Circulant preconditioners constructed from kernels

Circulant preconditioners for Hermitian Toeplitz systems are considered from the viewpoint of function theory. It is shown that some well-known circulant preconditioners can be derived from convoluting the generating function f of the Toeplitz matrix with famous kernels like the Dirichlet and the Fejer kernels. Several circulant preconditioners are then constructed using this approach. Finally, it is proven that if the convolution product converges of f uniformly, then the circulant preconditioned Toeplitz systems will have a clustered spectrum.