{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T06:02:33Z","timestamp":1771567353289,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T00:00:00Z","timestamp":1771372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62272079"],"award-info":[{"award-number":["62272079"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62572088"],"award-info":[{"award-number":["62572088"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Young Backbone Teachers in Higher Education Institutions of Henan Province","award":["2025GGJS178"],"award-info":[{"award-number":["2025GGJS178"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we analyze the relationship between relay fusion frames and standard fusion frames in finite-dimensional real Hilbert spaces. We propose an optimal design method for tight relay fusion frames in the setting of orthogonal subspaces. Additionally, we prove the existence of non-trivial relay operators and establish stability results for both subspaces and relay operators, showing that small perturbations preserve the relay fusion frame property with frame bounds converging to the original ones. We also present a sufficient condition for constructing relay fusion frames from scaled operators of existing fusion frames and show that invertible relay operators induce fusion frames.<\/jats:p>","DOI":"10.3390\/axioms15020148","type":"journal-article","created":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T09:22:46Z","timestamp":1771406566000},"page":"148","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Properties of Relay Fusion Frames in Finite Dimensions"],"prefix":"10.3390","volume":"15","author":[{"given":"Jianxia","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Intelligent Engineering, Henan Institute of Technology, Xinxiang 453003, China"}]},{"given":"Zuhua","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Henan Institute of Technology, Xinxiang 453003, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9382-5412","authenticated-orcid":false,"given":"Guoqing","family":"Hong","sequence":"additional","affiliation":[{"name":"School of Science, Henan Institute of Technology, Xinxiang 453003, China"}]},{"given":"Fugen","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]},{"given":"Xinyan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Science, Henan Institute of Technology, Xinxiang 453003, China"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1090\/conm\/706\/14220","article-title":"Fusion frames and distributed sparsity","volume":"Volume 706","author":"Aceska","year":"2018","journal-title":"Frames and Harmonic Analysis"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1090\/conm\/345\/06242","article-title":"Frames of subspaces","volume":"345","author":"Casazza","year":"2003","journal-title":"Contemp. 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An Introduction to Frames and Riesz Bases, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-8224-8"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"101596","DOI":"10.1016\/j.acha.2023.101596","article-title":"Representation of operators using fusion frames","volume":"68","author":"Balazs","year":"2024","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s00009-019-1318-x","article-title":"Non-orthogonal fusion frames of an analytic operator and application to a one-dimensional wave control system","volume":"16","author":"Ellouz","year":"2019","journal-title":"Mediterr. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1016\/j.jmaa.2008.01.062","article-title":"Some properties of frames of subspaces obtained by operator theory methods","volume":"343","author":"Ruiz","year":"2008","journal-title":"J. Math. Anal. 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