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Consistency conditions for a finite set of projections of a function

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer
K. J. Falconer
Affiliation:
Corpus Christi College, Cambridge
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Abstract

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 61 - 68
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Logan, B. F. and Shepp, L. A.Optimal reconstruction of a function from its projections. Duke Math. J. 42 (1975), 645–659.Google Scholar
(2)Nikol'skii, S. M.Approximation of junctions of several variables and imbedding theorems (Berlin, Heidelberg, New York; Springer-Verlag, 1975).Google Scholar
(3)Smith, K. T., Solmon, D. C. and Wagner, S. L.Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Amer. Math. Soc. 83 (1977), 1227–1270.CrossRefGoogle Scholar