Article

Journal of Fourier Analysis and Applications

, Volume 7, Issue 4, pp 419-433

Subspace Weyl-Heisenberg frames

  • Jean-Pierre GabardoAffiliated withDepartment of Mathematics and Statistics, McMaster University
  • , Deguang HanAffiliated withDepartment of Mathematics, University of Central Florida

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Abstract

A Weyl-Heisenberg frame (WH frame) for L2(ℝ) allows every square integrable function on the line to be decomposed into the infinite sum of linear combination of translated and modulated versions of a fixed function. Some sufficient conditions for g ∈ L2(ℝ) to be a subspace Weyl-Heisenberg frame were given in a recent work [3] by Casazza and Christensen. Obviously every invariant subspace (under translation and modulation) is cyclic if it has a subspace WH frame. In the present article we prove that the cyclicity property is also sufficient for a subspace to admit a WH frame. We also investigate the dilation property for subspace Weyl-Heisenberg frames and show that every normalized tight subspace WH frame can be dilated to a normalized tight WH frame which is “maximal” In other words, every subspace WH frame is the compression of a WH frame which cannot be dilated anymore within the L2(ℝ) space.

Math Subject Classifications

42A99 42C99 46N99 47N40 47N99

Keywords and Phrases

group-like unitary systems von Neumann algebras Weyl-Heisenberg systems subspace WH frames