For mutually localized families, the frame property is equivalent to invertibility, injectivity, surjectivity, or closed-range conditions on frame-related operators and on an associated R-dual sequence.
Localization of operator-valued frames
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abstract
We introduce a localization concept for operator-valued frames, where the quality of localization is measured by the associated operator-valued Gram matrix belonging to some suitable Banach algebra. We prove that intrinsic localization of an operator-valued frame is preserved by its canonical dual. Moreover, we show that the series associated to the perfect reconstruction of an operator-valued frame converges not only in the underlying Hilbert space, but also in a whole class of associated (quasi-)Banach spaces. Finally, we apply our results to irregular Gabor g-frames.
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Localized frames without inequalities
For mutually localized families, the frame property is equivalent to invertibility, injectivity, surjectivity, or closed-range conditions on frame-related operators and on an associated R-dual sequence.