Coordinate-free principles for extension of smooth functions

Date

2021-06-23

Authors

Frei-Pearson, Abraham

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Abstract

This dissertation will study interpolation of smooth functions, broadly defined, in two related contexts. Given a finite subset E of ℝ [superscript n] and a function f : E→ℝ , what is the smallest C [superscript m]-norm of a function F : ℝ [superscript n]→ℝ extending f? In chapter 2, we prove the following result: for every m,n, there exist constants k [superscript #] and C [superscript #] depending on m and n only such that the following holds. Suppose that for every set S ⊂ E with at most k [superscript #] points, there exists a function F [superscript S] : ℝ [superscript n] → ℝ such that F [superscript S] |[subscript S] = f|[subscript S], and [double bar]F [superscript S] [double bar] [subscript C superscript m][subscript parenthesis ℝ superscript n parenthesis] ≤ 1. Then there exists a function F extending f of C [superscript m]-norm at most C [superscript #]. Our approach to this theorem is coordinate-free, and establishes constants which are an exponential improvement over the constants previously established in the literature. Our results are proved by induction on a measure of the tameness of the set E. In order to control the number of steps in the induction argument, we must identify a certain quantity called the signature which has crucial monotonicity properties. In chapter 3, we investigate a related problem. Suppose (X, d) is a metric space, and Γ is a map from X into the compact, convex subsets of the hyperbolic plane ℍ². We are interested in constructing a Lipschitz selection of Γ, i.e. a Lipschitz map F : X → ℍ² such that F(x) ∈ Γ (x) for all x ∈ X. We establish the following results: there exist universal constants k [superscript #] and C [superscript #] independent of Γ and X such that the following holds. Suppose that, for all subsets S ⊂ X containing at most k [superscript #] points, there is a map F [superscript S] : S → ℍ² of Lipschitz constant 1 such that F [superscript S] (x) ∈ Γ (x) for all x ∈ S. Then there is a map F : X → ℍ² such that F(x) ∈ Γ(x) for all x ∈ X with Lipschitz constant at most C [superscript #].

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