Eigenvalue distribution of positive definite kernels on unbounded domains
type: Item
instance of: scholarly article
title: Eigenvalue distribution of positive definite kernels on unbounded domains
zbMATH Open document ID: 1132.45002
author: A. C. Paixão
author: Jorge Buescu
published in: Integral Equations and Operator Theory
publication date: 2007-05-03T00:00:00Z
review text: Results are obtained on the asymptotic behavior of eigenvalues of the integral operator \(K: L^2(I)\to L^2(I)\) with a positive definite continuous kernel \(k(x,y)\) where \(I\) is a real compact interval are extended to the case \(I\) is unbounded. Assume without loss of generality \(I=[0, \infty)\), and \(k(x, y): I^2\to\mathbb C\) is a positive definite kernel such that \(k(x, y)\) is continuous on \(I^2\), \(k(x,x)\in L^1(I)\) and \(k(x, x)\to 0\) as \(x\to\infty\). As shown by \textit{J. Buescu} [J. Math. Anal. Appl. 296, No. 1, 244--255 (2004; Zbl 1068.47060)] \(K\) is a Hilbert-Schmidt operator, and so its spectrum consists of a sequence \(\{\lambda_n: n\geq 1\}\) of eigenvalues \(\geq 0\), arranged in the decreasing order and repeated according to multiplicities. The eigenfunctions corresponding to nonzero eigenvalues of \(K\) are uniformly continuous on \(I\) and vanish at infinity. Moreover, the classical Mercer theorem remains valid, that is, the eigenfunction expansion of \(k(x, y)\) converges absolutely and uniformly on \(I^2\). A typical result goes as follows: Let \(m\geq 0\), \(\beta>1\) and the partial derivatives \(\frac{\partial^{i+j}}{\partial x^i\partial y^j}k(x,y)\) exist and are continuous on \(I^2\) for \(0\leq i, j\leq m\). Suppose that \(k_m(x, y)\equiv \frac{\partial^{2m}}{\partial x^m\partial y^m}k(x, y)\) is uniformly continuous with respect to \(y\) on the diagonal. It is shown that if \(\int_L^\infty k(x, x)\, dx=O(1/L^{\beta-1})\) as \(L\to\infty\) and \(\gamma\equiv\frac{(2m+1)\beta}{2m+\beta}\), then \(\lambda_n=O(1/n^\gamma)\). The version for little \(o\) is also given. Faster rates of decay of the eigenvalues are obtained if further smooth conditions are supposed such as \(k_m(x, y)\) is Lip\(^\alpha\) or \(\frac{\partial k_m}{\partial x}\) is uniformly continuous with respect to \(y\) on the diagonal.
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 5149190
zbMATH Keywords: integral operators
zbMATH Keywords: asymptotic behavior of eigenvalues
zbMATH Keywords: Hilbert-Schmidt operator
MaRDI profile type: Publication
full work available at URL: s00020-006-1445-1
OpenAlex ID: W2017645932
DOI: 10.1007/S00020-006-1445-1
description: scientific article; zbMATH DE number 5149190