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 have introduced $(p,q)$-Bessel and $(p,q)$-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $q$-Bessel sequence 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 have constructed a large number of examples of finite measures $\mu$ which admit infinite $(p,q)$-Bessel measures $\nu$. We have showed 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 the convolutions of probability measures, one can obtain other $(p,q)$-Bessel/frame measures for $\mu$. We have presented a general way of constructing a $(p,q)$-Bessel/frame measure for a given measure

Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P∈Ξ and X∈P, the one point compactification of topological space X belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T1, TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator