This paper addresses the sufficient and necessary conditions for constructing $C^r$ conforming finite element spaces from a superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline space and the finite element space. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $C^r$ conforming elements in $d$ dimensions should in general require an extra $C^{2^{s}r}$ continuity on $s$-codimensional simplices, and the polynomial degree is at least $(2^d r + 1)$.

翻译：本文探讨了在一般单纯形剖分上从超样条空间构造$C^r$协调有限元空间的充分必要条件。我们引入了预单元空间可扩展性的概念，该概念同时涵盖了超样条空间与有限元空间。通过考察两类空间的可扩展性条件，我们给出了关于构造条件的解答。我们结果的一个推论是：在$d$维空间中构造$C^r$协调元通常需要在$s$余维单纯形上额外满足$C^{2^{s}r}$连续性，且多项式次数至少为$(2^d r + 1)$。