ESTIMATES TO DISTANCES BETWEEN FUZZY WAVELET LIKE OPERATORS

Abstract

The basic fuzzy wavelet like operators $A_k, B_k, C_k, D_k, k \in {\mathbb Z}$ were first introduced in [14], see also Chapter 12, where they were studied among others for their pointwise/uniform convergence with rates to the fuzzy unit operator I. Here we continue this study by estimating the fuzzy distances between these operators. We give the pointwise convergence with rates of these distances to zero. The related approximation is of higher order since we involve these higher order fuzzy derivatives of the engaged fuzzy continuous function f. The derived Jackson type inequalities involve the fuzzy (first) modulus of continuity. Some comparison inequalities are also given so we get better upper bounds to the distances we study. The defining of these operators scaling function ϕ is of compact support in [–a, a], a > 0 and is not assumed to be orthogonal. This chapter is based on [23].