Approximation by Certain Complex Nevai Operators : Theory and Applications

TL;DR

The paper addresses the problem of approximating complex-valued functions by extending Nevai interpolation to the complex domain through Chebyshev-based kernels. It develops three operator families—complex generalized Nevai operators $N_{n,s}(f,z)$, complex Kantorovich type Nevai operators $K_{n,s}(f,z)$, and complex Hermite type Nevai operators $H^{(r)}_{n,s}(f,z)$—and proves convergence results in both $C(X)$ and $L^p(X)$ settings using modulus of continuity and Peetre's $K$-functional. Key contributions include explicit interpolation properties, stability bounds, and quantitative error estimates, along with comprehensive numerical validations on analytic, non-analytic, and integrable functions, and an image-reconstruction application with SSIM, PSNR, and RMSE metrics. The work sheds light on complex-domain approximation and demonstrates practical impact in signal processing and image analysis by preserving amplitude and phase information. Overall, the results advance complex approximation theory and provide versatile tools for complex-valued function approximation and applications such as complex image reconstruction.

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.

Figures (8)

• Figure 1: Approximation of the real and imaginary parts of $f_1$ by $(N_{n,s}f_1)$ for $s=2$ and different values of $n$
• Figure 2: Contour lines of $|g_1|$, $|N_{n,2}g_1|$ and absolute error for $n=10,20$
• Figure 3: Approximation of the real and imaginary parts of the function $f_2$ by $(K_{n,s}f_2)$ for $s=2$ and different values of $n$
• Figure 4: Contour lines of $|g_2|$, $|K_{n,2}g_2|$ and absolute error for $n=10,20$
• Figure 5: Approximation of the real and imaginary parts of the function $f_3$ by $(H^{(3)}_{n,s}f_3)$ for $s=2$ and different values of $n$

Theorems & Definitions (17)

• Lemma 3.1
• Lemma 3.2
• Remark 1
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 3.6
• Lemma 4.1
• Lemma 4.2
• Lemma 4.4