On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$

Author name not available (Why is that?)

Abstract: In this announcement we consider the following problem. Let

n, m g e q 1

,

U s u b s e t m a t h b b R^{n}

open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let

p h i : U o m a t h b b R^{n}

be a

C^{m}

map. If

E s u b s e t U

is compact (with some geometry) and the restriction of

p h i

to

E

is an almost isometry with small distortion, how to decide when there exists a

C^{m} (m a t h b b R^{n})

one-to-one and onto almost isometry

P h i : m a t h b b R^{n} o m a t h b b R^{n}

with small distortion which agrees with

p h i

in a neighborhood of

E

and a Euclidean motion

A : m a t h b b R^{n} o m a t h b b R^{n}

away from

E

. (b) Let

p h i : U o m a t h b b R^{n}

be

C^{i n f t y}

map. If

E s u b s e t U

is compact (with some geometry) and the restriction of

p h i

to

E

is an almost isometry with small distortion, how to decide when there exists a

C^{i n f t y} (m a t h b b R^{n})

one-to-one and onto almost isometry

P h i : m a t h b b R^{n} o m a t h b b R^{n}

with small distortion which agrees with

p h i

in a neighborhood of

E

and a Euclidean motion

A : m a t h b b R^{n} o m a t h b b R^{n}

away from

E

. Our results complement those of [14,15,20] where there,

E

is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in

m a t h b b R^{n}

.

Mathematics Subject Classification ID

No records found.

This page was built for publication: On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6507437)