DOI: 10.1017/s030500410005550x. Consistency conditions for a finite set of projections of a function

Consistency conditions for a finite set of projections of a function

10.1017/s030500410005550x

Crossref journal-article

Cambridge University Press (CUP)

Mathematical Proceedings of the Cambridge Philosophical Society (56)

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.

Bibliography

Falconer, K. J. (1979). Consistency conditions for a finite set of projections of a function. Mathematical Proceedings of the Cambridge Philosophical Society, 85(1), 61â68.

Authors 1

K. J. Falconer (first)

References 3 Referenced 5

Dates

Type	When
Created	16 years, 9 months ago (Nov. 7, 2008, 10:16 a.m.)
Deposited	6 years, 3 months ago (May 26, 2019, 6:18 p.m.)
Indexed	3 years, 5 months ago (April 2, 2022, 11:15 a.m.)
Issued	46 years, 8 months ago (Jan. 1, 1979)
Published	46 years, 8 months ago (Jan. 1, 1979)
Published Online	16 years, 10 months ago (Oct. 24, 2008)
Published Print	46 years, 8 months ago (Jan. 1, 1979)

Funders 0

None

BibTeX

@article{Falconer_1979, title={Consistency conditions for a finite set of projections of a function}, volume={85}, ISSN={1469-8064}, url={http://dx.doi.org/10.1017/s030500410005550x}, DOI={10.1017/s030500410005550x}, number={1}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Falconer, K. J.}, year={1979}, month=jan, pages={61–68} }

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