Continuous $ k $-Frames and their Dual in Hilbert Spaces

The notion of $k$-frames was recently introduced by Gu avruc ta in Hilbert  spaces to study atomic systems with respect to a bounded linear operator. A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous super positions. In this manuscript, we construct a continuous $k$-frame, so called c$k$-frame along with an atomic system for this version of frames. Also we introduce a new method for obtaining the dual of a c$k$-frame and prove some new results about it.