Frames and semi-frames in mathematical physics

Antoine, Jean-Pierre [UCL]

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if the sequence satisfies only the upper (resp. lower) frame bound. 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. Both concepts are extended to the continuous case. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing or mathematical physics. We present some results concerning the duality between lower and upper semi-frames, as well as two examples involving wavelets/coherent states.