Comprehensive Review of \(\lambda\)-Bernstein Operators

Abstract

In this chapter, we explore the historical development of the significant results surrounding \(\lambda\)-Bernstein operators within the field of approximation theory. The primary objective of this study is to review the progress in this area and evaluate both the rapidity of convergence, using the modulus of continuity, and the rate of convergence, utilizing Lipschitz functions and Peetre’s K-functional. Operator theory has garnered considerable interest over the past two decades, largely due to the widespread applicability of Bernstein polynomials in approximation theory. These polynomials are now integral to numerous fields, including fixed point theory, numerical analysis, image processing, neural networks, machine learning, and the solution of both ordinary and partial differential equations. This chapter also highlights a few significant outcomes and includes the authors’ pertinent opinions, tracing the development of these operators from their inception to the present day.