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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
DOI: https://doi.org/10.1090/S0002-9947-06-03910-9
Published electronically: July 20, 2006
MathSciNet review: 2238924
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Abstract | References | Similar Articles | Additional Information

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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  • 1. S. Alimov, On the eigenfunction expansion of a piecewise smooth function, Journal Fourier Analysis Applications 9 (2003), 67-76. MR 1953073 (2003m:42027)
  • 2. A. Beurling, Sur les ensembles exceptionnels, Acta Mathematica 72, 1940, 1-13. MR 0001370 (1:226a)
  • 3. S. Bochner, Summation of multiple Fourier series by spherical means, Transactions American Mathematical Society 40 (1936), 175-207. MR 1501870
  • 4. L. Brandolini, L. Colzani, Localization and convergence of eigenfunction expansions, Journal Fourier Analysis Applications 5 (1999), 431-447. MR 1755098 (2001g:42054)
  • 5. L. Brandolini, L. Colzani, Decay of Fourier transforms and summability of eigenfunction expansions, Annali Scuola Normale Superiore Pisa 29 (2000), 611-638. MR 1817712 (2002e:35178)
  • 6. L. Brandolini, L. Colzani, A. Iosevich, G. Travaglini, The rate of convergence of Fourier expansions in the plane: A geometric viewpoint, Mathematische Zeitschrift 242 (2002), 709-724. MR 1981194 (2004a:42009)
  • 7. A. Carbery, F. Soria, Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an L(2)-localisation principle, Revista Matemática Iberoamericana 4 (1988), 319-337. MR 1028744 (91d:42015)
  • 8. L. Colzani, M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, Journal Approximation Theory 80 (1995), 119-131. MR 1308597 (95k:42021)
  • 9. L. DeMichele, D. Roux, Approximate units and Gibbs phenomenon, Bollettino Unione Matematica Italiana (A) 7 (1997), 739-746. MR 1489045 (99c:42011)
  • 10. L. DeMichele, D. Roux, The Gibbs phenomenon for $ L_{loc}^{1}$ kernels, Journal Approximation Theory 100 (1999), 144-156. MR 1710557 (2000m:42012)
  • 11. A. Gray, M.A. Pinsky, Gibbs' phenomenon for Fourier-Bessel series, Expositiones Mathematicae 11 (1993), 123-135. MR 1214664 (94g:42043)
  • 12. H. Federer, On spherical summation of the Fourier transform of a distribution whose partial derivatives are representable by integration, Annals Mathematics 91 (1970), 136-143. MR 0410251 (53:14001)
  • 13. E. Hlawka, Über Integrale auf convexen Körpen, I, II, Monatshefte Mathematik 54 (1950), 1-36, 81-99.
  • 14. L. Hörmander, The analysis of linear partial differential operators, I, II, III, IV, Springer-Verlag, Berlin, 1983-1985. MR 0705278 (85g:35002b)
  • 15. V.A. Il'in, Theorem on the possibility of expansion of a piecewise smooth function according to eigenfunctions of an arbitrary two-dimensional region, Doklady Akademii Nauk SSSR (N.S.) 109 (1956), 442-445. MR 0087020 (19:288a)
  • 16. J.P. Kahane, Le phénomène de Pinsky et la géométrie des surfaces, Comptes Rendus Académie Sciences Paris (I) 321 (1995), 1027-1029. MR 1360566 (96m:42018)
  • 17. E. Montini, On the capacity of sets of divergence associated with the spherical partial integral operator, Transactions American Mathematical Society 355 (2003), 1415-1441. MR 1946398 (2003h:42020)
  • 18. C. Meaney, On almost-everywhere convergent eigenfunction expansions of the Laplace-Beltrami operator, Mathematical Proceedings Cambridge Philosophical Society 92 (1982), 129-131. MR 0662968 (83k:58093)
  • 19. M.A. Pinsky, Pointwise Fourier inversion and related eigenfunction expansions, Communications Pure Applied Mathematics 47 (1994), 653-681. MR 1278348 (95e:35145)
  • 20. M.A. Pinsky, Fourier inversion in the piecewise smooth category, in Fourier Analysis, analytic and geometric aspects, eds. W.O. Bray, P.S. Milojevic, C.V. Stanojevic, Marcel Dekker (1994). MR 1277831 (95h:42014)
  • 21. M.A. Pinsky, M.E. Taylor, Pointwise Fourier inversion: a wave equation approach, Journal Fourier Analysis Applications 3 (1997), 647-703. MR 1481629 (99d:42019)
  • 22. R. Salem, A. Zygmund, Capacity of sets and Fourier series, Transactions American Mathematical Society 59 (1946), 23-41. MR 0015537 (7:434h)
  • 23. M.E. Taylor, Pointwise Fourier inversion on tori and other compact manifolds, Journal Fourier Analysis Applications 5 (1999), 449-463. MR 1755099 (2001i:42018)
  • 24. M.E. Taylor, The Dirichlet-Jordan test and multidimensional extensions, Proceedings American Mathematical Society 129 (2001), 1031-1035. MR 1709767 (2001g:42020)
  • 25. M.E. Taylor, The Gibbs phenomenon, the Pinsky phenomenon, and variants for eigenfunction expansions, Communications Partial Differential Equations 27 (2002), 565-605. MR 1900555 (2003d:58047)
  • 26. G.V. Welland, Riesz-Bochner summability of Fourier integrals and Fourier series, Studia Mathematica 44 (1972), 229-238. MR 0313714 (47:2268)
  • 27. A. Zygmund, Trigonometric series I, II, Cambridge University Press, 1977. MR 0617944 (58:29731)

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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: https://doi.org/10.1090/S0002-9947-06-03910-9
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.

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