Abstract: In this paper we investigated the function space W₂² which has a regenerate nucleus. We obtained a finite expression of the regenerate nucleus and used it to construct a concrete expression of the optimum Hermitean interpolation operator. It was also proved that, as the density of the node system increases infinitely, (H_2nu)(x) and (H_2nu)'(x) uniformly converge to u(x) and u'(x), respectively, and the error decreases to zero monotonically, under the Sobolev norm.