ON THE CONVERGENCE OF THE SEQUENCE OF KANTOROVICH OPERATORS IN WEIGHTED LEBESGUE SPACES

• Main Author: Angga Taebenu, Erick
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/79889
• Institution: Institut Teknologi Bandung

Kantorovich operator is a certain modification of the Bernstein polynomial which is constructed to approximate functions that are integrable on closed and bounded intervals. The uniform boundedness and convergence of the sequence of Kantorovich operators have been studied in Lebesgue space. In this thesis, we examine the uniform boundedness and convergence of the sequence of Kantorovich operators in weighted Lebesgue space, which is one of the generalizations of Lebesgue space. If the weight satisfied certain condition, the sequence of Kantorovich operators is uniformly bounded on the weighted Lebesgue space (p > 1) and converges for any functions on the closure continuous functions in the Weighted Lebesgue spaces (p > 1). Meanwhile, for the case of p = 1 with a certain weight, the uniform boundedness of the sequence of Kantorovich operators is obtained on the intersection of the weighted Lebesgue space(p = 1) and the Lebesgue space (p = 1). This implies the convergence of the sequence of Kantorovich operators in the weighted Lebesgue space (p = 1) for each function on the intersection of closure of continuous functions in the weighted Lebesgue space (p = 1) and the Lebesgue space (p = 1). The estimation for rate of convergence of the sequence of Kantorovich operators in the weighted Lebesgue space with p > 1 is also studied for three different classes of functions.