arXiv subjects

A. Sathish Kumar

Publications and source records attributed to A. Sathish Kumar.

13 recordsLinked to original sources

arXiv2025

Approximation by Neural Network operators in $L^p$ spaces associated with an arbitrary measure

In this paper, we investigate the approximation behavior of both one and multidimensional neural network type operators for functions in $L^p(I^d,ρ)$, where $1\leq p<\infty$, associated with a general measure $ρ$ defined over a hypercube. First, we prove the uniform approximation for a continuous function and the $L^p$ approximation theorem by the NN operators in one and multidimensional settings. In addition, we also obtain the $L^p$ error bounds in terms of $\mathcal{K}$-functionals for these neural network operators. Finally, we consider the logistic and tangent hyperbolic activation functions and verify the hypothesis of the theorems. We also show the implementation of continuous and integrable functions by NN operators with respect to the Lebesgue and Jacobi measures defined on $[0,1]\times[0,1]$ with logistic and tangent hyperbolic activation functions.

math.FA↗

arXiv2025

On Hermite type sampling Kantorovich operators in the settings of mixed norm Spaces

In this paper, we analyze the convergence behavior of Hermite-type sampling Kantorovich operators in the context of mixed norm spaces. We prove certain direct approximation theorems, including the uniform convergence theorem, the Voronovskaja-type asymptotic formula, and an estimate of error in the approximation in terms of the modulus of continuity in mixed norm settings. Next, we estimate the rate of convergence of these sampling Kantorovich operators in terms of the modulus of continuity. In addition, we obtain simultaneous approximation results of these sampling Kantorovich operators, including the uniform approximation, the asymptotic formula, and the approximation error in terms of the modulus of continuity in mixed settings. Finally, using cardinal $B$-splines, the implementation of differentiable functions has been shown.

math.FA↗

arXiv2025

Approximation by Neural Network Sampling Operators in Mixed Lebesgue Spaces

In this paper, we prove the rate of approximation for the Neural Network Sampling Operators activated by sigmoidal functions with mixed Lebesgue norm in terms of averaged modulus of smoothness for a bounded measurable functions on bounded domain. In order to achieve the above result, we first establish that the averaged modulus of smoothness is finite for certain suitable subspaces of $L^{p,q}(\mathbb{R}\times\mathbb{R}).$ Using the properties of averaged modulus of smoothness, we estimate the rate of approximation of certain linear operators in mixed Lebesgue norm. Then, as an application of these linear operators, we obtain the Jackson type approximation theorem, in order to give a characterization for the rate of approximation of neural network operators in-terms of averaged modulus of smoothness in mixed norm. Lastly, we discuss some examples of sigmoidal functions and using these sigmoidal functions, we show the implementation of continuous and discontinuous functions by neural network operators.

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↗

arXiv2021

Approximation of discontinuous functions by Kantorovich exponential sampling series

The Kantorovich exponential sampling series at jump discontinuities of the bounded measurable signal f has been analysed. A representation lemma for the series is established and using this lemma certain approximation theorems for discontinuous signals are proved. The degree of approximation in terms of logarithmic modulus of smoothness for the series is studied. Further a linear prediction of signals based on past sample values has been obtained. Some numerical simulations are performed to validate the approximation of discontinuous signals f by the sampling series.

math.FA↗

arXiv2021

Approximation of Discontinuous Signals by Exponential Sampling Series

We analyse the behaviour of the exponential sampling series $S_{w}^χf$ at jump discontinuity of the bounded signal $f.$ We obtain a representation lemma that is used for analysing the series $S_{w}^χf$ and we establish approximation of jump discontinuity functions by the series $S_{w}^χf.$ The rate of approximation of the exponential sampling series $S_{w}^χf$ is obtained in terms of logarithmic modulus of continuity of functions and the round-off and time-jitter errors are also studied. Finally we give some graphical representation of approximation of discontinuous functions by $S_{w}^χf$ using suitable kernels.

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

On Bivariate Kantorovich Exponential Sampling Series

We analyse the approximation properties of the bivariate generalization of the family of Kantorovich type exponential sampling series. We derive the point-wise and Voronovskaya type theorem for these sampling type series. Using the modulus of smoothness, we obtain the quantitative estimate of order of convergence of these series. Further, we establish the degree of approximation for these series associated with generalized Boolean sum (GBS) operators. Finally, we provide a few examples of kernels to which the theory can be applied along with the graphical representation and error estimates.

math.FA↗

arXiv2020

On Approximation by Kantorovich Exponential Sampling Operators

In this article, we analyse the Kantorovich type exponential sampling operators and its linear combination. We derive the Voronovskaya type theorem and its quantitative estimates for these operators in terms of an appropriate K-functional. Further, we improve the order of approximation by using the convex type linear combinations of these operators. Subsequently, we prove the estimates concerning the order of convergence for these linear combinations. Finally, we give some examples of kernels along with the graphical representations.

math.FA↗

arXiv2019

Approximation by Exponential Type Neural Network Operators

In the present article, we introduce and study the behaviour of the new family of exponential type neural network operators activated by the sigmoidal functions. We establish the point-wise and uniform approximation theorems for these NN (Neural Network) operators in C[a; b]: Further, the quantitative estimates of order of approximation for the proposed NN operators in C(N)[a; b] are established in terms of the modulus of continuity. We also analyze the behaviour of the family of exponential type quasi-interpolation operators in C(R+): Finally, we discuss the multivariate extension of these NN operators and some examples of the sigmoidal functions.

math.NA↗

arXiv2019

Inverse approximation and GBS of bivariate Kantorovich type sampling series

In this paper, we derive an inverse result for bivariate Kantorovich type sampling series for the space of all continuous functions with upto second order partial derivatives are continuous and bounded on $R^2.$ Further, we prove the rate of approximation in the Bogel space of continuous functions for the GBS (Generalized Boolean Sum) of these operators. Finally, we give some examples for the kernel to which the theory can be applied

math.NA↗

arXiv2017

Approximation by generalized bivariate Kantorovich sampling type series

The purpose of this paper is to construct a bivariate generalization of new family of Kantorovich type sampling operators $(K_w^φf)_{w>0}.$ First, we give the pointwise convergence theorem and a Voronovskaja type theorem for these Kantorovich generalized sampling series. Further, we obtain the degree of approximation by means of modulus of continuity and quantitative version of Voronovskaja type theorem for the family $(K_w^φf)_{w>0}.$ Finally, we give some examples of kernels such as box spline kernels and Bochner-Riesz kernel to which the theory can be applied.

math.FA↗

arXiv2017

Approximation by generalized Kantorovich sampling type series

In the present article, we analyse the behaviour of a new family of Kantorovich type sampling operators $(K_w^φf)_{w>0}.$ First, we give a Voronovskaya type theorem for these Kantorovich generalized sampling series and a corresponding quantitative version in terms of the first order of modulus of continuity. Further, we study the order of approximation in $C({\mathbb{R}})$ (the set of all uniformly continuous and bounded functions on ${\mathbb{R}}$) for the family $(K_w^φf)_{w>0}.$ Finally, we give some examples of kernels such as B-spline kernels and Blackman-Harris kernel to which the theory can be applied.

math.CA↗