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Continuous g ‐frame and g ‐Riesz sequences in Hilbert spaces
Author(s) -
Zhang Yan,
Li YunZhang
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6191
Subject(s) - mathematics , orthonormal basis , hilbert space , measure (data warehouse) , sequence (biology) , pure mathematics , operator (biology) , absolute continuity , frame (networking) , discrete mathematics , telecommunications , biochemistry , chemistry , physics , repressor , quantum mechanics , database , biology , computer science , transcription factor , gene , genetics
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.