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

Abstract

The estimate $\lambda _n = o({1 /n})$ obtained by H. Weyl (1912) for the nth largest in modulus eigenvalue $\lambda _n $ of any symmetric Fredholm operator on $L^2 [0,1]^2 $ with kernel in $C^1 [0,1]^4 $ is shown to be best possible in the sense that for any increasing sequence $\alpha _n \to \infty $ there exist such operators whose nth eigenvalue is not $o({1 / {n\alpha _n }})$. The construction of the counterexample makes use of Rudin–Shapiro polynomials. The corresponding result for positive definite operators is proved with a simpler counterexample. The methods generalise to the case $L^2 [0,1]^m (m \geqq 3)$ without further difficulty.