Duke Mathematical Journal

Integration of oscillatory and subanalytic functions

Raf Cluckers, Georges Comte, Daniel J. Miller, Jean-Philippe Rolin, and Tamara Servi

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Abstract

We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This article extends the investigation started by Lion and Rolin and Cluckers and Miller to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.

Article information

Source
Duke Math. J., Volume 167, Number 7 (2018), 1239-1309.

Dates
Received: 20 July 2016
Revised: 6 November 2017
First available in Project Euclid: 14 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1521014410

Digital Object Identifier
doi:10.1215/00127094-2017-0056

Mathematical Reviews number (MathSciNet)
MR3799699

Zentralblatt MATH identifier
06892359

Subjects
Primary: 26B15: Integration: length, area, volume [See also 28A75, 51M25]
Secondary: 03C64: Model theory of ordered structures; o-minimality 14P15: Real analytic and semianalytic sets [See also 32B20, 32C05] 32B20: Semi-analytic sets and subanalytic sets [See also 14P15] 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 14P10: Semialgebraic sets and related spaces 33B10: Exponential and trigonometric functions

Keywords
stability under integration oscillatory integrals Fourier transforms globally subanalytic functions constructible functions preparation theorems uniformly distributed functions oscillation index families of exponential periods o-minimality

Citation

Cluckers, Raf; Comte, Georges; Miller, Daniel J.; Rolin, Jean-Philippe; Servi, Tamara. Integration of oscillatory and subanalytic functions. Duke Math. J. 167 (2018), no. 7, 1239--1309. doi:10.1215/00127094-2017-0056. https://projecteuclid.org/euclid.dmj/1521014410


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