2012/03/02 by J-P. Antoine, P. Balazs · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #msc:42C15 #msc:42C40 #msc:46C50 #msc:47A70 #msc:65T60
published as Numer. Funct. Anal. Optimiz. 33 (2012) 736-769 · 27 pages; Numerical Functional Analysis and Optimization, 33 (2012) in press. arXiv admin note: substantial text overlap with arXiv:1101.2859
arxiv created 2012/03/02 · arxiv updated 2012/10/12
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. 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. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.