  It is an established fact that a finite difference operator approximates a
derivative with a fixed algebraic rate of convergence. Nevertheless, we exhibit
a new finite difference operator and prove it has spectral accuracy. Its rate
of convergence is not fixed and improves with the function's regularity. For
example, the rate of convergence is exponential for analytic functions. Our new
framework is conceptually nonstandard, making no use of polynomial
interpolation, nor any other expansion basis, such as typically considered in
approximation theory. Our new method arises solely from the numerical
manipulation of singular integrals, through an accurate quadrature for Cauchy
Principal Value convolutions. The kernel is a distribution which gives rise to
multi-resolution grid coefficients. The respective distributional finite
difference scheme is spatially structured having stencils of different support
widths. These multi-resolution stencils test/estimate function variations in a
nonlocal fashion, giving rise to a highly accurate distributional finite
difference operator. Computational illustrations are presented, where the
accuracy and roundoff error structure are compared with the respective Fourier
based method. We also compare our method with a recent and popular complex-step
method.
