  We study the spectral properties of infinitely smooth multivariate kernel
matrices when the nodes form a single cluster. We show that the geometry of the
nodes plays an important role in the scaling of the eigenvalues of these kernel
matrices. For the multivariate Dirichlet kernel matrix, we establish a
criterion for the sampling set ensuring precise scaling of eigenvalues.
Additionally, we identify specific sampling sets that satisfy this criterion.
Finally, we discuss the implications of these results for the problem of
super-resolution, i.e. stable recovery of sparse measures from bandlimited
Fourier measurements.
