CONVERGENCE ANALYSIS OF THE JANARDAN OPERATOR USING KOROVKIN THEOREM AND VORONOVSKAJA ASYMPTOTIC FORMULATION

Abstract

We have seen many complex functions that are typical to handle in mathematical analysis directly and replacing them with simpler ones that exhibit their properties has become a neccessary part. In mathematical analysis one of the most well known and approachable tool is approximation with operators. This thesis is devoted to one such positive linear operator referred to as Janardan operator that is constructed on the basis of probabilistic distribution that helps to approximation continuous functions. This in vestigation begins with the computation of moments of Janardon operator over three test functions as constant function f(t) = 1, the linear function f(t) = t, and the quadratic function f(t) = t2. These moment estimates occupy a central place in the analysis, since they reveal how the operator responds to the most elementary inputs and thereby lay the groundwork for studying its behaviour on arbitrary continuous functions. Once the moments are established, the classical theorem of Korovkin is invoked. This theorem asserts that if a sequence of positive linear operators converges to the identity on the three test functions mentioned above, then it converges uniformly to any continuous function defined on a closed and bounded interval. The Bohman–Korovkin criterion is verified for the Janardhan operator, thereby establishing its uniform convergence on C[a,b]. Beyond establishing convergence, the thesis addresses the quantitative aspect of ap proximation, that is, how rapidly the operator output approaches the target function. For this purpose, the modulus of continuity is employed as a measure of the oscillation of a function, and explicit error bounds are derived. These bounds provide a guaran teed estimate of the maximum deviation between the operator and the function being approximated. A Voronovskaja-type asymptotic formula is also derived for the Janardhan opera tor. Unlike convergence theorems, which merely confirm that the error tends to zero, a Voronovskaja result describes the precise asymptotic behaviour of the error, scaled ap 4 propriately, as the parameter tends to infinity. This gives a sharper and more complete picture of how the approximation process unfolds in its final stages. The thesis concludes with numerical computations carried out for selected test func tions. These calculations illustrate the theoretical results in concrete terms and confirm that the approximation behaviour predicted by the analysis is indeed observed in prac tice. Taken together, the results establish the Janardhan operator as a mathematically well-founded and practically effective tool within the broader framework of approximation theory.