arXiv subjects

Shivam Bajpeyi

Publications and source records attributed to Shivam Bajpeyi.

7 recordsLinked to original sources

arXiv2025

Approximation by Certain Complex Nevai Operators : Theory and Applications

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.

math.NA↗

arXiv2025

Approximation by multivariate neural network operators in mixed norm space: theory and application

On the one hand, the framework of mixed norm spaces has potential applications in different areas of mathematics. On the other hand, neural network (NN) operators are well established as approximators, attracting significant attention in the fields of approximation theory and signal analysis. In this article, we aim to integrate both these concepts. Here, we analyze multivariate Kantorovich-type NN operators as approximators based on sigmoidal activation function within various mixed norm structures, namely mixed norm Lebesgue spaces and mixed norm Orlicz spaces. The Orlicz spaces consist of various significant function spaces, such as classical \textit{Lebesgue spaces, Zygmund spaces and Exponential spaces}. We establish the boundedness and convergence of multivariate Kantorovich-type NN operators in these mixed norm function spaces. At the end, the approximation abilities are illustrated through graphical representations and error-estimates along with application in image processing, including image reconstruction, image scaling, image inpainting and image denoising. Finally, we address the beneficial impact of mixed norm structure on aforementioned image processing tasks.

math.FA↗

arXiv2025

Constructive Approximation in Mixed norm Spaces

The concept of mixed norm spaces has emerged as a significant interest in fields such as harmonic analysis. In addition, the problem of function approximation through sampling series has been particularly noteworthy in the realm of approximation theory. This paper aims to address both these aspects. Here we deal with the problem of function approximation in diverse mixed norm function spaces. We utilise the family of Kantorovich type sampling operators as approximator for the functions in mixed norm Lebesgue space, and mixed norm Orlicz space. The Orlicz spaces are well-known as a generalized family that encompasses many significant function spaces. We establish the boundedness of the family of generalized as well as Kantorovich type sampling operators within the framework of these mixed norm spaces.Further, we study the approximation properties of Kantorovich-type sampling operators in both mixed norm Lebesgue and Orlicz spaces. At the end, we discuss a few examples of suitable kernel involved in the discussed approximation procedure.

math.FA↗

arXiv2023

Certain Approximation Results for Kantorovich Exponential Sampling Series

In this paper, we study a strong inverse approximation theorem and saturation order for the family of Kantorovich exponential sampling operators. The class of log-uniformly continuous and bounded functions, and class of log-Hölderian functions are considered to derive these results. We also prove some auxiliary results including Voronovskaya type theorem, and a relation between the Kantorovich exponential sampling series and the generalized exponential sampling series, to achieve the desired plan. Moreover, some examples of kernels satisfying the conditions, which are assumed in the hypotheses of our theorems, are discussed.

math.FA↗

arXiv2023

Random Sampling of Mellin Band-limited Signals

In this paper, we address the random sampling problem for the class of Mellin band-limited functions BT which is concentrated on a bounded cube. It is established that any function in BT can be approximated by an element in a finite-dimensional subspace of BT. Utilizing the notion of covering number and Bernstein's inequality to the sum of independent random variables, we prove that the random sampling inequality holds with an overwhelming probability provided the sampling size is large enough.

math.FA↗

arXiv2020

Approximation by Durrmeyer type Exponential Sampling Series

In this article, we analyze the approximation properties of the new family of Durrmeyer type exponential sampling operators. We derive the point-wise and uniform approximation theorem and Voronovskaya type theorem for these generalized family of operators. Further, we construct a convex type linear combination of these operators and establish the better approximation results. Finally, we provide few examples of the kernel functions to which the presented theory can be applied along with the graphical representation.

math.FA↗

arXiv2020

Direct and inverse results for Kantorovich type exponential sampling series

In this article, we analyze the behaviour of the new family of Kantorovich type exponential sampling series. We obtain the point-wise approxi mation theorem and Voronovskaya type theorem for the series. Further, we obtain a representation formula and an inverse result approximation for these operators. Finally, we give some examples of kernel functions to which the theory can be applied along with the graphical representation.

math.NA↗