Frames and semi-frames

Preprint, Article English OPEN
Antoine, Jean-Pierre ; Balazs, Peter (2011)
  • Related identifiers: doi: 10.1088/1751-8113/44/20/205201
  • Subject: Gabor frames | Mathematical Physics | Semi-frames | Mathematics - Functional Analysis | 42C15, 42C40, 65T60 | Unbounded frames | Bessel sequences

Loosely speaking, a semi-frame is a generalized frame for which one of the frame bounds is absent. More precisely, given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. We study mostly upper semi-frames, both in the continuous case and in the discrete case, and give some remarks for the dual situation. In particular, we show that reconstruction is still possible in certain cases.
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