{"title":"Quaternionic free metaplectic transformation","authors":"Mohra Zayed, Youssef El Haoui","doi":"10.1002/mma.10573","DOIUrl":null,"url":null,"abstract":"<p>The free metaplectic transform (FMT), a generalized form of the linear canonical transform (LCT), has proven to be a useful analytical tool in signal processing applications. This paper aims to generalize the FMT to a quaternionic framework involving the two-dimensional signals. We further study some properties that correspond to those of the standard ones, including linearity, uniform continuity, inversion, and Parseval's identity for this new integral transform, which we coin as the quaternionic free metaplectic transform (QFMT). Furthermore, utilizing the relationship between the general quaternionic Fourier transform and the QFMT, several uncertainty principles (UPs) for the QFMT are established, including the Heisenberg–Weyl UP, Hardy UP, logarithmic UP, Donoho–Stark UP, and entropic UP. We expect that this paper will open up avenues of promising research and applications involving this new transformation.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 4","pages":"4740-4756"},"PeriodicalIF":2.1000,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.10573","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The free metaplectic transform (FMT), a generalized form of the linear canonical transform (LCT), has proven to be a useful analytical tool in signal processing applications. This paper aims to generalize the FMT to a quaternionic framework involving the two-dimensional signals. We further study some properties that correspond to those of the standard ones, including linearity, uniform continuity, inversion, and Parseval's identity for this new integral transform, which we coin as the quaternionic free metaplectic transform (QFMT). Furthermore, utilizing the relationship between the general quaternionic Fourier transform and the QFMT, several uncertainty principles (UPs) for the QFMT are established, including the Heisenberg–Weyl UP, Hardy UP, logarithmic UP, Donoho–Stark UP, and entropic UP. We expect that this paper will open up avenues of promising research and applications involving this new transformation.
期刊介绍：
Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome.
Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted.
Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.
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