Approximation by Certain Complex Nevai Operators : Theory and Applications

TL;DR

This paper introduces and analyzes a family of complex Nevai operators for approximating complex-valued functions, with theoretical results and applications in image processing.

Contribution

It develops new complex Nevai interpolation operators, including Kantorovich and Hermite types, with proven approximation properties and practical numerical illustrations.

Findings

Operators effectively approximate complex functions.

Boundedness and convergence of operators established.

Numerical results demonstrate practical approximation capabilities.

Abstract

The approximation of complex-valued functions is of fundamental importance as it generalizes classical approximation theory to the complex domain, providing a rigorous framework for amplitude and phase-dependent phenomena. In this paper, we study the Nevai operator, a concept formulated by the distinguished mathematician Paul G. Nevai. We propose a family of complex Nevai interpolation operators to approximate analytic as well as non-analytic complex-valued functions along with real-life application in image processing. In this direction, the first operator is constructed using Chebyshev polynomials of the first kind, namely complex generalized Nevai operators for approximating complex-valued continuous functions. We establish the approximation results for the proposed operators utilizing the notion of a modulus of continuity. To approximate not necessary continuous but integrable function, we define complex Kantorovich type Nevai operators and establish their boundedness and convergence. Furthermore, in order to approximate functions preserving higher derivatives, we introduce complex Hermite type Nevai operators and study their approximation capabilities using higher order of modulus of continuity. To validate the theoretical results, we provide numerical illustrations of approximation abilities of proposed family of complex Nevai operators.