Efficient Algorithms for Approximate Smooth Selection

Charles Fefferman; Bernat Guillén Pegueroles

doi:10.1007/s12220-019-00242-y

Efficient Algorithms for Approximate Smooth Selection

Charles Fefferman, Bernat Guillén Pegueroles

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper, we provide efficient algorithms for approximate C^m(Rⁿ, R^D) - selection. In particular, given a set E, a constant M> 0 , and convex sets K(x) ⊂ R^D for x∈ E, we show that an algorithm running in C(τ) Nlog N steps is able to solve the smooth selection problem of selecting a point y∈ (1 + τ) ⧫K(x) for x∈ E for an appropriate dilation of K(x), (1 + τ) ⧫K(x) , and guaranteeing that a function interpolating the points (x, y) will be C^m(Rⁿ, R^D) with norm bounded by CM.

Original language	English (US)
Pages (from-to)	6530-6600
Number of pages	71
Journal	Journal of Geometric Analysis
Volume	31
Issue number	7
DOIs	https://doi.org/10.1007/s12220-019-00242-y
State	Published - Jul 2021

All Science Journal Classification (ASJC) codes

Geometry and Topology

Keywords

Approximate algorithms
Efficient algorithms
Partition of unity
Smooth selection

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10.1007/s12220-019-00242-y

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title = "Efficient Algorithms for Approximate Smooth Selection",

abstract = "In this paper, we provide efficient algorithms for approximate Cm(Rn, RD) - selection. In particular, given a set E, a constant M> 0 , and convex sets K(x) ⊂ RD for x∈ E, we show that an algorithm running in C(τ) Nlog N steps is able to solve the smooth selection problem of selecting a point y∈ (1 + τ) ⧫K(x) for x∈ E for an appropriate dilation of K(x), (1 + τ) ⧫K(x) , and guaranteeing that a function interpolating the points (x, y) will be Cm(Rn, RD) with norm bounded by CM.",

keywords = "Approximate algorithms, Efficient algorithms, Partition of unity, Smooth selection",

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TY - JOUR

T1 - Efficient Algorithms for Approximate Smooth Selection

AU - Fefferman, Charles

AU - Guillén Pegueroles, Bernat

PY - 2021/7

Y1 - 2021/7

N2 - In this paper, we provide efficient algorithms for approximate Cm(Rn, RD) - selection. In particular, given a set E, a constant M> 0 , and convex sets K(x) ⊂ RD for x∈ E, we show that an algorithm running in C(τ) Nlog N steps is able to solve the smooth selection problem of selecting a point y∈ (1 + τ) ⧫K(x) for x∈ E for an appropriate dilation of K(x), (1 + τ) ⧫K(x) , and guaranteeing that a function interpolating the points (x, y) will be Cm(Rn, RD) with norm bounded by CM.

AB - In this paper, we provide efficient algorithms for approximate Cm(Rn, RD) - selection. In particular, given a set E, a constant M> 0 , and convex sets K(x) ⊂ RD for x∈ E, we show that an algorithm running in C(τ) Nlog N steps is able to solve the smooth selection problem of selecting a point y∈ (1 + τ) ⧫K(x) for x∈ E for an appropriate dilation of K(x), (1 + τ) ⧫K(x) , and guaranteeing that a function interpolating the points (x, y) will be Cm(Rn, RD) with norm bounded by CM.

KW - Approximate algorithms

KW - Efficient algorithms

KW - Partition of unity

KW - Smooth selection

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Efficient Algorithms for Approximate Smooth Selection

Abstract

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