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.

翻译：复值函数的逼近具有基础性重要意义，它将经典逼近理论推广至复域，为依赖振幅与相位的现象提供了严谨的数学框架。本文研究由杰出数学家Paul G. Nevai提出的Nevai算子概念，构建了一类复Nevai插值算子族，用于逼近解析及非解析的复值函数，并探讨其在图像处理中的实际应用。在此方向上，首先利用第一类切比雪夫多项式构造了首个算子——复广义Nevai算子，用于逼近复值连续函数。我们借助连续模的概念建立了所提算子的逼近结果。为逼近未必连续但可积的函数，定义了复Kantorovich型Nevai算子，并证明了其有界性与收敛性。此外，为保持更高阶导数的函数逼近，引入复Hermite型Nevai算子，利用高阶连续模研究了其逼近能力。为验证理论结果，我们通过数值算例展示了所提复Nevai算子族的逼近性能。