On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1

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Abstract: In this paper, we study the following problem: Let

D g e q 2

and let

E s u b s e t m a t h b b R^{D}

be finite satisfying certain conditions. Suppose that we are given a map

p h i : E o m a t h b b R^{D}

with

p h i

a small distortion on

E

. How can one decide whether

p h i

extends to a smooth small distortion

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

which agrees with

p h i

on

E

. We also ask how to decide if in addition

P h i

can be approximated well by certain rigid and non-rigid motions from

m a t h b b R^{D} o m a t h b b R^{D}

. Since

E

is a finite set, this question is basic to interpolation and alignment of data in

m a t h b b R^{D}

.

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