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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fourier expansions of functions with bounded variation of several variables


Author: Leonardo Colzani
Journal: Trans. Amer. Math. Soc. 358 (2006), 5501-5521
MSC (2000): Primary 42B08, 43A50
Published electronically: July 20, 2006
MathSciNet review: 2238924
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Abstract: In the first part of the paper we establish the pointwise convergence as $ t\rightarrow +\infty $ for convolution operators $ \int_{\mathbb{R}^{d}}t^{d}K\left( ty\right) \varphi (x-y)dy$ under the assumptions that $ \varphi (y)$ has integrable derivatives up to an order $ \alpha $ and that $ \left\vert K(y)\right\vert \leq c\left( 1+\left\vert y\right\vert \right) ^{-\beta }$ with $ \alpha +\beta >d$. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.


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Additional Information

Leonardo Colzani
Affiliation: Dipartimento di Matematica, Università di Milano–Bicocca, Edificio U5, via R.Cozzi 53, 20125 Milano, Italia
Email: leonardo@matapp.unimib.it

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03910-9
PII: S 0002-9947(06)03910-9
Received by editor(s): April 26, 2004
Received by editor(s) in revised form: November 16, 2004
Published electronically: July 20, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.