Approximation by Szasz-type Operators Based on Bernoulli Polynomials of Order $(-1)$

Abstract

In this study, we define a Szasz-type operator sequence derived from Bernoulli polynomials of order (-1). By computing the moments and central moments of the operator, we show via the Korovkin-type method the uniform convergence of the operator. We then determine the operator's rate of convergence by means of the modulus of continuity, Peetre's K-function, and the Lipschitz class. We also look at their local as well as global convergence. For the smooth functions, we derive estimates of the rate of convergence and asymptotic behavior of the operator. Furthermore, we establish a generalization of the operator we are currently dealing with by using the Taylor series. We conclude by providing examples to validate and contrast the research's conclusions.