Frame bounds for subspaces of L 2(R)

Yeon Hyang Kim, Matthew J. Petro, Andrew Dugowson, Robert Fraser

Research output: Contribution to journalArticlepeer-review

Abstract

Let H be a separable Hilbert space with inner product (•,•). We say a set {f k: k ∈ ℤ} is a (fundamental) frame for H if there exist A,B > 0 such that for each f ∈ H, A ∥ f ∥ 2 H ≤ Σ n∈ℤ|(f,f k)| 2≤B∥f∥ 2 H In case {f k: k ∈ ℤ} is a frame for the subspace span{f k: k ∈ ℤ}, we say that {f k: k ∈ ℤ} is a frame sequence. A Weyl-Heisenberg frame sequence is a frame sequence which is generated by translated and modulated versions of L 2-functions. In this paper, we characterize Weyl-Heisenberg frame sequences using infinite Hermitian matrices and obtain the optimal frame bounds in terms of the operator norms of these matrices. This work is inspired by a paper by Casazza and Christensen, where sufficient conditions for a Weyl-Heisenberg system to be a frame sequence are studied.

Original languageEnglish
Pages (from-to)385-401
Number of pages17
JournalInternational Journal of Pure and Applied Mathematics
Volume75
Issue number4
StatePublished - 2012

Keywords

  • Frame sequences
  • Hermitian matrix
  • Modulation frames
  • Weyl-Heisenberg systems

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