Arie Israel

Sobolev extension by linear operators

Let L(R) be the Sobolev space of functions with mth derivatives lying in L(R). Assume that n < p < ∞. For E ⊂ R, let L(E) denote the space of restrictions to E of functions in L(R). We show that there exists a bounded linear map T : Lm,p(E)→ L(R) such that, for any f ∈ L(E), we have Tf = f on E. We also give a formula for the order of magnitude of ‖f‖Lm,p(E) for a given f : E→ R when E is finite.

Interpolation of data by smooth nonnegative functions

We prove a finiteness principle for interpolation of data by nonnegative Cm 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.

Finiteness Principles for Smooth Selection

In this paper we prove finiteness principles for

Fitting a Sobolev function to data II

{C^m{({\mathbb{R}^n},{\mathbb{R}^D)}}}

The structure of Sobolev extension operators

Cm(Rn,RD) and

A coordinate-free proof of the finiteness principle for the Whitney extension problem.

{C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)}