Whitney extensions and orthonormal expansions
Abstract
The Whitney near extension problem for finite sets in Rd,\, d≥ 2 asks the following: Let φ:E Rd be a near distortion on a finite set E⊂ Rd with certain geometry. How to decide whether φ extends to a smooth, one to one and onto near distortion : Rd Rd which agrees with φ on E and with Euclidean motions in Rd. The Whitney near extension problem for compact sets E⊂ U in open subsets U of Rn,\, n≥ 1 asks the following: Let U⊂ Rn be open and let E⊂ U be a compact set. Let φ:U Rn be a smooth near isometry. How to decide if there exists a smooth one-to-one and onto near isometry : Rn Rn which extends φ on E and agrees with Euclidean motions on Rn. The classical Whitney extension problem asks the following: Let φ:E R be a map defined on an arbitrary set E⊂ Rn. How can one decide whether φ extends to a map : Rn R which agrees with φ on E and is in Cm( Rn),\, m≥ 1, the space of functions from Rn to R whose derivatives of order m are continuous and bounded. In this paper, we survey some of our work on the near Whitney extension problem [2] in Rn. Thereafter, we survey some of our work on weighted Lp( R),\, 1<p≤ ∞ convergence of orthonormal expansions in R [3] and present a result of [13]. The motivation for doing this is motivated by interesting connections between Whitney extension theorems, Taylor series and Fourier expansions. Finally, we raise various open questions to study.