Continuous g ‐frame and g ‐Riesz sequences in Hilbert spaces

A continuous g ‐frame is a generalization of g ‐frames and continuous frames, but they behave much differently from g ‐frames due to the underlying characteristic of measure spaces. Now, continuous g ‐frames have been extensively studied, while continuous g ‐sequences such as continuous g ‐frame sequence, g ‐Riesz sequences, and continuous g ‐orthonormal systems have not. This paper addresses continuous g ‐sequences. It is a continuation of Zhang and Li, in Numer. Func. Anal. Opt., 40 (2019), 1268‐1290, where they dealt with g ‐sequences. In terms of synthesis and Gram operator methods, we in this paper characterize continuous g ‐Bessel, g ‐frame, and g ‐Riesz sequences, respectively, and obtain the Pythagorean theorem for continuous g ‐orthonormal systems. It is worth that our results are similar to the case of g ‐ones, but their proofs are nontrivial. It is because the definition of continuous g ‐sequences is different from that of g ‐sequences due to it involving general measure space.