Spectral Properties of Numerical Differentiation

TL;DR

This work analyzes numerical differentiation on grids with an arbitrary number of nodes by performing spectral analyses of the corresponding weight coefficients. In the infinite-point limit, it derives closed-form weight sequences $\alpha_m^{(1)}$ and $\alpha_m^{(2)}$, yielding Fourier-domain expressions $\beta_1(\omega)$ and $\beta_2(\omega)$ that make $f'(x)$ and $f''(x)$ effectively exact for all frequencies except the Nyquist edge for the first derivative, while the second derivative remains exact at that edge. It also investigates differentiation from odd-number-point grids, obtaining a half-point scheme with a piecewise spectrum that differentiates slow signals up to $\omega_{\max}/2$ and envelope functions at higher frequencies. A one-sided first-derivative formula is derived and analyzed spectrally, revealing reduced accuracy relative to central schemes but potential utility in time-stepping and differential-equation solvers. Overall, the results provide explicit, frequency-aware weight constructions for high-precision differentiation and insights into their applicability in numerical analysis and theoretical physics contexts.

Abstract

We study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. We investigate the case of the infinite number of points in the formulae for the calculation of the first and the second derivatives. The spectra of the corresponding weight coefficients sequences are obtained. We examine the first derivative calculation of a function given in odd-number points and analyze the spectra of the weight coefficients sequences in the cases of both finite and infinite number of nodes. We derive the one-sided approximation for the first derivative and examine its spectral properties.

Figures (4)

• Figure 1: The spectra of differentiating filters, (a) $\alpha_{m}^{(1)}$ and (b) $\alpha_{m}^{(2)}$, in the case of infinite number of points.
• Figure 2: (a) The spectra of various sequences $\alpha_{m}^{(1/2)}(n)$ at $N=2000$. (b) The example of the use of the sequence $\alpha^{(1/2)}_m(n)$. A rapidly oscillating function corresponds to the dashed lines, its envelope functions are shown by the dotted lines.
• Figure 3: The spectrum of the sequence $\alpha_{m}^{(1/2)}$ in the case of infinite number of points.
• Figure 4: The imaginary parts (a) and the real parts (b) of the spectra of various sequences $a_{m}^{(1)}(n)$ at $N=2000$.