The goal of the present paper is to investigate approximation and smoothness properties
of Larsen type single input single output (SISO) fuzzy systems, that is, fuzzy logic
systems using the maximum as aggregation for the individual rule outputs, product
(Goguen) t-norm as the conjunctive operator and center of gravity defuzzification.
We prove that the function providing the output of the above considered Larsen type
fuzzy system is capable of approximating any continuous function. Also, it is continuously
differentiable under very relaxed conditions as e. g. continuous differentiability
of the antecendents except at their core, and continuous differentiability of the
consequences of fuzzy rules except at their core and the enpoints of their support.
We show practical examples regarding the approximation and smoothness of the Larsen
type operators, showing also by an example that the conditions on the antecedent part
cannot be in general weakened further without loosing continuous differentiability.
The present paper provides a theoretical background for claims in literature stating
that the output of a fuzzy control system of this type is smooth.
