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On the existence of large subspaces of $C(K)$ that perform stable phase retrieval

Enrique García-Sánchez, David de Hevia, Mitchell Taylor

Abstract

The purpose of this article is to address an open problem posed by Freeman-Oikhberg-Pineau-T.~(\textit{Math.~Ann.}~2024) regarding the existence of large subspaces of $C(K)$ that perform stable phase retrieval (SPR). We begin by proving that for both the real and complex fields, the space $C(K)$ admits an infinite-dimensional SPR subspace if and only if the second Cantor-Bendixson derivative $K{''}$ is nonempty. We then show how to construct ``large" SPR subspaces of $C(K)$, where the size of the subspace depends quantitatively on the number of non-trivial Cantor-Bendixson derivatives that the compact Hausdorff space $K$ possesses.

On the existence of large subspaces of $C(K)$ that perform stable phase retrieval

Abstract

The purpose of this article is to address an open problem posed by Freeman-Oikhberg-Pineau-T.~(\textit{Math.~Ann.}~2024) regarding the existence of large subspaces of that perform stable phase retrieval (SPR). We begin by proving that for both the real and complex fields, the space admits an infinite-dimensional SPR subspace if and only if the second Cantor-Bendixson derivative is nonempty. We then show how to construct ``large" SPR subspaces of , where the size of the subspace depends quantitatively on the number of non-trivial Cantor-Bendixson derivatives that the compact Hausdorff space possesses.

Paper Structure

This paper contains 10 sections, 19 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A reformulation of the phase retrieval problem
  3. Summary of the results and an outline of the proof
  4. Summary of the results
  5. Obstructions to phase retrieval and an outline of the proof
  6. Existence of SPR subspaces in complex -spaces
  7. is a universal SPR container of all separable Banach spaces
  8. in the complex setting
  9. Quantitative SPR embeddings of -spaces
  10. A solution to the Question of Freeman et al. for countable ordinals

Key Result

Theorem 1.3

Let $K$ be a compact Hausdorff space. The following statements are equivalent over both the real and complex fields:

Figures (1)

  • Figure 1: Representation of $x^{(n)}$. The points in $[1,\omega^2]$ are ordered from left to right and from the top to the bottom.

Theorems & Definitions (38)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Lemma 2.1
  • proof
  • ...and 28 more