{"title":"一类多变量汉克尔矩阵的特征值渐近性","authors":"Christos Panagiotis Tantalakis","doi":"10.1515/conop-2022-0137","DOIUrl":null,"url":null,"abstract":"Abstract A one-variable Hankel matrix H a {H}_{a} is an infinite matrix H a = [ a ( i + j ) ] i , j ≥ 0 {H}_{a}={\\left[a\\left(i+j)]}_{i,j\\ge 0} . Similarly, for any d ≥ 2 d\\ge 2 , a d d -variable Hankel matrix is defined as H a = [ a ( i + j ) ] {H}_{{\\bf{a}}}=\\left[{\\bf{a}}\\left({\\bf{i}}+{\\bf{j}})] , where i = ( i 1 , … , i d ) {\\bf{i}}=\\left({i}_{1},\\ldots ,{i}_{d}) and j = ( j 1 , … , j d ) {\\bf{j}}=\\left({j}_{1},\\ldots ,{j}_{d}) , with i 1 , … , i d , j 1 , … , j d ≥ 0 {i}_{1},\\ldots ,{i}_{d},{j}_{1},\\ldots ,{j}_{d}\\ge 0 . For γ > 0 \\gamma \\gt 0 , Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices H a {H}_{a} with a ( j ) = j − 1 ( log j ) − γ a\\left(j)={j}^{-1}{\\left(\\log j)}^{-\\gamma } , for j ≥ 2 j\\ge 2 , obey the asymptotics λ n ( H a ) ∼ C γ n − γ {\\lambda }_{n}\\left({H}_{a})\\hspace{0.33em} \\sim \\hspace{0.33em}{C}_{\\gamma }{n}^{-\\gamma } , as n → + ∞ n\\to +\\infty , where the constant C γ {C}_{\\gamma } is calculated explicitly. This article presents the following d d -variable analogue. Let γ > 0 \\gamma \\gt 0 and a ( j ) = j − d ( log j ) − γ a\\left(j)={j}^{-d}{\\left(\\log j)}^{-\\gamma } , for j ≥ 2 j\\ge 2 . If a ( j 1 , … , j d ) = a ( j 1 + ⋯ + j d ) {\\bf{a}}\\left({j}_{1},\\ldots ,{j}_{d})=a\\left({j}_{1}+\\cdots +{j}_{d}) , then H a {H}_{{\\bf{a}}} is compact and its eigenvalues follow the asymptotics λ n ( H a ) ∼ C d , γ n − γ {\\lambda }_{n}\\left({H}_{{\\bf{a}}})\\hspace{0.33em} \\sim \\hspace{0.33em}{C}_{d,\\gamma }{n}^{-\\gamma } , as n → + ∞ n\\to +\\infty , where the constant C d , γ {C}_{d,\\gamma } is calculated explicitly.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Eigenvalue asymptotics for a class of multi-variable Hankel matrices\",\"authors\":\"Christos Panagiotis Tantalakis\",\"doi\":\"10.1515/conop-2022-0137\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract A one-variable Hankel matrix H a {H}_{a} is an infinite matrix H a = [ a ( i + j ) ] i , j ≥ 0 {H}_{a}={\\\\left[a\\\\left(i+j)]}_{i,j\\\\ge 0} . Similarly, for any d ≥ 2 d\\\\ge 2 , a d d -variable Hankel matrix is defined as H a = [ a ( i + j ) ] {H}_{{\\\\bf{a}}}=\\\\left[{\\\\bf{a}}\\\\left({\\\\bf{i}}+{\\\\bf{j}})] , where i = ( i 1 , … , i d ) {\\\\bf{i}}=\\\\left({i}_{1},\\\\ldots ,{i}_{d}) and j = ( j 1 , … , j d ) {\\\\bf{j}}=\\\\left({j}_{1},\\\\ldots ,{j}_{d}) , with i 1 , … , i d , j 1 , … , j d ≥ 0 {i}_{1},\\\\ldots ,{i}_{d},{j}_{1},\\\\ldots ,{j}_{d}\\\\ge 0 . For γ > 0 \\\\gamma \\\\gt 0 , Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices H a {H}_{a} with a ( j ) = j − 1 ( log j ) − γ a\\\\left(j)={j}^{-1}{\\\\left(\\\\log j)}^{-\\\\gamma } , for j ≥ 2 j\\\\ge 2 , obey the asymptotics λ n ( H a ) ∼ C γ n − γ {\\\\lambda }_{n}\\\\left({H}_{a})\\\\hspace{0.33em} \\\\sim \\\\hspace{0.33em}{C}_{\\\\gamma }{n}^{-\\\\gamma } , as n → + ∞ n\\\\to +\\\\infty , where the constant C γ {C}_{\\\\gamma } is calculated explicitly. This article presents the following d d -variable analogue. Let γ > 0 \\\\gamma \\\\gt 0 and a ( j ) = j − d ( log j ) − γ a\\\\left(j)={j}^{-d}{\\\\left(\\\\log j)}^{-\\\\gamma } , for j ≥ 2 j\\\\ge 2 . If a ( j 1 , … , j d ) = a ( j 1 + ⋯ + j d ) {\\\\bf{a}}\\\\left({j}_{1},\\\\ldots ,{j}_{d})=a\\\\left({j}_{1}+\\\\cdots +{j}_{d}) , then H a {H}_{{\\\\bf{a}}} is compact and its eigenvalues follow the asymptotics λ n ( H a ) ∼ C d , γ n − γ {\\\\lambda }_{n}\\\\left({H}_{{\\\\bf{a}}})\\\\hspace{0.33em} \\\\sim \\\\hspace{0.33em}{C}_{d,\\\\gamma }{n}^{-\\\\gamma } , as n → + ∞ n\\\\to +\\\\infty , where the constant C d , γ {C}_{d,\\\\gamma } is calculated explicitly.