Divergent Cesàro and Riesz means of Jacobi and Laguerre expansions

Meaney, Christopher

doi:10.1090/S0002-9939-02-06853-3

Divergent Cesàro and Riesz means of Jacobi and Laguerre expansions
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by Christopher Meaney

Proc. Amer. Math. Soc. 131 (2003), 3123-3128

DOI: https://doi.org/10.1090/S0002-9939-02-06853-3

Published electronically: December 30, 2002

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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 $\delta$.

References

Aline Bonami and Jean-Louis Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223–263 (French). MR 338697, DOI 10.1090/S0002-9947-1973-0338697-5
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 1132609, DOI 10.1090/memo/0487
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 763947, DOI 10.1512/iumj.1984.33.33047
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 663667, DOI 10.1016/0021-9045(82)90003-X
J.J. Gergen, Summability of double Fourier series, Duke Math. J. 3 (1937), 133–148.
G. H. Hardy and M. Riesz, A general theory of Dirichlet series, Cambridge University Press, Cambridge, 1915.
Yūichi Kanjin, Convergence almost everywhere of Bochner-Riesz means for radial functions, Ann. Sci. Kanazawa Univ. 25 (1988), 11–15. MR 964084
Yūichi Kanjin, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1063–1069. MR 954984, DOI 10.1090/S0002-9939-1988-0954984-8
C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), no. 1, 19–37 (English, with Russian summary). MR 662701, DOI 10.1007/BF02073769
C. Markett, Cohen type inequalities for Jacobi, Laguerre and Hermite expansions, SIAM J. Math. Anal. 14 (1983), no. 4, 819–833. MR 704495, DOI 10.1137/0514063
Christopher Meaney, Divergent Jacobi polynomial series, Proc. Amer. Math. Soc. 87 (1983), no. 3, 459–462. MR 684639, DOI 10.1090/S0002-9939-1983-0684639-4
Christopher Meaney and Elena Prestini, Bochner-Riesz means on symmetric spaces, Tohoku Math. J. (2) 50 (1998), no. 4, 557–570. MR 1653434, DOI 10.2748/tmj/1178224898
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 1804415, DOI 10.1090/S0002-9947-00-02729-X
Krzysztof Stempak, Divergent Laguerre series, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1123–1126. MR 1709766, DOI 10.1090/S0002-9939-00-05657-4
Gábor Szegő, Orthogonal polynomials, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1967. MR 0310533
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587