Equiangular frames and generalizations of the Welch bound to dual pairs of frames

Abstract

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Continuous Welch bounds with Applications

“…In a recent work, Christensen, Datta and Kim derived Welch bounds for dual frames [9]. We now extend this result to continuous frames.…”

“…Therefore it is desirable to improve Theorem 1.2.and to get a continuous version of Inequality (1) by replacing maximum by supremum. For the sake of completeness, we note that there are some further refinements of Theorem 1.1, see [9,13,37]. The goal of this article is to derive Theorem 1.1 for arbitrary measure spaces (Theorem 2.7).…”

Discrete and Continuous Welch Bounds for Banach Spaces with Applications

“…There are several theoretical and practical applications of Theorem 1.1 such as in the study of root-meansquare (RMS) absolute cross relation of unit vectors [57], frame potential [7,12,15], correlations [56], codebooks [25], numerical search algorithms [71,72], quantum measurements [58], coding and communications [61,67], code division multiple access (CDMA) systems [41,42], wireless systems [53], compressed sensing [64], 'game of Sloanes' [37], equiangular tight frames [62], etc. Some improvements of Theorem 1.1 has been done in [18,23,68,69]. It is in the paper [22] where the following generalization of Theorem 1.1 has been done for continuous collections.…”

Modular Welch Bounds with Applications

“…Over the time, Theorem 1.1 has been generalized for dual frames, vectors which may not be normalized, continuous frames indexed by complex projective space, continuous Bessel sequences indexed by arbitrary measure spaces, natural number m replaced by arbitrary reals [16,19,20,24,33,48,[83][84][85]. But to the best of our knowledge, Theorem 1.1 is not known for Hilbert C*-modules.…”