Riemann-Liouville Type Fractional Generalized Λ-Bernstein Operators

Abstract

Approximation theory, with roots in the Weierstrass theorem, has evolved to encompass various types of polynomial operators, among which the Bernstein polynomials have become particularly influential due to their stability and convergence characteristics. With the addition of shape parameters such as λ and fractional parameters inspired by fractional calculus, modern approximation theory now accommodates complex function behaviors that traditional integer-order operators could not capture. This chapter focuses on the development of new Riemann-Liouville-type fractional λ-Bernstein-Kantorovich operators. It provides detailed proofs of their convergence, pointwise estimates, and asymptotic behavior. Using thorough mathematical analysis, we present direct approximation results, highlighting how these operators effectively approximate functions in different function spaces. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.