Closed-form approximation of Hilbert transforms of Gaussian derivatives based on weighted polynomial fitting

Abstract

Hilbert transforms of Gaussian derivatives are related to Dawson's integral. Since this integral cannot be expressed in a closed form, various methods for the approximation of the derivatives have been developed. A closed-form approximation can be obtained by using the weighted polynomial fitting in which Gaussian function is used as the weighting function. Such an approach results in explicit approximation formulas. In literature, they are available only for the derivatives of the second, third, and fourth order. Furthermore, they utilize only low-order polynomials. In this paper, we propose an approximation of the Hilbert transforms of the Gaussian derivatives of arbitrary orders, which utilize high-order polynomials. The coefficients of these polynomials are obtained by using the least-squares error criterion. Closed-form expressions are provided for their calculation.