Properties of fuzzy transform obtained from Lp minimization and a connection with Zadeh’s extension principle

Abstract

In 2006 Perfilieva proved that the components Fk of the fuzzy transform are actually minimizers of certain weighted L2 type distances, obtained from a fuzzy partition. Inspired by this property, we consider and analyze the properties of a fuzzy transform when the components Fk are obtained as minimizers with respect to weighted Lp type distances, where p ≥ 1. In this way we can generate new approximation operators based on the fuzzy transform method. For these operators we obtain good rates of uniform convergence and what is more, they satisfy all the properties required for an effective approximation operator, such as additivity and homogeneity. If we consider the extended fuzzy transform and inverse fuzzy transform, respectively, in this setting of Lp type distances, then we get some nice supplementary properties like monotonicity of the operator and preservation of monotone functions. Finally, we consider fuzzy-valued versions of the L1 and L2 type fuzzy transforms, proposed recently as quantile and expectile smoothing tools, and we prove that the L1 type fuzzy-valued transform of a continuous monotonic function f coincides with the Zadeh’s fuzzy extension fEP evaluated at suitable fuzzy numbers, obtained from the basic functions of the used fuzzy partition.