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Efficient Algorithms for Approximate Smooth Selection

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Abstract

In this paper, we provide efficient algorithms for approximate \({\mathcal {C}}^m({\mathbb {R}}^n, {\mathbb {R}}^D)-\)selection. In particular, given a set E, a constant \(M_0 > 0\), and convex sets \(K(x) \subset {\mathbb {R}}^D\) for \(x \in E\), we show that an algorithm running in \(C(\tau ) N \log N\) steps is able to solve the smooth selection problem of selecting a point \(y \in (1+\tau )\blacklozenge K(x)\) for \(x \in E\) for an appropriate dilation of K(x), \((1+\tau )\blacklozenge K(x)\), and guaranteeing that a function interpolating the points (x, y) will be \({\mathcal {C}}^m({\mathbb {R}}^n, {\mathbb {R}}^D)\) with norm bounded by CM.

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Authors and Affiliations

  1. Princeton University, Princeton, NJ, USA

    Charles Fefferman & Bernat Guillén Pegueroles

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  1. Charles Fefferman
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  2. Bernat Guillén Pegueroles
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Correspondence to Bernat Guillén Pegueroles.

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In fond memory of Elias Stein.

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Charles Fefferman: Supported by AFOSR FA9550-18-1-069, NSF DMS-1700180, BSF 2014055. Supported by the US-Israel BSF. Bernat Guillén Pegueroles: Supported by Fulbright-Telefónica, AFOSR FA9550-18-1-069, NSF.

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Fefferman, C., Guillén Pegueroles, B. Efficient Algorithms for Approximate Smooth Selection. J Geom Anal 31, 6530–6600 (2021). https://doi.org/10.1007/s12220-019-00242-y

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