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On the problem of well-posedness for the Radon transform. (English) Zbl 0555.46020
Mathematical aspects of computerized tomography, Proc., Oberwolfach 1980, Lect. Notes Med. Inf. 8, 36-44 (1981).
[For the entire collection see Zbl 0538.00034.]
Some inverse continuity and discontinuity properties of the classical Radon transform R are discussed. First, a compactly supported sequence $$(f_ k)$$ of $$L^ 1$$ functions is exhibited, such that $$Rf_ k$$ converges uniformly, but $$f_ k$$ does not converge weakly (thus inversion of R cannot be properly posed within a function space set-up). On the other hand, partial results on inverse continuity for some Sobolev, measure, and distribution spaces are given. - In subsequent papers, the continuity behaviour of $$R^{-1}$$ (and its consequences for the range of R) has been described completely for Sobolev and classical test function and distribution spaces [A. Hertle, Math. Z. 184, 164-192 (1983; Zbl 0507.46036) and Math. Ann. 267, 91-99 (1984)], and for Sobolev and measure spaces by M. G. Hahn and E. T. Quinto [Z. Wahrscheinlichkeitstheorie (to appear; Zbl 0555.28005)].

##### MSC:
 46F12 Integral transforms in distribution spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 44A05 General integral transforms 44A15 Special integral transforms (Legendre, Hilbert, etc.)