Approximate singular values of the fractional difference and summation operators

We obtain sharp bounds on the singular values of the fractional difference and summation operators on Rn. 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.