Reconstructing Geometric Objects from the Measures of Their Intersections with Test Sets
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- 作者：Márton Elekes (1) (2)
Tamás Keleti (2)
András Máthé (3) - 关键词：Reconstruction ; Intersection ; Lebesgue measure ; Fourier transform ; Radon transform ; Convex set ; Random construction ; 28A99 ; 42A61 ; 26A46 ; 42A38 ; 42B10 ; 51M05
- 刊名：Journal of Fourier Analysis and Applications
- 出版年：2013
- 出版时间：June 2013
- 年：2013
- 卷：19
- 期：3
- 页码：545-576
- 全文大小：659KB
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Tamás Keleti (2)
András Máthé (3)
1. Alfréd Rényi Institute of Mathematics, PO Box 127, 1364, Budapest, Hungary
2. Institute of Mathematics, E?tv?s Loránd University, Pázmány Péter s. 1/c, 1117, Budapest, Hungary
3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK - ISSN：1531-5851
文摘
Let us say that an element of a given family $\mathcal{A}$ of subsets of ?sup class="a-plus-plus"> d can be reconstructed using n test sets if there exist T 1,-T n ?? d such that whenever $A,B\in\mathcal{A}$ and the Lebesgue measures of A?em class="a-plus-plus">T i and B?em class="a-plus-plus">T i agree for each i=1,-n then A=B. Our goal will be to find the least such n. We prove that if $\mathcal{A}$ consists of the translates of a fixed reasonably nice subset of??sup class="a-plus-plus"> d then this minimum is n=d. To obtain this we prove the following two results. (1)?A?translate of a fixed absolutely continuous function of one variable can be reconstructed using one test set. (2)?Under rather mild conditions the Radon transform of the characteristic function of K (that is, the measure function of the sections of K), (R θ χ K )(r)=λ d?(K∩{x∈? d :?em class="a-plus-plus">x,θ?r}) is absolutely continuous for almost every direction θ. These proofs are based on techniques of harmonic analysis. We also show that if $\mathcal{A}$ consists of the enlarged homothetic copies rE+t (r?,t∈? d ) of a fixed reasonably nice set E?? d , where d?, then d+1 test sets reconstruct an element of $\mathcal{A}$ , and this is optimal. This fails in ? we prove that a closed interval, and even a closed interval of length at least 1 cannot be reconstructed using two test sets. Finally, using randomly constructed test sets, we prove that an element of a reasonably nice k-dimensional family of geometric objects can be reconstructed using 2k+1 test sets. An example from algebraic topology shows that 2k+1 is sharp in general.