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.
Citation
Patrick Sargos. "The multidimensional van der Corput transformation." Funct. Approx. Comment. Math. 52 (1) 133 - 176, March 2015. https://doi.org/10.7169/facm/2015.52.1.11
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