In this paper we define and discuss a new non-Lagrangean procedure for theweakinterpolation of a non-linear mapping on the space of continuous functions. We suppose that the mappingis defined by an empirical data setfor eachwhere the functionsxrandyrare known only by evaluation vectorsand. In this situation the Lagrangean interpolation proposed by Prenter cannot be applied. We construct a mappingsuch thatS[u]=S[x] whenu(si)=x(si) for eachi= 1,2, …,m and such thatfor eachr= 1,2,$hellip;,pand eachk= 1,2,…,n. We show that in some special circumstances theweakinterpolation becomes astronginterpolation and we also show that the interpolation operator is continuous in the strong topology at each point of the empirical data set.