The Radon Transform by Sigurdur Helgason (auth.)

By Sigurdur Helgason (auth.)

The first version of this publication has been out of print for it slow and i've made up our minds to stick with the publisher's sort recommendation to arrange a brand new variation. Many examples with specific inversion formulation and variety theo­ rems were extra, and the group-theoretic point of view emphasised. for instance, the indispensable geometric point of view of the Poisson imperative for the disk ends up in fascinating analogies with the X-ray rework in Euclidean 3-space. to maintain the introductory taste of the booklet the fast and self-contained bankruptcy Von Schwartz' distributions has been extra. the following §5 offers proofs of the wanted effects in regards to the Riesz potentials whereas §§3-4 improve the instruments from Fourier research following heavily the account in Hormander's books (1963] and [1983]. there's a few overlap with my books (1984] and [1994b] which despite the fact that depend seriously on Lie team conception. the current booklet is way extra undemanding. i'm indebted to Sine Jensen for a severe studying of components of the manuscript and to Hilgert and Schlichtkrull for concrete contributions males­ tioned at particular locations within the textual content. ultimately I thank Jan Wetzel and Bonnie Friedman for his or her sufferer and skillful education of the manuscript.

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Taking arbitrary derivative ak I axil JfR Jsn-1 dw Jsn-1 0 0 0 8xik at X = 0 we deduce (Sf)(w,p)wil···wik)cakcp)(p)dp=O for each k and each cp E V( t 0 , oo). Integrating by parts in the p variable we conclude that the function (98) p-+ { Jsn-1 (8/)(w,p)wit···Wikdw, pElR kth derivative = 0 for p > t 0 . Thus it equals a polynomial for > to. However, if n is odd the function (98) has compact suppprt so it has its p must vanish identically for p > to. On the other hand, if n is even and F E V(JR) then as remarked at the end of §3, limltl-+oo(1iF)(t) = 0.

For k E Z + we consider the polynomial (57) Pk(u) If u = jR,. f f(x)(x, u)k dx. 6. L this can be written r jR,. L ,. L of the polynomial P,.. In analogy with the space 'DH(Pn) in No. (x") =

Rn We also conclude that f(w,p) exists for almost all (w,p) E sn- 1 we consider the measurable function (x,w) -t j(x)cp(w, (w,x}) on JRn X X lit Next sn- 1 • We have 1sn- 1 xRn lf(x)cp(w, (w, x} )I dw dx = = lsn- (Ln lf(x)cp(w,(w,x})idx) dw lsn- (L lff(w,p)lcp(w,p)j dp) dw, 1 1 which in both cases is finite. Thus f (x) · cp( w, (w, x}) is integrable on JRn x sn- 1 and its integral can be calculated by removing the absolute values above. This gives the left hand side of (46). Reversing the integrations we conclude that

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