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The convergence of the Bochner-Riesz means at the critical index

Author(s): Lung-Kee Chen; Dashan Fan
Journal: Proc. Amer. Math. Soc. 124 (1996), 2717-2726.
MSC (1991): Primary 42A20, 42B05
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Abstract: In this paper, we study the pointwise convergence of the Bochner-Riesz means at the critical index on the space $L\log ^+ L (Q_n).$ We weaken the hypothesis, $`` x$ is a Lebesgue point", which is required on some research results by instead considering the convergence of averages of the function over balls when the radials of the balls approach to 0.

References:

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S. Bochner, Summation of Multiple Fourier Series by Spherical Means, Trans. Amer. Math. Soc. 40 (1936), 175--207.
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K. Davis and Y. Chang, Lectures on Bochner-Riesz means, London Math. Soc. Lecture Note Series, Vol. 114, Cambridge University Press, Cambridge, London, New York, Melbourne, 1987. MR 88m:42031
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S. Lu, Spherical Integrals and the Convergence of Spherical Riesz Means, Acta Math. Sinica 23 (4) (1980), 609--623. (Chinese) MR 82h:42021
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R. Salem, New Theorems on the Convergence of Fourier Series, Nederl. Akad. Wetensch. Proc. Ser. A57 = Indag. Math. 16 (1954), 550-555. MR 17:845
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E. M. Stein, On Certain Exponential Sums Arising in Multiple Fourier Series, Annals of Math. 73 (1961), 87--109. MR 23:A2715
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E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1970. MR 46:4102
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A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, London, New York, Melbourne, 1977. MR 58:29731

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Additional Information:

Lung-Kee Chen
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: chen@math.orst.edu

Dashan Fan
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: fan@csd.uwm.edu

DOI: 10.1090/S0002-9939-96-03333-3
PII: S 0002-9939(96)03333-3
Received by editor(s): August 9, 1994
Received by editor(s) in revised form: February 28, 1995
Additional Notes: The second author was supported in part by a grant of the graduate school research committee in the University of Wisconsin-Milwaukee.
Communicated by: J. Marshall Ash
Copyright of article: Copyright 1996, American Mathematical Society

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