\",\"PeriodicalId\":53800,\"journal\":{\"name\":\"Concrete Operators\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Concrete Operators\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/conop-2022-0137\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2022-0137","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
摘要单变量汉克尔矩阵H A {H_a}是一个无限矩阵H A = [A (i + j)] i,j≥0 {H_a}= {}{}{\left[a\left(i+j)]} _i,j{\ge 0}。同样,对于任意d≥2 d\ge 2, d变量汉克尔矩阵定义为Ha = [a (i + j)] {H_}={{\bf{a}}}\left[{\bf{a}}\left({\bf{i}}+{\bf{j}})],其中i = (i 1,…,i d) {\bf{i}}=\left ({i_1}, {}\ldots,{i_d}),j = (j 1,…,j d) {=}{\bf{j}}\left ({j_1}, {}\ldots,{j_d}),其中i 1,…,i d,j 1,…,j d≥0 {i_1}, {}{}\ldots,{i_d},{j_1}, {}{}\ldots,{j_d}{}\ge 0。对于γ > \gamma\gt 0, Pushnitski和Yafaev证明了a (j)=j−1 (log j)−γ a {}{}\left (j)=j^-1 {}{}{\left (\log j}){^-\gamma,}对于j≥2 j \ge 2,服从渐近性λ n (H a) ~ C γ n−γ {\lambda _n}{}\left (H_a){}{}\hspace{0.33em}\sim _\hspace{0.33em}{C}{\gamma n}{^}-{\gamma,}为n→+∞n\to + \infty,其中常数C γ {C_}{\gamma显式计算。本文介绍了下面的d变量模拟。设γ > 0}\gamma\gt 0, a (j)=j−d (log j)−γ a \left (j)={j}^{-d}{\left (\log j)}^{- \gamma,}对于j≥2 j\ge 2。如果a (j 1,…,j d)=a (j 1+⋯+j d) {\bf{a}}\left ({j_1}, {}\ldots,{j_d})=a {}\left ({j_1}+ {}\cdots +{j_d}),则H a {H_}是紧致的,其特征值遵循渐近性λ n (H a) ~ C d, γ n−{γ }{{\bf{a}}}{\lambda _n}{}\left (H_{)}{{\bf{a}}}\hspace{0.33em}\sim _d\hspace{0.33em}{C}, {\gamma n}{^}-{\gamma,为}n→+∞n\to + \infty,其中常数C d, γ {C_d}, {\gamma是显式}计算的。
Eigenvalue asymptotics for a class of multi-variable Hankel matrices
Abstract A one-variable Hankel matrix H a {H}_{a} is an infinite matrix H a = [ a ( i + j ) ] i , j ≥ 0 {H}_{a}={\left[a\left(i+j)]}_{i,j\ge 0} . Similarly, for any d ≥ 2 d\ge 2 , a d d -variable Hankel matrix is defined as H a = [ a ( i + j ) ] {H}_{{\bf{a}}}=\left[{\bf{a}}\left({\bf{i}}+{\bf{j}})] , where i = ( i 1 , … , i d ) {\bf{i}}=\left({i}_{1},\ldots ,{i}_{d}) and j = ( j 1 , … , j d ) {\bf{j}}=\left({j}_{1},\ldots ,{j}_{d}) , with i 1 , … , i d , j 1 , … , j d ≥ 0 {i}_{1},\ldots ,{i}_{d},{j}_{1},\ldots ,{j}_{d}\ge 0 . For γ > 0 \gamma \gt 0 , Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices H a {H}_{a} with a ( j ) = j − 1 ( log j ) − γ a\left(j)={j}^{-1}{\left(\log j)}^{-\gamma } , for j ≥ 2 j\ge 2 , obey the asymptotics λ n ( H a ) ∼ C γ n − γ {\lambda }_{n}\left({H}_{a})\hspace{0.33em} \sim \hspace{0.33em}{C}_{\gamma }{n}^{-\gamma } , as n → + ∞ n\to +\infty , where the constant C γ {C}_{\gamma } is calculated explicitly. This article presents the following d d -variable analogue. Let γ > 0 \gamma \gt 0 and a ( j ) = j − d ( log j ) − γ a\left(j)={j}^{-d}{\left(\log j)}^{-\gamma } , for j ≥ 2 j\ge 2 . If a ( j 1 , … , j d ) = a ( j 1 + ⋯ + j d ) {\bf{a}}\left({j}_{1},\ldots ,{j}_{d})=a\left({j}_{1}+\cdots +{j}_{d}) , then H a {H}_{{\bf{a}}} is compact and its eigenvalues follow the asymptotics λ n ( H a ) ∼ C d , γ n − γ {\lambda }_{n}\left({H}_{{\bf{a}}})\hspace{0.33em} \sim \hspace{0.33em}{C}_{d,\gamma }{n}^{-\gamma } , as n → + ∞ n\to +\infty , where the constant C d , γ {C}_{d,\gamma } is calculated explicitly.