Functiones et Approximatio Commentarii Mathematici

The multidimensional van der Corput transformation

Patrick Sargos

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 20 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L07: Estimates on exponential sums

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


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.

Export citation


  • J. Guo, On lattices points in large convex bodies, Acta Arithmetica 151.1 (2012), 83–108.
  • S. Graham and G. Kolesnik, Van der Corput's Method of Exponential Sums, vol. 126 of London Mathematical Society Lecture Note Series, Cambridge University Press, 1991.
  • L. Hörmander, The analysis of linear partial differential operators I, Springer, Berlin, 1983.
  • M.N. Huxley, Area, lattice points and exponential sums, Clarendon Press, Oxford, 1996.
  • E. Krätzel, Lattice points, Kluwer, 1988.
  • E. Krätzel, Double exponential sums, Analysis 16 (1996), 109–123.
  • S. Lang, Undergraduate analysis, Springer, 2005.
  • H.Q. Liu, On the estimates of double exponential sums, Acta Arithmetica 129.3 (2007), 203–247.
  • W. Müller, Lattice points in large convex bodies, Mh. Math. 128 (1999), 315–330.
  • M. Redouaby and P. Sargos, Sur la transformation B de Van der Corput, Expo. Math. 17 (1999), 207–232.
  • B.R. Srinivasan, The lattice point problem of many-dimensional hyperboloïds II, Acta Arithmetica VIII (1963), 173–204.
  • B.R. Srinivasan, The lattice point problem of many-dimensional hyperboloids III, Math. Annalen 160 (1965), 280–311.
  • E.M. Stein, Harmonic Analysis : real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, 1999.
  • J. Vandehey, Error term improvements for Van der Corput transforms, preprint, 2012, at arXiv:1205.0090.