Approximate singular values of the fractional difference and summation operators

Abstract

We obtain sharp bounds on the singular values of the fractional difference and summation operators on R n . These bounds allow us to establish that the convergence rates of the distributions of the singular values of these operators are O(1/ n). Since the fractional difference operator of order α is associated with the Toeplitz matrix with Fisher–Hartwig symbol (2 − 2 cos u) α , α > 0, we are able to obtain similar bounds on the eigenvalues of this Toeplitz matrix and a similar result on the convergence rate of the distribution of its eigenvalues.