an:03884774
Zbl 0555.46020
Hertle, Alexander
On the problem of well-posedness for the Radon transform
EN
Mathematical aspects of computerized tomography, Proc., Oberwolfach 1980, Lect. Notes Med. Inf. 8, 36-44 (1981).
1981
a
46F12 46E35 44A05 44A15
inverse continuity and discontinuity properties of the classical Radon transform; inverse continuity for some Sobolev, measure, and distribution spaces
[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 [\textit{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 \textit{M. G. Hahn} and \textit{E. T. Quinto} [Z. Wahrscheinlichkeitstheorie (to appear; Zbl 0555.28005)].
Zbl 0538.00034; Zbl 0507.46036; Zbl 0555.28005