On Basic Fuzzy Korovkin Theory

We prove the basic fuzzy Korovkin theorem via a fuzzy Shisha– Mond inequality given here. This determines the degree of convergence with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The surprising fact is that only the real case Korovkin assumptions are enough for the validity of the fuzzy Korovkin theorem, along with a natural realization condition fulfilled by the sequence of fuzzy positive linear operators. The last condition is fulfilled by almost all operators defined via fuzzy summation or fuzzy integration. 0. Introduction Motivation for this work are the references [1], [2], [5], [6]. Our results Theorems 3 and 4 are simple, basic and very general, directly transferring the real case of the convergence with rates of positive linear operators to the unit, to the fuzzy one. The same real assumptions are kept here in the fuzzy setting, and they are the only assumptions we make along with the very natural and general realization condition (1). Condition (1) is fulfilled by almost all example — fuzzy positive operators, that is, by most fuzzy summation and fuzzy integration operators. At each step of our work we provide an example to justify our method. To the best of our knowledge our theorems are the first general fuzzy Korovkin type results. 1. Background We start with Received by the editors: 09.06.2005. 2000 Mathematics Subject Classification. 26E50, 41A17, 41A25, 41A36, 47S40.