The Dirichlet-Jordan test and multidimensional extensions
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- by Michael Taylor PDF
- Proc. Amer. Math. Soc. 129 (2001), 1031-1035 Request permission
Abstract:
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.References
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Additional Information
- Michael Taylor
- Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3902
- MR Author ID: 210423
- Email: met@math.unc.edu
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1031-1035
- MSC (1991): Primary 42B08, 35P10
- DOI: https://doi.org/10.1090/S0002-9939-00-05658-6
- MathSciNet review: 1709767