Frames and semi-frames
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.