Proceedings of the American Mathematical Society

Author(s): Christopher Meaney
Journal: Proc. Amer. Math. Soc. 131 (2003), 3123-3128.
MSC (2000): Primary 42C05, 33C45, 42C10
Posted: December 30, 2002
Retrieve article in: PDF

Abstract: We show that for $\delta$ below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesàro and Riesz means of order $\delta$ .

1.: Aline Bonami and Jean-Louis Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223-263. MR 49:3461
2.: Sagun Chanillo and Benjamin Muckenhoupt, Weak type estimates for Cesàro sums of Jacobi polynomial series, Mem. Amer. Math. Soc. 102 (1993), no. 487, viii+90. MR 93g:42018
3.: Leonardo Colzani, Mitchell H. Taibleson, and Guido Weiss, Maximal estimates for Cesàro and Riesz means on spheres, Indiana Univ. Math. J. 33 (1984), no. 6, 873-889. MR 86g:43012
4.: B. Dreseler and P. M. Soardi, A Cohen-type inequality for Jacobi expansions and divergence of Fourier series on compact symmetric spaces, J. Approx. Theory 35 (1982), no. 3, 214-221. MR 84d:42029
5.: J.J. Gergen, Summability of double Fourier series, Duke Math. J. 3 (1937), 133-148.
6.: G. H. Hardy and M. Riesz, A general theory of Dirichlet series, Cambridge University Press, Cambridge, 1915.
7.: Yuichi Kanjin, Convergence almost everywhere of Bochner-Riesz means for radial functions, Ann. Sci. Kanazawa Univ. 25 (1988), 11-15. MR 90b:42034
8.: -, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1063-1069. MR 89i:42030
9.: C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), no. 1, 19-37. MR 83j:40004
10.: -, Cohen type inequalities for Jacobi, Laguerre and Hermite expansions, SIAM J. Math. Anal. 14 (1983), no. 4, 819-833. MR 85d:42025
11.: Christopher Meaney, Divergent Jacobi polynomial series, Proc. Amer. Math. Soc. 87 (1983), no. 3, 459-462. MR 84c:42040
12.: Christopher Meaney and Elena Prestini, Bochner-Riesz means on symmetric spaces, Tohoku Math. J. (2) 50 (1998), no. 4, 557-570. MR 99j:43003
13.: Benjamin Muckenhoupt and David W. Webb, Two-weight norm inequalities for Cesàro means of Laguerre expansions, Trans. Amer. Math. Soc. 353 (2001), no. 3, 1119-1149. MR 2001m:42051
14.: Krzysztof Stempak, Divergent Laguerre series, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1123-1126. MR 2001g:42057
15.: Gábor Szego, Orthogonal polynomials, third ed., American Mathematical Society, Providence, R.I., 1967, American Mathematical Society Colloquium Publications, Vol. 23. MR 46:9631
16.: A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London, 1968. MR 38:4882

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C05, 33C45, 42C10

Christopher Meaney
Affiliation: Department of Mathematics, Macquarie University, North Ryde, New South Wales 2109, Australia
Email: chrism@maths.mq.edu.au

DOI: 10.1090/S0002-9939-02-06853-3
PII: S 0002-9939(02)06853-3
Keywords: Jacobi polynomial, Laguerre function, Ces\`aro mean, Riesz mean, Cantor-Lebesgue Theorem, uniform boundedness
Received by editor(s): February 26, 2002
Received by editor(s) in revised form: April 29, 2002
Posted: December 30, 2002
Communicated by: Andreas Seeger
Copyright of article: Copyright 2002, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS