Eigenvalue distribution of positive definite kernels on unbounded domains
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.
scientific article; zbMATH DE number 5149190