Generalized Bessel and Frame Measures

Considering a finite Borel measure $\mu$ on $\mathbb{R}^d$, a pair of conjugate exponents $p, q$, and a compatible semi-inner product on $L^p(\mu)$, 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 $L^p(\mu)$. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $\mu$. We construct a large number of examples of finite measures $\mu$ which admit infinite $(p,q)$-Bessel measures $\nu$. We show that if $\nu$ is a $(p,q)$-Bessel/frame measure for $\mu$, then $\nu$ is $\sigma$-finite and it is not unique. In fact, by using convolutions of probability measures, one can obtain other $(p,q)$-Bessel/frame measures for $\mu$. We present a general way of constructing a $(p,q)$-Bessel/frame measure for a given measure.