Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function

Abstract

a b s t r a c tA new sampling methodology based on an incomplete cosine expansion series is presented as an alternative to the traditional sinc function approach. Numerical integration shows that this methodology is efficient and practical. Applying the incomplete cosine expansion we obtain a rational approximation of the complex error function that with the same number of the summation terms provides an accuracy exceeding the Weideman's approximation accuracy by several orders of the magnitude. Application of the expansion results in an integration consisting of elementary function terms only. Consequently, this approach can be advantageous for accurate and rapid computation.

A Single-Domain Implementation of the Voigt/Complex Error Function by Vectorized Interpolation

“…Using a new method of sampling based on incomplete cosine expansion of the sinc function (A.1) we can obtain [41] (see also cited literature in context therein) w (z) ≈ Ω (z + iς/2)…”

“…In our publications [13,41] we have introduced the following product-to-sum identity M m=1 cos t 2 m = Appendix C…”

Pricing early-exercise and discrete barrier options by Shannon wavelet expansions

“…This issue is not new in numerical integration, and was adressed in [1,2]. There, a combination of Vieta's formula and a cosine product-to-sum identity was used to approximate the sinc-function by a socalled incomplete cosine expansion.…”

“…Besides the derivation based on Vieta's formula in [2], a derivation based on Parseval's identity, see [15], was given, which is a discretization of the integral that arises in Corollary 1. If that integral is approximated by the midpoint rule with J subintervals, it coincides once more with the result in Lemma 4, which was the missing link between the two approaches.…”

A rational approximation of the sinc function based on sampling and the Fourier transforms

“…respectively. In our earlier works we have applied sampling by incomplete cosine expansion of the sinc function and obtained a rapid and highly accurate rational approximation for the Voigt/complex error function that with only 16 summation terms can provide accuracy ∼ 10 −14 [1,2]. As a further development, in this work we generalize the methodology described in our paper [1] by an example showing how sampling and the Fourier transforms (2), (3) can be implemented to obtain a rational approximation of the sinc function.…”