{"title":"由展开式给出的矩阵值正定核:严格正定性","authors":"W. Franca, V. Menegatto","doi":"10.7153/mia-2021-24-79","DOIUrl":null,"url":null,"abstract":"Matrix functions of the form (x,y) ∈ Ω × Ω → ∑α Aα fα (x,y) , in which Ω is a nonempty set, the Aα are positive semi-definite matrices of the same fixed order, the fα are complex-valued positive definite kernels on Ω , and the series is convergent for all x and y in Ω define matrix-valued positive definite kernels on Ω . Here, the sum may be multi-indexed, Ω may be endowed with either a topological or a metric structure, and { fα} may inherit properties attached to the setting. In this paper, we present a criterion that establishes an abstract necessary and sufficient condition in order that the kernel is strictly positive definite on Ω . We point some implications and connections of the criterion in some relevant and concrete settings in order to motivate future work on the topic. Mathematics subject classification (2020): 42A82, 42C10, 43A35.","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Matrix-valued positive definite kernels given by expansions: strict positive definiteness\",\"authors\":\"W. Franca, V. Menegatto\",\"doi\":\"10.7153/mia-2021-24-79\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Matrix functions of the form (x,y) ∈ Ω × Ω → ∑α Aα fα (x,y) , in which Ω is a nonempty set, the Aα are positive semi-definite matrices of the same fixed order, the fα are complex-valued positive definite kernels on Ω , and the series is convergent for all x and y in Ω define matrix-valued positive definite kernels on Ω . Here, the sum may be multi-indexed, Ω may be endowed with either a topological or a metric structure, and { fα} may inherit properties attached to the setting. In this paper, we present a criterion that establishes an abstract necessary and sufficient condition in order that the kernel is strictly positive definite on Ω . We point some implications and connections of the criterion in some relevant and concrete settings in order to motivate future work on the topic. Mathematics subject classification (2020): 42A82, 42C10, 43A35.\",\"PeriodicalId\":49868,\"journal\":{\"name\":\"Mathematical Inequalities & Applications\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Inequalities & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7153/mia-2021-24-79\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Inequalities & Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7153/mia-2021-24-79","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
形式为(x,y)∈Ω × Ω→∑α a α fα (x,y)的矩阵函数,其中Ω为非空集,a α为同定阶的正半定矩阵,fα为Ω上的复值正定核,且级数收敛于Ω中所有x和y在Ω上定义了矩阵值正定核。在这里,和可以是多索引的,Ω可以被赋予拓扑结构或度量结构,{fα}可以继承附加到设置的属性。本文给出了在Ω上核是严格正定的一个抽象充分必要条件的判据。我们在一些相关和具体的设置中指出了标准的一些含义和联系,以激励未来关于该主题的工作。数学学科分类(2020):42A82、42C10、43A35。
Matrix-valued positive definite kernels given by expansions: strict positive definiteness
Matrix functions of the form (x,y) ∈ Ω × Ω → ∑α Aα fα (x,y) , in which Ω is a nonempty set, the Aα are positive semi-definite matrices of the same fixed order, the fα are complex-valued positive definite kernels on Ω , and the series is convergent for all x and y in Ω define matrix-valued positive definite kernels on Ω . Here, the sum may be multi-indexed, Ω may be endowed with either a topological or a metric structure, and { fα} may inherit properties attached to the setting. In this paper, we present a criterion that establishes an abstract necessary and sufficient condition in order that the kernel is strictly positive definite on Ω . We point some implications and connections of the criterion in some relevant and concrete settings in order to motivate future work on the topic. Mathematics subject classification (2020): 42A82, 42C10, 43A35.
期刊介绍:
''Mathematical Inequalities & Applications'' (''MIA'') brings together original research papers in all areas of mathematics, provided they are concerned with inequalities or their role. From time to time ''MIA'' will publish invited survey articles. Short notes with interesting results or open problems will also be accepted. ''MIA'' is published quarterly, in January, April, July, and October.