Positive linear operators and approximation properties

A fundamental result in approximation theory is the Weierstrass approximation theorem, stating that every continuous function on [0,1] (or continuous 2π periodic function on [0,2π]) can be approximated by an algebraic (or trigonometric) polynomial. Bernstein provided a concise and elegant proof of this theorem by introducing Bernstein polynomials. Szász and Mirakjan independently studied the Szász-Mirakjan operators to handle functions in C[0,∞). Various extensions of Bernstein operators have been devised to approximate functions in 𝐿ᵖ[0,1] (1 ≤ 𝑝 <∞), including Bernstein-Kantorovich and Bernstein-Durrmeyer operators. This thesis focuses on classical calculus, introducing Stancu-type generalizations of Baskakov-Durrmeyer operators and α-Stancu-Schurer-Kantorovich operators by exploring new operators using q-calculus, starting with Stancu-type modifications of generalized Baskakov-Szász operators and Phillips operators. Then, shifting knots of q-Bernstein-Kantorovich operators, a family of summation-integral type hybrid operators with shape parameter α, and the q-Baskakov-Kantorovich operators via wavelets are introduced. The computation of moments and central moments, determining the order of approximation, discussion of basic results and approximation properties and also study of estimates and rate of convergence for these operators by employing various useful tools like Voronovskaja type result, Peetre's K-functional and asymtotic error constant.