Interpolation of data by smooth nonnegative functions

Abstract

We prove a finiteness principle for interpolation of data by nonnegative C^m and C^{m−1,1} functions. Our result raises the hope that one can start to understand constrained interpolation problems in which, e.g., the interpolating function F is required to be nonnegative.

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.…”

Min–max formulas for nonlocal elliptic operators

“…However, when considering the extension among functions inG n that are sufficiently regular (in the sense that they are the restriction of smooth functions) then E β n preserves the ordering up to a small correcting function whose C β norm vanishes as n goes to infinity. It is worthwhile to point out to a recent preprint of Fefferman, Israel, and Luli [22], where a closely related question, the interpolation of functions with a positivity constraint, is studied.…”

Smooth Selection for Infinite Sets

“…The following nonnegative variation of Whitney's extension problem has attracted some recent attention (see [9,10,[20][21][22]).…”

“…The pioneer papers [5,9,28] studied the analogous version of Problem 1.3 for finite sets E. In particular, for finite sets E ⊂ R n , the authors in [9] gave an answer to the finite set version of the problem by means of proving a finiteness principle. See also [16,[25][26][27][29][30][31] for related Lipschitz selection problems and [10,[20][21][22] for related problems of nonnegative interpolation.…”