Approximation by a new sequence of operators involving Laguerre polynomials

TL;DR

This paper introduces a new sequence of operators based on Laguerre polynomials for function approximation over [0,∞), providing theoretical convergence analysis and numerical validation.

Contribution

A novel integral operator involving Laguerre polynomials is proposed, with established approximation properties and convergence analysis using various analytical tools.

Findings

Operators effectively approximate functions on [0,∞)

Convergence proven using Korovkin's theorem and other methods

Numerical examples confirm theoretical results

Abstract

This paper offers a newly created integral approach for operators employing the orthogonal modified Laguerre polynomials and P\u{a}lt\u{a}nea basis. These operators approximate the functions over the interval $[0,\infty)$. Further, the moments are established for the proposed operators, and the universal Korovkin's theorem is used to derive the approximation properties of the operators. We examine convergence using a variety of analytical methods, including the Lipschitz class, Peetre's K-functional, the second-order modulus of smoothness, and the modulus of continuity. Moreover, an asymptotic formula associated with the Voronovskaja-type is established. The approximation is estimated through the weighted modulus of continuity, and convergence of the proposed operators in weighted spaces of functions is investigated as well. Ultimately, we employ numerical examples and visual representations to validate the theoretical findings.