In this paper, we introduced and characterized the controlled $g$-duals of a frame in separable Hilbert space $\mathcal{H}$ . Afterwards, obtained new $C$-controlled $g$-dual frames from given frames. addition, approximation for was defined some their properties were investigated. Finally, relationship between approximately dual $g$-dual.

‎in this paper‎, ‎first we develop the duality concept for $g$-bessel sequences‎ ‎and bessel fusion sequences in hilbert spaces‎. ‎we obtain some results about dual‎, ‎pseudo-dual ‎and approximate dual of frames and fusion frames‎. ‎we also expand every $g$-bessel ‎sequence to a frame by summing some elements‎. ‎we define the restricted isometry property for ‎$g$-frames and generalize some resu...

In this technical report we build Discrete Wavelet Frames on the sphere S, discretizing the existing Spherical Continuous Wavelet Transform (CWT). We first explore the spherical half-continuous frames, i.e. where the position remains a continuous variable; and then we proceed to the fully discrete frames. We introduce the notion of controlled frames, which reflects the particular nature of the ...