The Jet of an Interpolant on a Finite Set

We study functions F ∈ Cm(Rn) having norm less than a given constant M, and agreeing with a given function f on a finite set E. Let Γf(S,M) denote the convex set formed by taking the (m − 1)jets of all such F at a given finite set S ⊂ Rn. We provide an efficient algorithm to compute a convex polyhedron Γ̃f(S,M), such that Γf(S, cM) ⊂ Γ̃f(S,M) ⊂ Γf (S,CM) , where c and C depend only on m and n. Fix m, n ≥ 1, and let f : E −→ R be given, where E ⊂ R. For a given real number M > 0, we are interested in functions F ∈ C(R) with norm at most M, such that F = f on E. To understand how F behaves, we fix a finite set S ⊂ R, and compute the (m − 1) order Taylor polynomial of F at each point of S. What can we say about the resulting family of polynomials? To answer this question, we introduce some notation and definitions. As usual, C(R) consists of all m times continuously differentiable functions F : R −→ R for which the norm ‖ F ‖= sup x∈Rn max |α|≤m |∂F(x)| is finite. For F ∈ C(R) and x ∈ R, we write Jx(F) (the “jet” of F at x) to denote the (m−1) order Taylor polynomial of F at x. Thus, Jx(F) belongs to P, the vector space of (real-valued) (m − 1) degree polynomials on R. 2000 Mathematics Subject Classification: 49K24, 52A35.