Eigenvalue asymptotics for a class of multi-variable Hankel matrices

IF 0.3 Q4 MATHEMATICS Concrete Operators Pub Date : 2022-06-25 DOI:10.1515/conop-2022-0137
Christos Panagiotis Tantalakis
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引用次数: 0

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.
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一类多变量汉克尔矩阵的特征值渐近性
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。
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来源期刊
Concrete Operators
Concrete Operators MATHEMATICS-
CiteScore
1.00
自引率
16.70%
发文量
10
审稿时长
22 weeks
期刊最新文献
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