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On pointwise convergence of Fourier series of radial functions in several variables
Author(s):
Shigehiko
Kuratsubo
Journal:
Proc. Amer. Math. Soc.
127
(1999),
2987-2994.
MSC (1991):
Primary 42B05
Posted:
April 28, 1999
MathSciNet review:
1605996
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Abstract:
We prove the pointwise convergence of the Fourier series for radial functions in several variables, which in the case  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:
- 1.
- L. Brandolini and L. Colzani, A convergence theorem for multiple Fourier series (preprint).
- 2.
- F. Fricker, Einführung in die Gitterpunktlehre, Birkhäuser Verlag, 1982. MR 84b:10001
- 3.
- G. H. Hardy and E. Landau, The lattice points of a circle, Proc. Roy. Soc. London Ser. A 105 (1924), 244-258.
- 4.
- S. Kuratsubo, On pointwise convergence of Fourier series of the indicator function of n dimensional ball, Sci. Report Hirosaki 43 (1996), 199-208. MR 98e:42010
- 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.
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- 7.
- M. A. Pinsky, Pointwise Fourier Inversion and Related Eigenfunction Expansions, Comm. Pure and Appl. Math. 47 (1994), 653-681. MR 95e:35145
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- M. A. Pinsky, N. K. Stanton and P. E. Trapa, Fourier Series of Radial Functions in Several Variables, J. Funct. Anal. 116 (1993), 111-132. MR 94j:42019
- 9.
- E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. MR 46:4102
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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:
10.1090/S0002-9939-99-04886-8
PII:
S 0002-9939(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
Posted:
April 28, 1999
Communicated by:
Frederick W. Gehring
Copyright of article:
Copyright
1999,
American Mathematical Society
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