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The Dirichlet-Jordan test and multidimensional extensions

Author: Michael Taylor
Journal: Proc. Amer. Math. Soc. 129 (2001), 1031-1035
MSC (1991): Primary 42B08, 35P10
Published electronically: October 10, 2000
MathSciNet review: 1709767
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If $\mathcal{F}$ is a foliation of an open set $\Omega\subset \mathbb{R}^n$ by smooth $(n-1)$-dimensional surfaces, we define a class of functions $\mathcal{B}(\Omega,\mathcal{F})$, supported in $\Omega$, that are, roughly speaking, smooth along $\mathcal{F}$ and of bounded variation transverse to $\mathcal{F} $. We investigate geometrical conditions on $\mathcal{F}$ that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.

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Additional Information

Michael Taylor
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3902

Keywords: Fourier series, Dirichlet-Jordan test, Gibbs phenomenon
Received by editor(s): April 29, 1999
Received by editor(s) in revised form: June 22, 1999
Published electronically: October 10, 2000
Additional Notes: The author was partially supported by NSF grant DMS-9600065
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society