Uniform Approximation of Functions with Discrete Approximation Functionals

Abstract

Hoffman, Kouri, and collaborators have calculated nonrelativistic quantum scattering amplitudes by numerically evaluating Feynman path integrals. They observed that the errors introduced by their numerical scheme were uniform in coordinate space, implying that their scheme accurately reproduces both the shape and the phase of functions. Furthermore, they observed that the size and the uniform nature of the errors were preserved when the functions were allowed to evolve in time under the action of the kinetic energy operator. In this paper it is established that these observed properties of the errors are not numerical artifacts but follow from analytical properties of a general class of approximations that include those of Hoffman, Kouri, and collaborators as a special case. Academic Press

Particular and homogeneous solutions of time-independent wavepacket Schrödinger equations: calculations using a subset of eigenstates of undamped or damped Hamiltonians

“…The DAF approximation did not encounter this problem and gave rise to stable plateaus even at extremely low energies when fewer than one-tenth of the eigenstates were used. This dierence in behavior is due to the fundamental dierence between a DAF representation and the usual standard basis set expansion [47], of which the Chebychev polynomial approximation, considered earlier, is a very good example. While the leading error to a truncated Chebychev approximation of order N is a polynomial of order N 1, which is oscillatory, this is not the case for the DAF expansion.…”

“…While the leading error to a truncated Chebychev approximation of order N is a polynomial of order N 1, which is oscillatory, this is not the case for the DAF expansion. The DAF approximation to a wide class of functions has been proved [47] to exhibit uniform convergence. This is unlike a standard basis set expansion approximation to a function which converges in the sense of the norm of the dierence between the function and its approximation in the complete domain of de®nition of the function.…”

“…Since the DAF is not a standard basis set expansion [21,47], it has properties that dier, for example, from the Chebychev expansion. Of particular interest is the fact that the truncation error is not a higher-order polynomial for the DAF (as it is for the standard Chebychev expansion or for any other similar basis set expansion).…”

Effect of time-dependent basis functions and their superposition error on atom-centered density matrix propagation (ADMP): Connections to wavelet theory of multiresolution analysis

“…53 One of the main reasons for this is, when the scaling and wavelet functions are appropriately chosen, the multiresolution analysis theory provides a framework to accurately represent nonuniform functions that may portray a diverse spatial ͑or temporal͒ frequency dependence. 57 The normed convergence is a weak convergence criterion as compared to the uniform convergence.͔ Alternately, such artifacts in the plane-wave bases may be eliminated by using a ''smoothing'' or ''windowing'' process to reduce the contribution of high frequency noise. This leads to the so-called Gibb's phenomenon 54 where the approximated function constantly intertwines around the real function.…”

Non-Cartesian distributed approximating functional

“…Armed with the expressions for the NCDAF function and derivative approximations and the results of the previous sections, it is now possible to discuss a number of the numerical properties of the NCDAF approximation. In a few special cases, these properties can be proven for the continuous NCDAF approximation (see for example [22], [113], and [114]); however, none rigorously carry over to the NCDAF approximation for functions that are known only as sampled points. It is thus the approach of this discussion to demonstrate through example calculations a number of properties of the approximation that are commonly observed.…”