Pseudo-duals of continuous frames in Hilbert spaces

In this paper, we get some properties and characterizations of Q-pseudo-duals, Q-duals and Q-approximate duals of continuous frames, where Q is a bounded operator inserted between the synthesis and analysis operators of the continuous Bessel mappings. At first, some equivalent conditions for a continuous Bessel mapping to possess a Q-pseudo-dual are presented, then nearly Parseval frames and scalable frames for vector-valued functions defined on a measure space are focused and some relationships between them and approximate duals of continuous frames are collected. Also, the stability of Q-approximate duals under different kinds of perturbations are considered and it is shown that there are close relationships between the concept of closeness (also, nearness) of continuous frames and the approximate duality. Moreover, some results for Riesz-type frames and Gabor frames are obtained.