Skip to Main Content
Near Extensions and Alignment of Data in R(superscript)n
book

Near Extensions and Alignment of Data in R(superscript)n

by Steven B. Damelin
November 2023
Intermediate to advanced content levelIntermediate to advanced
192 pages
3h 29m
English
Wiley
Content preview from Near Extensions and Alignment of Data in R(superscript)n

12 Proofs: Gluing and Whitney Machinery

12.1 Theorem 11.23

We begin with Theorem 11.23.

Proof: Using the Lojasiewicz inequality, (see Chapter 8), there exists a Euclidean motion upper A for which we have

StartAbsoluteValue phi left-parenthesis x right-parenthesis minus upper A left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to c Subscript upper K Baseline delta Superscript 1 slash upper C Super Subscript upper K Superscript Baseline comma x element-of upper E period

Without loss of generality, we may replace phi by phi o upper A Superscript negative 1. Hence, we may suppose that

StartAbsoluteValue phi left-parenthesis x right-parenthesis minus x EndAbsoluteValue less-than-or-equal-to upper C prime Subscript upper K Baseline delta Superscript 1 slash upper C Super Subscript k Superscript Baseline comma x element-of upper E period

Now we will employ a similar technique to the proof of Lemma 9.15.

Let theta left-parenthesis y right-parenthesis be a smooth cut off function on double-struck upper R Superscript d such that theta left-parenthesis y right-parenthesis equals 1 for StartAbsoluteValue y EndAbsoluteValue less-than-or-equal-to 1 slash 100, theta left-parenthesis y right-parenthesis equals 0 for and with for all . Then set

The summands are smooth and have pairwise disjoint supports ...

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

American Signal Intelligence in Northwest Africa and Western Europe

American Signal Intelligence in Northwest Africa and Western Europe

George F. Howe

Publisher Resources

ISBN: 9781394196777Purchase Link