Spectral decay for definite kernels and functions

A classical theorem of Weyl [7] guarantees that the eigenvalues, ordered according to decreasing absolute values, of a symmetric kernel of class Cm (m ≥ 0) satisfy λn = o(n−m−½). Reade [5, 6] recently proved that if K is, in addition, positive definite, then λn = o(n−m−1;). He has also in [4] made similar improvements of classical spectral estimates for kernels of class Lip α. James Cochran pointed out to me that allied theorems for trigonometric Fourier coefficients seem to have been neglected in the literature. The trigonometric versions turn out to be elementary; nevertheless, in their conclusions concerning the decreasing rearrangement {f^*(n)} they generalize known results about the behaviour of monotone trigonometric transforms. Furthermore they suggest that the Cm hypothesis of Reade's theorem could be relaxed.