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1991 | 100 | 2 | 129-147
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Almost everywhere summability of Laguerre series

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Abstrakty
EN
We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions $ℓ_n^a(x) = (n!/Γ(n+a+1))^{1/2} e^{-x/2} L_n^a(x)$, n = 0,1,2,..., in $L^2(ℝ_+, x^adx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f ∈ L^p(x^adx)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.
Twórcy
  • Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
  • [1] R. Askey and I. I. Hirschman, Jr., Mean summability for ultraspherical polynomials, Math. Scand. 12 (1963), 167-177.
  • [2] R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708.
  • [3] C. P. Calderón, On Abel summability of multiple Laguerre series, Studia Math. 33 (1969), 273-294.
  • [4] R. Coifman et G. Weiss, Analyse harmonique non commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin 1971.
  • [5] J. Długosz, Almost everywhere convergence of some summability methods for Laguerre series, Studia Math. 82 (1985), 199-209.
  • [6] G. Freud and S. Knapowski, On linear processes of approximation (III), ibid. 25 (1965), 373-383.
  • [7] E. Görlich and C. Markett, Mean Cesàro summability and operator norms for Laguerre expansions, Comment. Math., Tomus specialis II (1979), 139-148.
  • [8] E. Görlich and C. Markett, A convolution structure for Laguerre series, Indag. Math. 44 (1982), 161-171.
  • [9] 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.
  • [10] C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), 19-37.
  • [11] C. Markett, Norm estimates for generalized translation operators associated with a singular differential operator, Indag. Math. 46 (1984), 299-313.
  • [12] B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231-242.
  • [13] B. Muckenhoupt, Mean convergence of Hermite and Laguerre series. II, ibid. 147 (1970), 433-460.
  • [14] J. Peetre, The Weyl transform and Laguerre polynomials, Le Matematiche 27 (1972), 301-323.
  • [15] E. L. Poiani, Mean Cesàro summability of Laguerre and Hermite series, Trans. Amer. Math. Soc. 173 (1972), 1-31.
  • [16] H. Pollard, The mean convergence of orthogonal series. II, ibid. 63 (1948), 355-367.
  • [17] K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math. 88 (1988), 245-256.
  • [18] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc. 322 (1990), 671-690.
  • [19] S. Thangavelu, Multipliers for the Weyl transform and Laguerre expansions, Proc. Indian Acad. Sci. 100 (1990), 9-20.
  • [20] S. Thangavelu, On almost everywhere and mean convergence of Hermite and Laguerre expansions, Colloq. Math. 60/61 (1990), 21-34.
  • [21] G. N. Watson, Another note on Laguerre polynomials, J. London Math. Soc. 14 (1939), 19-22.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv100i2p129bwm
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