A Gaussian Regularization for Higher-Order Derivative Sampling Expansion in the Linear Canonical Transform Domain

Rashad M. Asharabi

Session: General

Keywords: Derivative sampling theorem; Generalized Hermite interpolation, Linear canonical transform, Contour integral, Gaussian regularization.

Abstract: The derivative sampling theorem for bandlimited functions within the fractional Fourier transform (FrFT) domain, which involves samples from the function and its $r$-derivatives, was introduced by Jing and his collaborators in 2019. In this paper, we extend this type of sampling expansion to the linear canonical transform (LCT) domain and propose an alternative representation of this sampling using a contour integral. The convergence rate of the reformulated sampling expansion remains slow, particularly at the order of $O(\ln(N)/N)$. To address this slow convergence, we develop a Gaussian regularization method for higher-order derivative sampling in the LCT domain. This regularization sampling method applies to a broader range of functions within the Paley-Wiener space in the LCT domain, including entire functions that may not necessarily belong to the space $L^{2}(\mathbb{R})$ when their domain is restricted to $\mathbb{R}$. Notably, this regularized sampling still achieves an exponential convergence rate and requires only a finite number of samples from the original function and its first $r$ derivatives. The results of Asharabi (2016) within the classical Fourier transform domain and Annaby et al. (2023) within the LCT domain will be special cases of the findings presented in this paper. Furthermore, we present a numerical example that demonstrates excellent consistency with our theoretical analysis.

Submission Number: 18