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Stable summation methods for a class of singular Sturm-Liouville expansions


Authors: Harvey Diamond, Mark Kon and Louise Raphael
Journal: Proc. Amer. Math. Soc. 81 (1981), 279-286
MSC: Primary 34B25
DOI: https://doi.org/10.1090/S0002-9939-1981-0593472-1
MathSciNet review: 593472
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Abstract: Given the Sturm-Liouville eigenfunction expansion of an $ {L_2}$ function $ f(x)$, summability theory provides means for recovering the value of $ f({x_0})$ at points $ {x_0}$ where $ f$ is sufficiently regular. If the coefficients in the expansion are perturbed slightly (in the $ {L_2}$ norm), a stable summation method will recover from the perturbed expansion a good approximation to $ f({x_0})$. In this paper we develop stable summation methods for expansions in eigenfunctions of the singular Sturm-Liouville system $ u'' - q(x)u = - \lambda u,u(0)\cos \beta + u'(0)\sin \beta = 0,u(\infty ) < \infty $; where $ q(x)$ and continuous. Given a summability method known to work at $ {x_0}$ for a particular expansion, our results say that if the summation parameter is appropriately scaled with the $ {L_2}$ error in the perturbed expansion, a stable summation method is obtained. We obtain a sharp scaling requirement for guaranteeing stability. We apply our results to Riesz and Stieltjes summability.


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  • [1] Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, 3rd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1978. MR 507190
  • [2] N. M. Krukovskiĭ, The Tihonov stable summability of Fourier series with perturbed coefficients by certain regular methods, Vestnik Moskov. Univ. Ser. I Mat. Meh. 28 (1973), no. 3, 22–29 (Russian, with English summary). MR 0338670
  • [3] B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, American Mathematical Society, Providence, R.I., 1975. Translated from the Russian by Amiel Feinstein; Translations of Mathematical Monographs, Vol. 39. MR 0369797
  • [4] Louise A. Raphael, The Stieltjes summability method and summing Sturm-Liouville expansions, SIAM J. Math. Anal. 13 (1982), no. 4, 676–689. MR 661599, https://doi.org/10.1137/0513046
  • [5] A. N. Tihonov, Stable methods for the summation of Fourier series, Soviet Math. Dokl. 5 (1964), 641-644
  • [6] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, Second Edition, Clarendon Press, Oxford, 1962. MR 0176151
  • [7] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0593472-1
Keywords: Sturm-Liouville expansions, stable summability, singular
Article copyright: © Copyright 1981 American Mathematical Society