duals of controlled k-g-frames and some equalities

Controlled frames for spherical wavelets were introduced to get a numerically more efficient approximationalgorithm. the notion of k-frames has been introduced to study atomic systems with respect to a boundedlinear operator k on a hilbert space h. in this paper motivating the concept of duals of controlled and kframes, the concept of controlled k-g-dual pair and dual operator are introduced and some known equalitiesand inequalities satisfied with duals are presented. the applications of these duals are in signal and imageprocessing