Sadri, Vahid; Rahimlou, Gholamreza; Ahmadi, Reza; Farfar, Ramazan Zarghami Construction of \(g\)-fusion frames in Hilbert spaces. (English) Zbl 1454.42029 Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23, No. 2, Article ID 2050015, 18 p. (2020). A frame in a Hilbert space is a redundant set of vectors which allow, in a stable way, a representation of each vector in that space. For theory and some applications, see the book [O. Christensen, An introduction to frames and Riesz bases. Boston, MA: Birkhäuser (2003; Zbl 1017.42022)]. \(g\)-frames (or operator-values frames) are generalizations of the usual frames and are equivalent to stable space splittings. In the paper under review, the authors consider generalized fusion frames (or \(g\)-fusion frames) and give their basic properties. Moreover, the authors introduce dual \(g\)-fusion frames and define \(g\)-fusion frame sequence. Some of their interesting properties are given. Beautiful paper! Reviewer: Paşc Găvruţă (Timişoara) Cited in 9 Documents MSC: 42C15 General harmonic expansions, frames 46C99 Inner product spaces and their generalizations, Hilbert spaces 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) Keywords:fusion frame; \(g\)-fusion frame; dual \(g\)-fusion frame; \(g\)-fusion frame sequence Citations:Zbl 1017.42022 PDFBibTeX XMLCite \textit{V. Sadri} et al., Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23, No. 2, Article ID 2050015, 18 p. (2020; Zbl 1454.42029) Full Text: DOI arXiv References: [1] Balazs, P., Antoine, J. P. and Grybos, A., Weighted and controlled frames: mutual relationship and first numerical properties, Int. J. Wavelets Multiresol. Inf. Process.14 (2010) 109-132. · Zbl 1192.42016 [2] Blocsli, H., Hlawatsch, H. F., and Fichtinger, H. G., Frame-Theoretic analysis of oversampled filter bank, IEEE Trans. Signal Process.46 (1998) 3256-3268. [3] Benedetto, J., Powell, A. and Yilmaz, O., Sigma-Delta quantization and finite frames, Acoust. Speech Signal Process.3 (2004) 937-940. [4] Candes, E. J. and Donoho, D. L., New tight frames of curvelets and optimal representation of objects with piecewise \(C^2\) singularities, Comm. Pure and App. Math.57 (2004) 219-266. · Zbl 1038.94502 [5] Casazza, P. G. and Kovac̆ević, J., Equal-norm tight frames with erasures, Adv. Comput. Math.18 (2003) 387-430. · Zbl 1035.42029 [6] Casazza, P. G., and Kutyniok, G., Frames of Subspaces, Contemp. Math.345 (2004) 87-114. · Zbl 1058.42019 [7] Casazza, P. G., Kutyniok, G. and Li, S., Fusion Frames and distributed processing, Appl. comput. Harmon. Anal.25 (2008) 114-132. · Zbl 1258.42029 [8] Christensen, O., An Introduction to Frames and Riesz Bases, 2nd edn. (Birkhäuser, 2016). · Zbl 1348.42033 [9] Diestel, J., Sequences and Series in Banach Spaces (Springer-Verlag, 1984). · Zbl 0542.46007 [10] Duffin, R. J., and Schaeffer, A. C., A class of nonharmonik Fourier series, Trans. Amer. Math. Soc.72 (1952) 341-366. · Zbl 0049.32401 [11] Faroughi, M. H., Ahmadi, R. and Afsar, Z., Z. Some properties of c-frames of subspaces, J. Nonlinear Sci. Appl.1 (2008) 155-168. · Zbl 1171.42017 [12] Faroughi, M. H. and Osgooei, E., C-Frames and C-Bessel Mappings, Bull. Iranian Math. Soc.38 (2012) 203-222. [13] Feichtinger, H. G. and Werther, T., Atomic systems for subspaces, in Proc. SampTA, Orlando, FL, , 2001, pp. 163-165. [14] Găvruţa, L., Frames for operators, Appl. Comp. Harm. Annal.32 (2012) 139-144. · Zbl 1230.42038 [15] Găvruţa, P., On the duality of fusion frames, J. Math. Anal. Appl.333 (2007) 871-879. · Zbl 1127.46016 [16] Hassibi, B., Hochwald, B., Shokrollahi, A. and Sweldens, W., Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory.47 (2001) 2335-2367. · Zbl 1017.94501 [17] Heuser, H., Functional Analysis (John Wiley, 1991). · Zbl 0721.26001 [18] Khayyami, M. and Nazari, A., Construction of continuous g-frames and continuous fusion frames, Sahand Comm. Math. Anal.4 (2016) 43-55. · Zbl 1413.42057 [19] Najati, A., Faroughi, M. H. and Rahimi, A., g-frames and stability of g-frames in Hilbert spaces, Methods Func. Anal. Top.14 (2008) 305-324. · Zbl 1174.46012 [20] Sadri, V. and Ahmadi, R., Some results on frames and g-frames, U. P. B. Sci. Bull. Series A80 (2018) 99-108. · Zbl 1424.42065 [21] Sadri, V., Ahmadi, R., Jafarizadeh, M. A. and Nami, S., Continuous \(k\)-fusion frames in Hilbert spaces, Sahand Comm. Math. Anal.17 (2020) 39-55. [22] Sun, W., G-frames and g-Riesz bases, J. Math. Anal. Appl.326 (2006) 437-452. · Zbl 1129.42017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.