Approximation of complex-valued functions by rational interpolation operators: Theory and applications

Abstract

The approximation of complex-valued functions is of fundamental importance as it provides a framework for amplitude and phase-dependent phenomena. In this paper, we propose a family of complex rational interpolation operators to approximate analytic as well as non-analytic functions along with real-life application. In this direction, the first operator is constructed using Chebyshev polynomials of the first kind, namely classical complex rational interpolation operators for approximating complex-valued continuous functions. We establish approximation results for these operators utilizing the notion of the modulus of continuity. To approximate not-necessarily analytic but integrable function, we define Kantorovich-type complex rational interpolation operators and establish their boundedness and convergence. Furthermore, in order to approximate functions preserving higher derivatives, we introduce Hermite-type complex rational interpolation operators and study their approximation capabilities using higher order of modulus of continuity. To validate the theoretical results, we provide numerical simulations illustrating approximation ability of proposed family of complex rational interpolation operators. Moreover, we present the utility of our construction in denoising of signals corrupted by Gaussian noise. In addition, we demonstrate its real-life application in brain image reconstruction by applying these operators to real brain imaging data, particularly magnetic resonance imaging (MRI) dataset. The algorithms are provided at the end to formally explain the reconstruction procedure of the proposed family of complex rational interpolation operators.