The generalization of approximation by max-min operators

In this thesis, we systematically study the approximation properties of the maximum-minimum operators defined by Bede et.al. in 2008. Since the max-min operators satisfy the pseudo-linearity condition that is a weaker concept than the usual linearity, the classical Korovkin approximation theorem does not hold for these operators. Hence, we first obtain a general approximation theorem for max-min operators and compute the rates of convergence in this approximation. Especially we give an error estimation for Hölder continuous functions. By considering some special cases of our approximation theorem, we investigate the approximation properties of max-min Shepard and max-min Bernstein operators. In this way, we get a classical approximation to univariate and bivariate continuous functions by means of max-min operators and confirm it by graphical illustrations. We also approximate to quasi-concave functions by these operators and verify it by graphs. Then, we also study some shape preserving properties. We show that the max-min Bernstein operators preserve the monotonicity while we give some examples indicating that the max-min Shepard operators do not preserve the monotonicity. With some applications, we also show that the approximation is valid for continuous functions not only on the unit interval [0, 1] but also on any closed and bounded interval [a,b]. With the help of regular summability methods, we improve the classical approximation results. Thus, we give some alternative ways where the classical approach fails. By using some special regular methods, we approximate to both univariate and bivariate continuous functions by max-min Shepard operators. We prove the existence of approximation for summability methods, such as the arithmetic mean convergence and the almost convergence, which are weaker than the convergence in the usual sense. We also compute the rates of convergence for this approximation by summability methods.