Growth of Lebesgue constants for convex polyhedra and other regions
Authors:
J. Marshall Ash and Laura De Carli
Journal:
Trans. Amer. Math. Soc. 361 (2009), 4215-4232
MSC (2000):
Primary 42B15, 42A05; Secondary 42B08, 42A45
DOI:
https://doi.org/10.1090/S0002-9947-09-04627-3
Published electronically:
March 4, 2009
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For any convex polyhedron in
,
, and
, there are constants
and
such that











- [A]
J. M. Ash, Triangular Dirichlet kernels and growth of
Lebesgue constants, preprint.
- [AAJRS]
B. Anderson, J. M. Ash, R. Jones, D. G. Rider, B. Saffari, Exponential sums with coefficients 0 or
and concentrated
norms, to appear in Annales de l'Institut Fourier.
- [C] C. K. Chui, An introduction to wavelets. Wavelet Analysis and its Applications, 1. Academic Press, Inc., Boston, MA, 1992. MR 1150048 (93f:42055)
- [E] A. Erdélyi et al., Higher transcendental functions. Vol. II. Based, in part, on notes left by Harry Bateman. McGraw-Hill, New York-Toronto-London, 1953. Page 85, 7.13.1(3).
- [GKP] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics. A foundation for computer science. Second edition. Addison-Wesley Publishing Company, Reading, MA, 1994. MR 1397498 (97d:68003)
- [I] V. A. Ilyin, Problems of localization and convergence for Fourier series in fundamental systems of the Laplace operator, Uspekhi Mat. Nauk 23(1968), 61-120. (Translation in Russian Mathematical Surveys 23, 59-116.) MR 0223823 (36:6870)
- [K] A. Ya. Khinchin, Continued Fractions, Nauka, Moscow (1978). Also translated from the third (1961) Russian edition. Reprint of the 1964 translation. Dover Publications, Inc., Mineola, NY, 1997. MR 1451873 (98c:11008)
- [L] E. R. Liflyand, Lebesgue constants of multiple Fourier series, Online Journal of Analytic Combinatorics, Issue 1 (2006), # 5, 1-112. MR 2249993 (2007g:42017)
- [NP] F. L. Nazarov and A. N. Podkorytov, On the behavior of the Lebesgue constants of two-dimensional Fourier sums over polygons, St. Petersburg Math. J. 7 (1996), 663-680. MR 1356537 (96m:42019)
- [NZ] Niven and Zuckerman, An introduction to the theory of numbers, 3rd edition, Wiley, 1972. MR 0344181 (49:8921)
- [Sh] H. S. Shapiro, Lebesgue constants for spherical partial sums, J. of Approximation theory 13(1975), 40-44. MR 0374808 (51:11004)
- [S] E M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993, pp. 362-363. MR 1232192 (95c:42002)
- [SW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32. Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
- [TB] R. M. Trigub and E. S. Belinsky, Fourier analysis and approximation of functions, Kluwer Academic Publishers, Dordrecht, 2004. MR 2098384 (2005k:42002)
- [Y] V. A. Yudin, Behavior of Lebesgue constants, Translated from Matematicheskie Zametki, 17(1975), 401-405. (Translation is in Mathematical Notes 17 on pages 233-235.) MR 0417690 (54:5739)
- [YY] A. A. Yudin and V. A. Yudin, Polygonal Dirichlet kernels and growth of Lebesgue constants, Translated from Matematicheskie Zametki, 37(1985), 220-236. (Translation is in Mathematical Notes 37 on pages 124-135.) MR 784367 (86k:42039)
- [Z] A. Zygmund, Trigonometric Series, v.1, 2nd ed., Cambridge University Press, New York, 1959. MR 0107776 (21:6498)
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B15, 42A05, 42B08, 42A45
Retrieve articles in all journals with MSC (2000): 42B15, 42A05, 42B08, 42A45
Additional Information
J. Marshall Ash
Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614
Email:
mash@math.depaul.edu
Laura De Carli
Affiliation:
Department of Mathematics, Florida International University, University Park, Miami, Florida 33199
Email:
decarlil@fiu.edu
DOI:
https://doi.org/10.1090/S0002-9947-09-04627-3
Keywords:
Lebesgue constant,
Dirichlet kernel,
kernels for convex sets,
kernels for polyhedra
Received by editor(s):
January 16, 2007
Received by editor(s) in revised form:
August 3, 2007
Published electronically:
March 4, 2009
Additional Notes:
The first author’s research was partially supported by a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.