Functiones et Approximatio Commentarii Mathematici

The multidimensional van der Corput transformation

Patrick Sargos

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Abstract

We make a systematic study of van der Corput's $B$-process for multiple exponential sums. We study directly the important case where the determinant of the Hessian $H_{f}(\mathbf{x})$ of the phase $f$ may be abnormally small. This requires a work on multidimensional stationary phase integrals uniform in $\delta$, the lower bound for $||\det H_{f}(\mathbf{x})||$. In passing, we obtain an independent result on the asymptotic behaviour of the stationary phase integral when the critical point of the phase is also a singular point of the boundary of the domain of integration. The whole paper is self-contained.

Article information

Source
Funct. Approx. Comment. Math., Volume 52, Number 1 (2015), 133-176.

Dates
First available in Project Euclid: 20 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.facm/1426857041

Digital Object Identifier
doi:10.7169/facm/2015.52.1.11

Mathematical Reviews number (MathSciNet)
MR3326130

Zentralblatt MATH identifier
06425019

Subjects
Primary: 11L07: Estimates on exponential sums

Keywords
van der Corput's $B$-process multidimensional stationary phase integrals oscillatory integrals change of variables

Citation

Sargos, Patrick. The multidimensional van der Corput transformation. Funct. Approx. Comment. Math. 52 (2015), no. 1, 133--176. doi:10.7169/facm/2015.52.1.11. https://projecteuclid.org/euclid.facm/1426857041


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