{"title":"Scalability of Generalized Frames for Operators","authors":"Varinder Kumar, Sapna Malhotra, Nikhil Khanna","doi":"10.1155/2024/8358987","DOIUrl":null,"url":null,"abstract":"In this paper, the Parseval <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>-frames are constructed from a given <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frame by scaling the elements of the <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frame with the help of diagonal operators, and these frames are named scalable <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frames. Also, we prove some properties of scalable <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frames and construct new scalable <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frames from a given <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frame. The necessary and sufficient conditions for a <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frame to be scalable are given. Further, equivalent conditions for the scalability of <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frames and the <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span>frames induced by <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frames are obtained. Finally, it is shown that the direct sum of two scalable <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frames is again a scalable <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>-</span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.52435 9.39034\" width=\"7.52435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-104\"></use></g></svg>-frame for some suitable bounded linear operator <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.95144 8.68572\" width=\"9.95144pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-76\"></use></g></svg>.</span>","PeriodicalId":15840,"journal":{"name":"Journal of Function Spaces","volume":"27 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Function Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/8358987","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, the Parseval --frames are constructed from a given --frame by scaling the elements of the --frame with the help of diagonal operators, and these frames are named scalable --frames. Also, we prove some properties of scalable --frames and construct new scalable --frames from a given --frame. The necessary and sufficient conditions for a --frame to be scalable are given. Further, equivalent conditions for the scalability of --frames and the -frames induced by --frames are obtained. Finally, it is shown that the direct sum of two scalable --frames is again a scalable --frame for some suitable bounded linear operator .
期刊介绍：
Journal of Function Spaces (formerly titled Journal of Function Spaces and Applications) publishes papers on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. As well as original research, Journal of Function Spaces also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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