I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
Here the kernel $K$ is positive symmetric with finite smoothness, i.e. the Fourier transform $$\widehat{K}(\omega) \sim ||\omega||^{-\beta - d},$$ where $\beta >0$ is the smoothness parameter, and $d$ is the dimension. According to 'Error estimates and condition numbers for radial basis function interpolation' (Schaback), the minimum eigenvalue $$c n^{-\beta/d}\le \lambda^n_n \le C n^{-\beta/d} \quad \mbox{for some }c, C >0.$$ My question is whether there is any result regarding the convergence of $n^{\beta/d} \lambda_n^n$ ? i.e. $n^{\beta/d} \lambda_n^n \rightarrow A$ as $n \rightarrow \infty$ ? Is there any way to prove this result ?
There is a closely related topic on the eigenvalues of continuous operator, say $Tf: = \int K(x - y)f(y) dy$. The kernel matrix can be regarded as a discretization of the continuum operator. Let $\lambda_1 > \lambda_2 \ldots$ be the eigenvalues of $T$. It is known that $\lambda_i$ can be written as Kolmogorov n-width, and classical results of Joseph Jerome imply that $$\lambda_i \sim Ci^{-(\beta+d)/d} \quad \mbox{for some } C >0.$$ So it is natural to expect a similar result for the kernel matrix. Also there has been some work on quantifying $|\lambda^n_i/n - \lambda_i|$, e.g. 'Accurate Error Bounds for the Eigenvalues of the Kernel Matrix' (Braun). However, the estimates are too large to conclude.
