continuous frames

In this paper, we present a study of relatively connected sublocales. Connected sublocales are relatively connected, not conversely. We study conditions under which relatively connected sublocales are connected. The development of this study is subsequently utilized to characterize what we call C-normal frames. We show that normal frames are C-normal but not conversely. Some results concerning J-frames are presented; amongst other things, we prove that regular continuous frames are rim-compact. A rim-compact J-frame is regular continuous. The latter is used to show that the least compactification of a regular continuous J-frame coincides with its Freudenthal compactification. In turn, this contributes to the known conditions under which the least compactification is perfect.

K-g-frames, as an extension of g-frames and K-frames are one of the active fields in frame theory. In this paper, we  consider continuous K-g-frames which are a generalization of discrete K-g-frames. We give the necessary and sufficient conditions to characterize their duals. For example, the canonical dual  of a given K-g-frame is described by both its frame operator and its alternate duals.

In this paper we consider  (extended) metaplectic representation of the  semidirect product  $G_{mathbb{J}}=mathbb{R}^{2d}timesmathbb{J}$  where $mathbb{J}$ is a closed subgroup of $Sp(d,mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Finally, we rewrite the dual conditions, by using the Wigner distribution and obtain more reconstruction formulas.

In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ with respect to $ mu $ is given. Also, a sufficient condition for a generalized continuous $K$-frame is given.  Further, among other results, we prove that generalized continuous $K$-frames are invariant under a linear  homeomorphism. Finally, keeping in mind the importance of perturbation theory in various branches of applied mathematics, we study perturbation of $K$-frames and obtain conditions for the stability of generalized continuous $K$-frames.

In this paper we introduce continuous g-Bessel multipliers in Hilbert spaces and investigate some of their properties. We provide some conditions under which a continuous g-Bessel multiplier is a compact operator. Also, we show the continuous dependency of continuous g -Bessel multipliers on their parameters.