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Reconstruction of functions from their integrals overk-planes

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Abstract

Thek-plane Radon transform assigns to a functionsf(x) on ℝn the collection of integralsf(τ)=∫ τ f over allk-dimensional planesτ. We give a systematic treatment of two inversion methods for this transform, namely, the method of Riesz potentials, and the method of spherical means. We develop new analytic tools which allow to invertf(τ) under minimal assumptions forf. It is assumed thatfεL p, 1≤p<n/k, orf is a continuous function with minimal rate of decay at infinity. In the framework of the first method, our approach employs intertwining fractional integrals associated to thek-plane transform. Following the second method, we extend the original formula of Radon for continuous functions on ℝ2 tofεL p(ℝn) and all 1≤k<n. New integral formulae and estimates, generalizing those of Fuglede and Solmon, are obtained.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Boris Rubin

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  1. Boris Rubin
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Correspondence to Boris Rubin.

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The work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).

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Rubin, B. Reconstruction of functions from their integrals overk-planes. Isr. J. Math. 141, 93–117 (2004). https://doi.org/10.1007/BF02772213

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  • DOI: https://doi.org/10.1007/BF02772213

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