Preconditioning techniques in frame theory and probabilistic frames
نویسندگان
چکیده
These notes have a dual goal. On the one hand we shall give an overview of the recently introduced class of scalable frames. These are (finite) frames with the property that each frame vector can be rescaled in such a way that the resulting frames are tight. This process can be viewed as a preconditioning method for finite frames. In particular, we shall: (1) describe the class of scalable frames; (2) formulate various equivalent characterizations of scalable frames, and relate the scalability problem to the Fritz John ellipsoid theorem. On the other hand, we shall give an overview of a probabilistic interpretation of frames. This point of view allow us to: (4) define probabilistic frames as a generalization of frames and as a subclass of continuous frames; (5) review the properties of certain potential functions whose minimizers are frames with certain optimality properties.
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