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Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems


Authors: J. J. Guadalupe, M. Pérez, F. J. Ruiz and J. L. Varona
Journal: Proc. Amer. Math. Soc. 116 (1992), 457-464
MSC: Primary 42C10; Secondary 33C10, 33C45, 42A20
DOI: https://doi.org/10.1090/S0002-9939-1992-1096211-0
MathSciNet review: 1096211
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Abstract: We study some problems related to convergence and divergence a.e. for Fourier series in systems $ \{ {\phi _k}\} $, where $ \{ {\phi _k}\} $ is either a system of orthonormal polynomials with respect to a measure $ d\mu $ on $ [-1,1]$ or a Bessel system on $ [0,1]$. We obtain boundedness in weighted $ {L^p}$ spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series. On the other hand, we find general results about divergence a.e. of the Fourier series associated to Bessel systems and systems of orthonormal polynomials on $ [-1,1]$.


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  • [1] R. Askey, A transplantation theorem for Jacobi series, Illinois J. Math. 13 (1969), 583-590. MR 0246038 (39:7344)
  • [2] V. M. Badkov, Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval, Math. USSR Sb. 24 (1974), 223-256. MR 0355464 (50:7938)
  • [3] A. I. Benedek and R. Panzone, Pointwise convergence of series of Bessel functions, Rev. Un. Mat. Argentina 26 (1972), 167-186. MR 0340936 (49:5686)
  • [4] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 0199631 (33:7774)
  • [5] J. E. Gilbert, Maximal theorems for some orthogonal series. I, Trans. Amer. Math. Soc. 145 (1969), 495-515. MR 0252941 (40:6156)
  • [6] R. Hunt, On the convergence of Fourier series, Proc. Conf. Orthogonal Expansions and Continuous Analogues, Southern Illinois Univ. Press, Carbondale, IL, 1968, pp. 235-255. MR 0238019 (38:6296)
  • [7] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 0312139 (47:701)
  • [8] R. Hunt and W. S. Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274-277. MR 0338655 (49:3419)
  • [9] A. Máté, P. Nevai, and V. Totik, Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials, J. Approx. Theory 46 (1986), 314-322. MR 840398 (87j:42074)
  • [10] C. Meaney, Divergent Jacobi polynomial series, Proc. Amer. Math. Soc. 87 (1983), 459-462. MR 684639 (84c:42040)
  • [11] B. Muchenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • [12] -, Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. Math. Soc., vol. 64, no. 356, Amer. Math. Soc., Providence, RI, 1986. MR 858466 (88c:42035)
  • [13] M. Pérez, Series de Fourier respecto de sistemas ortogonales: estudio de la convergencia en espacios de Lebesgue y de Lorentz, Doctoral Dissertation, Sem. Mat. García de Galdeano, Zaragoza, 1989.
  • [14] H. Pollard, The convergence almost everywhere of Legendre series, Proc. Amer. Math. Soc. 35 (1972), 442-444. MR 0302973 (46:2115)
  • [15] G Szegö, Orthogonal polynomials, 3rd ed., Amer. Math. Soc, Providence, RI, 1967.
  • [16] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, Vol. 1, 2nd ed., Oxford Univ. Press, Oxford, 1962. MR 0176151 (31:426)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1096211-0
Keywords: Fourier series, orthonormal polynomials, maximal operators, $ {A_p}$-weights
Article copyright: © Copyright 1992 American Mathematical Society

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