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On pointwise convergence of Fourier series
of radial functions in several variables


Author: Shigehiko Kuratsubo
Journal: Proc. Amer. Math. Soc. 127 (1999), 2987-2994
MSC (1991): Primary 42B05
DOI: https://doi.org/10.1090/S0002-9939-99-04886-8
Published electronically: April 28, 1999
MathSciNet review: 1605996
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the pointwise convergence of the Fourier series for radial functions in several variables, which in the case $n=1$ is the Dirichlet-Jordan theorem itself. In our proof the method for the case of the indicator function of the ball is very useful.


References [Enhancements On Off] (What's this?)

  • 1. L. Brandolini and L. Colzani, A convergence theorem for multiple Fourier series (preprint).
  • 2. François Fricker, Einführung in die Gitterpunktlehre, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 73, Birkhäuser Verlag, Basel-Boston, Mass., 1982 (German). MR 673938
  • 3. G. H. Hardy and E. Landau, The lattice points of a circle, Proc. Roy. Soc. London Ser. A 105 (1924), 244-258.
  • 4. Shigehiko Kuratsubo, On pointwise convergence of Fourier series of indicator function of 𝑛-dimensional ball, Sci. Rep. Hirosaki Univ. 43 (1996), no. 2, 199–208. MR 1445025
  • 5. E. Landau, Zur Analytischen Zahlentheorie der definiten quadratischen Formen (Über die Gitterpunkte in einem mehrdimensional Ellipsoid. ), Sitzungsber. Kgl. Preuss. Akad. Wiss. 31 (1915), 11-29.
  • 6. Břetislav Novák, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Czechoslovak Math. J. 22(97) (1972), 495–507 (German). MR 0308055
  • 7. Mark A. Pinsky, Pointwise Fourier inversion and related eigenfunction expansions, Comm. Pure Appl. Math. 47 (1994), no. 5, 653–681. MR 1278348, https://doi.org/10.1002/cpa.3160470504
  • 8. Mark A. Pinsky, Nancy K. Stanton, and Peter E. Trapa, Fourier series of radial functions in several variables, J. Funct. Anal. 116 (1993), no. 1, 111–132. MR 1237988, https://doi.org/10.1006/jfan.1993.1106
  • 9. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972

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Additional Information

Shigehiko Kuratsubo
Affiliation: Department of Mathematics, Hirosaki University, Hirosaki 036, Japan
Email: kuratubo@cc.hirosaki-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-99-04886-8
Keywords: Fourier series, spherical partial sum, bounded variation, indicator function, lattice point problem
Received by editor(s): September 30, 1997
Received by editor(s) in revised form: January 7, 1998
Published electronically: April 28, 1999
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society