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Reconstructing currents from their projections

Aidan Backus

Abstract

We prove an inversion formula for the exterior $k$-plane transform. As a consequence, we show that if $m < k$ then an $m$-current in $\mathbf R^n$ can be reconstructed from its projections onto $\mathbf R^k$, which proves a conjecture of Solomon.

Reconstructing currents from their projections

Abstract

We prove an inversion formula for the exterior -plane transform. As a consequence, we show that if then an -current in can be reconstructed from its projections onto , which proves a conjecture of Solomon.

Paper Structure

This paper contains 5 theorems, 34 equations.

Key Result

Theorem 1

For every $\alpha \in \mathscr S$ and $P \in G$, there exists a smooth $m$-form $\alpha_P$ on $P$ such that for every $x \in \mathbf{R}^n$, and for every compactly supported $m$-current $T$ on $\mathbf{R}^n$,

Theorems & Definitions (10)

  • Theorem 1
  • Definition 2: Solomon11
  • Lemma 3: Solomon11
  • Theorem 4: inversion formula
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['thm: decomposition of currents']}