Inversion of Radon transforms using wavelet transforms generated by wavelet measures

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The classical inversion formulae for the Radon transform R on R involve the dual transform R and the operator Dny1 ˆ y † ny1†=2 which represents the positive power of the Laplacian. It is shown that wavelet type representations of Dny1, generated by finite measures with a certain number of vanishing moments, lead to explicit inversion formulae for Rf ; f 2 Lp Rn†; and enable one to characterize the range R Lp Rn††. The same method can be applied to explicit inversion and characterization of Radon transforms of finite Borel measures. 1. Introduction The Radon transform of sufficiently nice function f on R is defined by R f † s† Rf † ; s† ˆ Z ? f s ‡ u†du; ; s† 2 ~ R ˆ ny1 R; 1:1† where ny1 is the unit sphere in R, du stands for the euclidean measure on the subspace ? orthogonal to . Among the basic problems related to (1.1) are explicit inversion of the operator R and a characterization of its range. The relations f ˆ nR y 1†' and f ˆ n y n†R#'; ' ˆ Rf ; 1:2† n ˆ 2 †1yn=2 (see e.g. [5]), involving the dual transform R#'† x† ˆ Z ny1 ' ; hx; i†d ; x 2 R; 1:3† and the ny 1†=2th power of the Laplacian (in one and n dimensions respectively) are usually employed for solution of these problems. In spite of the elegance and simplicity of (1.2), the practical implementation of these MATH. SCAND. 85 (1999), 285^300 * Partially sponsored by the Edmund Landau Center for research in Mathematical analysis, supported by the Minerva Foundation (Germany). Received January 9, 1997; in revised form March 11, 1997. {orders}ms/990839/rubin.3d -21.11.00 11:44 formulae entails difficulties connected with the realization of powers of the Laplacian. These difficulties increase when dealing with nonsmooth functions or measures, the differentiation of which can be performed only in the distribution sense. Additional difficulties arise in the case of n even, when the operator y † ny1†=2 is not local. In order to reduce these difficulties, in [1, 3, 8, 14] it was suggested to employ continuous wavelet transforms. In the present paper we show that wavelet constructions of the inverse Radon transform can be obtained directly from (1.2) if one replaces the powers of the Laplacians by their wavelet representations (see [7, 9]) generated by suitable wavelet measures. For example, the first formula from (1.2) gives rise to the following statement. Theorem 1.1. Assume that is a finite Borel measure on R satisfying the following conditions: Z jsj>1 jsj dj j s† ny1; Z 1 y1 sd s†ˆ0 for all jˆ0; 2; :::; 2‰ ny1†=2Š where ‰ ny1†=2Š designates the integer part of ny1†=2. (i) Let ' ˆ Rf , f 2 Lp Rn†, 1 p 0; ' t† ;s†ˆ Z 1 y1 ' ;syt †d †: 1:4† Then Z 1 0 R# ' t† dt tn lim !0 T'† x† ˆ c f ; 1:5† the limit being interpreted in the Lp-norm and in the a.e. sense, c ˆ n‡1=2 y1†n=2 y n=2†y n‡ 1†=2† Z 1 y1 jsjny1d s† if n is even; 2 ny1=2 y1† n‡1†=2 y n=2†y n‡ 1†=2† Z 1 y1 jsjny1 log jsjd s† if n is odd: 8> >: 1:6†