On the Sharpness of Weyl’s Estimate for Eigenvalues of Smooth Kernels

It is shown that Weyl’s estimate of $o({1 / {n^{{3 / 2}} }} )$ for the eigenvalues of any symmetric continuously differentiable kernel on a bounded region cannot be improved to $o({1 / {n^{{3 / 2}} }}\alpha _n )$ for any increasing $\alpha _n \to \infty $. The counter-example is constructed from Rudin–Shapiro polynomials.