Almost everywhere summability of Laguerre series. II

K. Stempak

Studia Mathematica (1992)

  • Volume: 103, Issue: 3, page 317-327
  • ISSN: 0039-3223

Abstract

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Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions L n a ( x ) = ( n ! / Γ ( n + a + 1 ) ) 1 / 2 e - x / 2 x a / 2 L n a ( x ) , n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the A p weights theory. We also take the opportunity to comment and slightly improve on our results from [9].

How to cite

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Stempak, K.. "Almost everywhere summability of Laguerre series. II." Studia Mathematica 103.3 (1992): 317-327. <http://eudml.org/doc/215955>.

@article{Stempak1992,
abstract = {Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L^a_n(x) = (n!/Γ(n+a+1))^\{1/2\} e^\{-x/2\} x^\{a/2\} L_n^a(x)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].},
author = {Stempak, K.},
journal = {Studia Mathematica},
keywords = {Laguerre expansions; Cesàro means; almost everywhere convergence; Laguerre expansion; Laguerre series; weights},
language = {eng},
number = {3},
pages = {317-327},
title = {Almost everywhere summability of Laguerre series. II},
url = {http://eudml.org/doc/215955},
volume = {103},
year = {1992},
}

TY - JOUR
AU - Stempak, K.
TI - Almost everywhere summability of Laguerre series. II
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 3
SP - 317
EP - 327
AB - Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L^a_n(x) = (n!/Γ(n+a+1))^{1/2} e^{-x/2} x^{a/2} L_n^a(x)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].
LA - eng
KW - Laguerre expansions; Cesàro means; almost everywhere convergence; Laguerre expansion; Laguerre series; weights
UR - http://eudml.org/doc/215955
ER -

References

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  1. [1] R. Askey and I. I. Hirschman, Jr., Mean summability for ultraspherical polynomials, Math. Scand. 12 (1963), 167-177. Zbl0132.29501
  2. [2] A. Bonami et J.-L. Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223-263. 
  3. [3] J. Długosz, Almost everywhere convergence of some summability methods for Laguerre series, Studia Math. 82 (1985), 199-209. Zbl0574.42020
  4. [4] C. Markett, Norm estimates for Cesàro means of Laguerre expansions, in: Approximation and Function Spaces (Proc. Conf. Gdańsk 1979), North-Holland, Amsterdam 1981, 419-435. 
  5. [5] C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), 19-37. Zbl0515.42023
  6. [6] B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231-242. Zbl0175.12602
  7. [7] B. Muckenhoupt, Mean convergence of Hermite and Laguerre series. II, ibid. 147 (1970), 433-460. 
  8. [8] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton 1971. Zbl0232.42007
  9. [9] K. Stempak, Almost everywhere summability of Laguerre series, Studia Math. 100 (1991), 129-147. Zbl0731.42027
  10. [10] S. Thangavelu, Summability of Laguerre expansions, Anal. Math. 16 (1990), 303-315. Zbl0737.41033

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