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Abstract: A geometric perspective involving Grammian and frame operators is used toderive the entire family of Welch bounds. This perspective unifies a number ofobservations that have been made regarding tightness of the bounds and theirconnections to symmetric k-tensors, tight frames, homogeneous polynomials, andt-designs. In particular. a connection has been drawn between sampling ofhomogeneous polynomials and frames of symmetric k-tensors. It is also shownthat tightness of the bounds requires tight frames. The lack of tight frames insymmetric k-tensors in many cases, however, leads to consideration of sets thatcome as close as possible to attaining the bounds. The geometric derivation isthen extended in the setting of generalized or continuous frames. The Welchbounds for finite sets and countably infinite sets become special cases of thisgeneral setting.

Autor: Somantika Datta, Stephen Howard, Douglas Cochran


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