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The determination of a function from its projections

Published online by Cambridge University Press:  24 October 2008

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

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ and perpendicular distance t from the origin, that is H(t, θ) = {x ε Rn: x.θ = t} (Note that H(t, θ) and H(−t, −θ) are the same hyperplane.) If f(x)ε L1(Rn) we will denote by F(t, θ) the projection of f perpendicular to θ, that is the integral of f(x) over H(t, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ε L1 (Rn), F(t, θ) exists for almost all t for every θ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 351 - 355
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

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