Fariba Zeinal Zadeh Farhadi

arXiv2019

Frame measures for infinitely many measures

For every frame spectral measure $ μ$, there exists a discrete measure $ ν$ as a frame measure. Since if $ μ$ is not a frame spectral measure, then there is not any general statement about the existence of frame measures $ ν$ for $ μ$, we were motivated to examine Bessel and frame measures. We construct infinitely many measures $ μ$ which admit frame measures $ ν$, and we show that there exist infinitely many frame spectral measures $ μ$ such that besides having a discrete frame measure, they admit continuous frame measures too.

math.FA↗

arXiv2019

Generalized Bessel and Frame Measures

Considering a finite Borel measure $ μ$ on $ \mathbb{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.

math.FA↗