Frames, semi-frames, and Hilbert scales
(2012) Numerical Functional Analysis and Optimization : an international journal — Vol. 33, n° 7-9, p. 736-769 (2012)
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- Authors
Antoine, Jean-Pierre 
- Abstract
- 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.
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Citations
Antoine, J.-P., & Balazs, P. (2012). Frames, semi-frames, and Hilbert scales. Numerical Functional Analysis and Optimization : an international journal, 33(7-9), 736-769. https://doi.org/10.1080/01630563.2012.682128 (Original work published 2012)