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Mean Cesàro summability of Laguerre and Hermite series


Author: Eileen L. Poiani
Journal: Trans. Amer. Math. Soc. 173 (1972), 1-31
MSC: Primary 42A56; Secondary 33A65
DOI: https://doi.org/10.1090/S0002-9947-1972-0310537-9
MathSciNet review: 0310537
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Abstract: The primary purpose of this paper is to prove inequalities of the type $ \vert\vert{\sigma _n}(f,x)W(x)\vert{\vert _p} \leqslant C\vert\vert f(x)W(x)\vert{\vert _p}$ where $ {\sigma _n}(f,x)$ is the $ n$th $ (C,1)$ mean of the Laguerre or Hermite series of $ f, W(x)$ is a suitable weight function of particular form, $ C$ is a constant independent of $ f(x)$ and $ n$, and the norm is taken over $ (0,\infty )$ in the Laguerre case and $ ( - \infty ,\infty )$ in the Hermite case for $ 1 \leqslant p \leqslant \infty $. Both necessary and sufficient conditions for these inequalities to remain valid are determined. For $ p < \infty $ and $ f(x)W(x) \in {L^p}$, mean summability results showing that $ \mathop {\lim }\nolimits_{n \to \infty } \vert\vert[{\sigma _n}(f,x) - f(x)]W(x)\vert{\vert _p} = 0$ are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.


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  • [1] R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708. MR 32 #316. MR 0182834 (32:316)
  • [2] R. Campbell, Détermination effective de toutes les moyennes de Cesàro d'ordre entier pour les séries de polynomes orthogonaux comprenant ceux de Laguerre et de Hermite, C. R. Acad. Sci. Paris 243 (1956), 882-885. MR 18, 125. MR 0079669 (18:125d)
  • [3] A. Erdélyi et al., Higher transcendental functions. Vol. II, Bateman Manuscript Project, McGraw-Hill, New York, 1953. MR 15, 419.
  • [4] A. Erdélyi, Asymptotic forms for Laguerre polynomials, J. Indian Math. Soc. 24 (1960), 235-250. MR 23 #A1073. MR 0123751 (23:A1073)
  • [5] D. Ernst, Über die Konvergenz der $ (C,1)$-Mittel von Fourier-Laguerre-Reihen, Compositio Math. 21 (1969), 81-101. MR 39 #3231. MR 0241894 (39:3231)
  • [6] H. Jeffreys and B. Swirles Jeffreys, Methods of mathematical physics, 3rd ed., Cambridge Univ. Press, Cambridge, 1956. MR 17, 590. MR 1744997 (2000i:00004)
  • [7] E. Kogbetliantz, Sur les moyennes arithmétiques des séries-noyaux des développements en séries d'Hermite et de Laguerre et sur celles de ces séries-noyaux derivées terms à terme, J. Math. Phys. 14 (1935), 37-99.
  • [8] B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231-242. MR 40 #3158. MR 0249917 (40:3158)
  • [9] -, Asymptotic forms for Laguerre polynomials, Proc. Amer. Math. Soc. 24 (1970), 288-292. MR 40 #4503. MR 0251272 (40:4503)
  • [10] -, Mean convergence of Hermite and Laguerre series. I, Trans. Amer. Math. Soc. 147 (1970), 419-431. MR 41 #711.
  • [11] -, Mean convergence of Hermite and Laguerre series. II, Trans. Amer. Math. Soc. 147 (1970), 433-460. MR 41 # 711. MR 0256051 (41:711)
  • [12] G. Szegö, Orthogonal polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1967.
  • [13] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, New York, 1966. MR 1349110 (96i:33010)
  • [14] A. Zygmund, Trigonometric series. Vols. I, II, 2nd ed., Cambridge Univ. Press, New York, 1968. MR 38 #4882. MR 0236587 (38:4882)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0310537-9
Keywords: Mean Cesàro summability, Laguerre series, Hermite series, norm inequalities, weight function, Cesàro kernel estimates
Article copyright: © Copyright 1972 American Mathematical Society

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