Whitney’s extension problems and interpolation of data

Abstract

Abstract. Given a function f : E → R with E ⊂ R n , we explain how to decide whether f extends to a C m function F on R n . If E is finite, then one can efficiently compute an F as above, whose C m norm has the least possible order of magnitude (joint work with B. Klartag).Let f : E → R be a function defined on a given (arbitrary) set E ⊂ R n , and let m ≥ 1 be a given integer. How can we decide whether f extends to a function The finite, effective versions of the above problems are basic questions about interpolation of data: Let E ⊂ R n be a finite set, and let f : E → R be given. Fix m ≥ 1. We want to find an interpolant, i.e., a function F ∈ C m (R n ) such that F = f on E. How small can we take the C m norm of an interpolant? How can we compute an interpolant whose C m norm is close to least possible? What if we require only that F and f agree approximately on E? What if we are allowed to discard a few points from E? An efficient solution to these problems would likely have practical applications. Joint work [19], [20] of B. Klartag and the author shows how to compute efficiently an interpolant F whose C m norm is within a factor C of least possible, where C is a constant depending only on m and n. Unfortunately, we do not know whether that constant C is absurdly large; we suspect that it is. To remedy this defect, and hopefully obtain results with practical applications, we pose the following sharper version of the interpolation problem.

Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

“…On the negative side, the formulas in [21] depending on Wells's construction are more complicated than the proof of [20], which uses Zorn's lemma and in particular is not constructive. For more information about Whitney extension problems and extension operators see [3,10,11,12,14,13,15,26,18,21,8,25] and the references therein.…”

Interpolation of data by smooth nonnegative functions

“…We then have ϕk(x) = 1 by (10), hence P(x) + bkϕk(x) ≥ 0 by (14). Since also b k ϕ k (x) ≥ 0 for all k, it follows that…”

“…This paper is part of a literature on extension, interpolation, and selection of functions, going back to H. Whitney's seminal work [33], and including fundamental contributions by G. Glaeser [19], Y, Brudnyi and P. Shvartsman [4,[6][7][8][9][23][24][25][26][27][28][29][30][31], J. Wells [32], E. Le Gruyer [21], and E. Bierstone, P. Milman, and W. Paw lucki [1][2][3], as well as our own papers [10][11][12][13][14][15][16][17]. See e.g.…”

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

“…[122] provides the necessary and sufficient conditions for a function f from a closed subset P of R k (k ∈ N) into R to have a C n (n ∈ N) extensionf : R k → R. This theorem has been studied extensively, see e.g. [7,49,50]. Here, we discuss it only for k = 1 and P ⊂ R being perfect.…”