{"title":"海森堡群上的左平移和斜对偶系统","authors":"Santi DAS, Radha RAMAKRİSHNAN, Peter MASSOPUST","doi":"10.33205/cma.1382306","DOIUrl":null,"url":null,"abstract":"In this paper, we characterize the system of left translates $\\{L_{(2k,l,m)}g:k,l,m\\in\\mathbb{Z}\\}$, $g\\in L^2(\\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\\lambda$. Here, $\\mathbb{H}$ denotes the Heisenberg group and $g^\\lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\\{L_{(2k,l,m)}g:k,l,m\\in\\mathbb{Z}\\}$ on $\\mathbb{H}$. This result is also illustrated with an example.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Systems of left translates and oblique duals on the Heisenberg group\",\"authors\":\"Santi DAS, Radha RAMAKRİSHNAN, Peter MASSOPUST\",\"doi\":\"10.33205/cma.1382306\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we characterize the system of left translates $\\\\{L_{(2k,l,m)}g:k,l,m\\\\in\\\\mathbb{Z}\\\\}$, $g\\\\in L^2(\\\\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\\\\lambda$. Here, $\\\\mathbb{H}$ denotes the Heisenberg group and $g^\\\\lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\\\\{L_{(2k,l,m)}g:k,l,m\\\\in\\\\mathbb{Z}\\\\}$ on $\\\\mathbb{H}$. This result is also illustrated with an example.\",\"PeriodicalId\":36038,\"journal\":{\"name\":\"Constructive Mathematical Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2023-11-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Constructive Mathematical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.33205/cma.1382306\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Constructive Mathematical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.33205/cma.1382306","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
在本文中,我们刻画了左平移$\{L_{(2k,l,m)}g:k,l,m\在\mathbb{Z}\}$中,$g\在l ^2(\mathbb{H})$中,根据相应函数$g^\lambda$的扭平移来表示的一个帧序列或Riesz序列。这里,$\mathbb{H}$表示海森堡群,$g^\ λ $表示$g$关于中心变量的傅里叶反变换。Riesz序列的这种特征使我们能够找到一些具体的例子。我们还研究了左平移系统$\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$ on $\mathbb{H}$的斜对偶结构。最后给出了一个算例。
Systems of left translates and oblique duals on the Heisenberg group
In this paper, we characterize the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$, $g\in L^2(\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\lambda$. Here, $\mathbb{H}$ denotes the Heisenberg group and $g^\lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$ on $\mathbb{H}$. This result is also illustrated with an example.