Generalized Bessel and Frame Measures

2019/02/18 by Farhadi, Fariba Zeinal Zadeh, Asgari, Mohammad Sadegh, Mardanbeigi, Mohammad Reza +1
#28A99 #42C15 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1902.06434

Abstract

Considering a finite Borel measure μ on ℝ^d , a pair of conjugate exponents p, q , and a compatible semi-inner product on L^p(μ) , we introduce (p,q) -Bessel and (p,q) -frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we define notions of q -Bessel and q-frame in the semi-inner product space L^p(μ) . Every finite Borel measure ν is a (p,q)-Bessel measure for a finite measure μ. We construct a large number of examples of finite measures μ which admit infinite (p,q) -Bessel measures ν. We show that if ν is a (p,q) -Bessel/frame measure for μ, then ν is σ-finite and it is not unique. In fact, by using convolutions of probability measures, one can obtain other (p,q) -Bessel/frame measures for μ. We present a general way of constructing a (p,q) -Bessel/frame measure for a given measure.

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