KANTOROVICH OPERATOR IN MORREY SPACE

• Main Author: Purwa Arsana, Mu'afa
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/55005
• Institution: Institut Teknologi Bandung

For a continuous function ???? on the interval [0,1], the Bernstein polynomial ????????(????)(????), ???? ? ?, is known to be a good approximation to ????(????). More precisely, the sequence of functions {????????(????)}????=1 ? converges uniformly to ????. One substitute for the Bernstein polynomial to approximate the discontinuous function at [0,1] is the Kantorovich operator ????????. In particular, for an integrable function ???? on [0,1], the sequence of function {????????(????)}????=1 ? converges to ???? in ????1([0,1]). The proof of this convergence uses the uniform boundedness of Kantorovich operator ???????? in Lebesgue space. In this thesis, we discuss the convergence of Kantorovich operators in Morrey space. This result is proved by Burenkov et al [17]. The key to proving the convergence of the Kantorvich operator on the Morrey space is to use the estimation of the Kantorovich operator with the Hardy-Littlewood maximal operator. However, the Hardy-Littlewood maximal operator norm in Morrey space ????? ????([0,1]) is infinite at ???? = 1. By considering the boundedness of the Kantorovich operator on ????1 and choosing a function on the Morrey space, it will be proved that the Kantorovich operator converges uniformly on the Morrey space ????? ????([0,1]) for 1 ? ???? < ?.