Suppose we are given a finite subset E⊂ℝn and a function f:E→ℝ. How to extend f to a Cm function F:ℝn→ℝ with Cm norm of the smallest possible order of magnitude? In this paper and in  we tackle this question from the perspective of theoretical computer science. We exhibit algorithms for constructing such an extension function F, and for computing the order of magnitude of its Cm norm. The running time of our algorithms is never more than CNlogN, where N is the cardinality of E and C is a constant depending only on m and n.