Continuous Welch bounds with Applications

Abstract

Let (Ω, µ) be a measure space and {τ α } α∈Ω be a normalized continuous Bessel family for a finite dimensional Hilbert space H of dimension d. If the diagonal ∆ := {(α, α) : α ∈ Ω} is measurable in the measure space Ω × Ω, then we show that supThis improves 47 years old celebrated result of Welch [IEEE Transactions on Information Theory, 1974 ].We introduce the notions of continuous cross correlation and frame potential of Bessel family and give applications of continuous Welch bounds to these concepts. We also introduce the notion of continuous Grassmannian frames.

Modular Welch Bounds with Applications

“…Remark 3.14. Using the Kasparov theory of integration in Hilbert C*-modules [43,81] over Lie groups [13,22], we can derive results obtained in [48] for Hilbert C*-modules (in a similar line) whenever the measure space is a Lie group and the collection is a modular continuous Bessel family defined as follows.…”

“…Here we list some sample results, conjectures and concepts done in [48] for Hilbert C*-modules. In the remaining part, (G, µ G ) is a Lie group with Haar measure.…”

“…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.…”

Discrete and Continuous Welch Bounds for Banach Spaces with Applications

“…Theorem 1.2 has been recently generalized in full generality for Hilbert spaces and σ-finite measure spaces by the author in [43].…”

“…Theorem 1.3. [43] Let (Ω, µ) be a measure space and {τ α } α∈Ω be a normalized continuous Bessel family for H of dimension d. If the diagonal ∆ := {(α, α) : α ∈ Ω} is measurable in the measure space Ω × Ω,…”