈ L1(Rn). To prove the second statement we invoke the Bessel-McDonald kernel K (|ξ|) 21−(α+n)/2 (n−α)/2 G (ξ) = λ , λ = , (3.2) α α |ξ|(n−α)/2 α πn/2Γ(α/2) ELEMENTARY INVERSION OF RIESZ POTENTIALS 5 where K (·) denotes the modified Bessel function (or the McDonald ν function) of order ν with the property K (z) = K (z); (3.3) ν −ν see, e.g., [31]. It is known that |ξ|(α−n−1)/2e−|ξ| if |ξ| > 1, |ξ|α−n if |ξ| < 1, α < n, G (ξ) ≤ c  (3.4) α α  1 if |ξ| < 1, α > n, 1+log(1/|ξ|) if |ξ| < 1, α = n,     where c is a continuous function of α; see, e.g., [21, p. 257], [16, p. α 285]. Furthermore, for any α > 0 theFourier transform of(1+|x|2)−α/2 in the sense of distributions is computed by the formula ((1+|x|2)−α/2,φ(x)) = (2π)−n(G (ξ),φˆ(ξ)), φ ∈ S(Rn), (3.5) α [16, Section 8.1]. Let us prove that (Iαw,φ) = a ((1+|x|2)(α−2m−n)/2,φ(x)) (3.6) α,m for any test function φ in the space Φ(Rn). Once this is done, the required pointwise equality will follow owing to Proposition 2.1. We have (Iαw,φ) = (w,Iαφ) = ((1+|x|2)(2m−n−α)/2,(−∆)m(Iαφ)(x)) = (2π)−n(G (ξ),|ξ|2m−αφˆ(ξ)) n+α−2m K (|ξ|) = (2π)−nλ m−α/2 ,|ξ|2m−αφˆ(ξ) . n+α−2m (cid:18) |ξ|m−α/2 (cid:19) Using (3.3) and (3.5), we continue: (2π)−nλ (Iαw,φ) = n+α−2m (G (ξ),φˆ(ξ)) 2m+n−α λ 2m+n−α = a ((1+|x|2)(α−2m−n)/2,φ(x)). α,m Evaluation of c is straightforward. To complete the proof, we note α,m that c 6= 0 if (n+α−2m)/2 6= 0,−1,−2,.... The latter is guaran- α,m teed under the afore-mentioned choice of m. (cid:3) Proof of Theorem A. The first statement follows immediately from Lemmas 3.1 and 3.2. To prove the second statement, we first note that w˜ = −∆w belongs to L1. Hence, the convolution Iαf ∗ t−αw˜ is t ˜ well-defined in the Lebesgue sense and can be written as f ∗ h with t 6 B. RUBIN h˜ = Iαw˜ = Iα(−∆)w. Furthermore, for any test function φ ∈ Φ(Rn), (3.6) yields (h˜,φ) = (Iα(−∆)w,φ) = −(Iαw,∆φ) = −a (∆[(1+|x|2)(α−n)/2−m],φ). α,m Since ∆[(1+|x|2)(α−n)/2−m] = O((1+|x|2)(α−n)/2−m−1) ∈ L1, then, by Proposition 2.1, we have a pointwise equality h˜(x) = −a ∆[(1+|x|2)(α−n)/2−m]. α,m Denote ∞ Iαf ∗w˜ t J = dt, ε > 0. ε Z t1+α ε Then ∞ f ∗h˜ ∞ h˜ (x) t t J = dt = f ∗ψ , ψ (x) = dt. ε ε ε Z t Z t ε ε Setting h˜(x) = h˜ (|x|2), we get 0 ∞ h˜ (|x|2/t2) 1 x|2/ε2 ψ (x) = 0 dt = h˜ (r)rn/2−1dr. ε Z tn+1 2|x|n Z 0 ε 0 Thisisascaledversionofthefunctionψ(x)=2−1|x|−n |x|2h˜ (r)rn/2−1dr. 0 0 Observing that R d d h˜ (r) = −a L[(1+r)(α−n)/2−m], L=4r1−n/2 rn/2 , 0 α,m dr dr we have 2a x|2 d d ψ(x) = − α,m rn/2 [(1+r)(α−n)/2−m]dr |x|n Z dr dr 0 2a d = − α,m rn/2 [(1+r)(α−n)/2−m] |x|n (cid:26) dr (cid:27) r=|x|2 = −a (α−n−2m)(1+|x|2)(α−n)/2−m−1. α,m This gives lim J = d f(x), where ε α,m ε→0 22m−α+1πn/2Γ(m+1−α/2) d = ψ(x)dx = = (2m−α)γ (2m−α), α,m n ZRn Γ((n+α−2m)/2) and the limit is understood in the required sense. ELEMENTARY INVERSION OF RIESZ POTENTIALS 7 4. Inversion of the Radon-John Transform We recall basic definitions. More information can be found in [5, 7, 23]. Let G and G be the affine Grassmann manifold of all non- n,k n,k oriented k-dimensional planes τ in Rn and the ordinary Grassmann manifold of k-dimensional linear subspaces ζ of Rn, respectively. Each k-plane τ ∈ G is parameterized as τ = (ζ, u), where ζ ∈ G and n,k n,k u ∈ ζ⊥ (the orthogonal complement of ζ in Rn). The k-plane Radon- John transform of a function f on Rn is defined by (R f)(τ) ≡ (R f)(ζ, u) = f(y +u)dy, (4.1) k k Z ζ where dy is the induced Lebesque measure on the subspace ζ ∈ G . n,k This transform assigns to a function f a collection of integrals of f over all k-planes in Rn. The corresponding dual transform of a function ϕ on G is defined as the mean value of ϕ(τ) over all k-planes τ through n,k x ∈ Rn: (R∗ϕ)(x) = ϕ(σζ +x)dσ, x ∈ Rn. (4.2) k 0 Z O(n) Here ζ ∈ G is an arbitrary fixed k−plane through the origin. If 0 n,k f ∈ L (Rn), then R f is finite a.e. on G if and only if 1 ≤ p < n/k p k n,k [29, 23]. AvarietyofinversionproceduresareknownforR f; see[5,7,22,23]. k Oneof the most important algorithms relies onthe Fuglede formula [4], R∗R f = d Ikf, d = (2π)kσ /σ , (4.3) k k k,n k,n n−k−1 n−1 with the Riesz potential Ikf on the right-hand side. Hence, Theorem A yields the following result. We denote w(x)=(−∆)m[(1+|x|2)m−(n+k)/2], m=[(k+2)/2], w˜(x)=(−∆w)(x); 4mπ(n+k)/2Γ(n/2)Γ(m−k/2) λ = , δ = (2m−k)λ . k k k Γ((n−k)/2)Γ((n+k)/2−m) Theorem B. If ϕ = R f, f ∈ Lp(Rn), 1 ≤ p < n/k, then k ∞ R∗ϕ∗w˜ λ f = lim(R∗ϕ∗t−kw ), δ f = k t dt, (4.4) k t→0 k t k Z t1+k 0 ∞ ∞ where (...) = lim (...). The limit in both formulae exists in the 0 ε→0 ε Lp-norRm and in the aR.e. sense. If, moreover, f ∈ C (Rn), then it exists 0 in the sup-norm. 8 B. RUBIN References [1] M. Agranovsky,Integralgeometryand spectral analysis.Topics in mathematical analysis,281-320,Ser. Anal. Appl. Comput., 3,World Sci. 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