On the Bochner-Riesz means of critical order

Author:
E. R. Liflyand

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1443-1450

MSC (1991):
Primary 42A24

DOI:
https://doi.org/10.1090/S0002-9939-97-03742-8

MathSciNet review:
1371133

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Abstract | References | Similar Articles | Additional Information

Abstract: Stein's well-known logarithmic asymptotics of the Lebesgue constants of the Bochner-Riesz means of critical order is extended to Lebesgue constants of more general linear means of multiple Fourier series. These means are generated by certain class of functions supported in convex domains with boundaries of non-vanishing Gaussian curvature.

**[Ba]**K. I. Babenko,*On the Mean Convergence of Multiple Fourier Series and the Asymptotics of the Dirichlet Kernel of Spherical Means*, Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow, Preprint No. 52, 1971 (Russian).**[Bc]**S. Bochner,*Summation of multiple Fourier series by spherical means*, Trans. Amer. Math. Soc.**40**(1936), 175 - 207.**[Be]**E. S. Belinskii,*Behavior of Lebesgue constants for some methods of summation of multiple Fourier series*, Metric Questions in the Theory of Functions and Mappings, Naukova Dumka, Kiev, 1977, pp. (19 - 39) (Russian). MR**58:29816****[BL]**E. S. Belinskii, E. R. Liflyand,*Lebesgue constants and integrability of Fourier transform of radial function*, Dokl. of the Academy of Sciences of Ukraine (6) (1980), 5 - 10 (Russian). MR**81i:42016****[CoS]**L. Colzani, P. M. Soardi,*norms of certain kernels of the -dimensional torus*, Trans. Amer. Math. Soc**266**(2) (1981), 617 - 627. MR**84e:42008****[Gi]**E. Giusti,*Minimal surfaces and functions of bounded variation*, Birkhäuser, Boston, 1984. MR**87a:58041****[IA]**V. A. Ilyin, Sh. A. Alimov,*Conditions for the convergence of expansions corresponding to self-adjoint extensions of elliptic operators. I. A self-adjoint extension of the Laplace operator with a point spectrum*, Diff. Urav.**7**(1971), no. 4, 670-710 (Russian); English translation in Diff. Eq.**7**(1971), 615-650.**[LRZ]**E. R. Liflyand, A. I. Zaslavsky, A. G. Ramm,*Estimates from below for Lebesgue constants*, J. of Fourier Analysis and Applications.**2**(1996), 287-301. CMP**96:09****[RZ1]**A. G. Ramm, A. I. Zaslavsky,*Singularities of the Radon transform*, Bull. Amer. Math. Soc.,**28**(1) (1993), 109 - 115. MR**93i:44003****[RZ2]**-,*Reconstructing singularities of a function from its Radon transform*, Math. Comp. Modelling**18**(1) (1993), 109-138. MR**94j:44006****[RZ3]**-,*Asymptotic behavior of the Fourier transform of piecewise-smooth functions*, C. R. Acad. Sci. Paris**316**(6) (1993), 541-546. MR**94d:42019****[S]**E. M. Stein,*On certain exponential sums arising in multiple Fourier series*, Ann. of Math.**73**(2) (1961), 87 - 109. MR**23:A2715****[V]**A. N. Varchenko,*Number of lattice points in families of homothetic domains*, Funk. Anal. i Ego Pril.**17**(1983), no, 2, 1-6 (Russian); English Translation in Funct. Anal. Appl.**17**(1983), 79-83. MR**85e:11077****[Z]**A. Zygmund,*Trigonometric series, Vol. I, II*, Camb. Univ. Press, Cambridge, 1959. MR**21:6498**

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Additional Information

**E. R. Liflyand**

Affiliation:
Department of Mathematics and Computer Science, Bar Ilan University, Ramat Gan, 52900, Israel

Email:
liflyand@bimacs.cs.biu.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-97-03742-8

Keywords:
Lebesgue constants,
Bochner-Riesz means,
critical order

Received by editor(s):
January 4, 1994

Received by editor(s) in revised form:
November 27, 1995

Additional Notes:
The author acknowledges the support of the Minerva Foundation in Germany through the Emmy Noether Institute in Bar-Ilan University.

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1997
American Mathematical Society