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

by E. García-Sánchez and D. de Hevia and M. Taylor

(Report number 2025-32)

Abstract
The purpose of this article is to address an open problem posed by Freeman-Oikhberg-Pineau-T. (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.

Keywords: Phase retrieval; stable phase retrieval; C(K)-space.

@Techreport{GdT25_1153,
  author = {E. García-Sánchez and D. de Hevia and M. Taylor},
  title = {On the existence of large subspaces of C(K) that perform stable phase retrieval},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-32.pdf },
  year = {2025}
}

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