Cesaro-One Summability and Uniform Convergence of Solutions of a Sturm-Liouville System

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Galerkin methods are used in separable Hilbert spaces to construct and compute L{sup 2} [0,{pi}] solutions to large classes of differential equations. In this note a Galerkin method is used to construct series solutions of a nonhomogeneous Sturm-Liouville problem defined on [0,{pi}]. The series constructed are shown to converge to a specified du Bois-Reymond function f in L{sup 2} [0,{pi}]. It is then shown that the series solutions can be made to converge uniformly to the specified du Bois-Reymond function when averaged by the Ces{'a}ro-one summability method. Therefore, in the Ces{'a}ro-one sense, every continuous function f on [0,{pi}] is the ... continued below

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Baty, R.S. & Tucker, D.H. January 26, 1999.

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This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided by UNT Libraries Government Documents Department to Digital Library, a digital repository hosted by the UNT Libraries. It has been viewed 88 times , with 4 in the last month . More information about this article can be viewed below.

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  • Sandia National Laboratories
    Publisher Info: Sandia National Laboratories, Albuquerque, NM, and Livermore, CA
    Place of Publication: Albuquerque, New Mexico

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Galerkin methods are used in separable Hilbert spaces to construct and compute L{sup 2} [0,{pi}] solutions to large classes of differential equations. In this note a Galerkin method is used to construct series solutions of a nonhomogeneous Sturm-Liouville problem defined on [0,{pi}]. The series constructed are shown to converge to a specified du Bois-Reymond function f in L{sup 2} [0,{pi}]. It is then shown that the series solutions can be made to converge uniformly to the specified du Bois-Reymond function when averaged by the Ces{'a}ro-one summability method. Therefore, in the Ces{'a}ro-one sense, every continuous function f on [0,{pi}] is the uniform limit of solutions of nonhomogeneous Sturm-Liouville problems.

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  • Journal Name: Real Analysis Exchange

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  • Other: DE00003223
  • Report No.: SAND99-0177J
  • Grant Number: AC04-94AL85000
  • Office of Scientific & Technical Information Report Number: 3223
  • Archival Resource Key: ark:/67531/metadc675557

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  • January 26, 1999

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  • July 25, 2015, 2:20 a.m.

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  • Nov. 22, 2016, 10:07 p.m.

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Baty, R.S. & Tucker, D.H. Cesaro-One Summability and Uniform Convergence of Solutions of a Sturm-Liouville System, article, January 26, 1999; Albuquerque, New Mexico. (digital.library.unt.edu/ark:/67531/metadc675557/: accessed September 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.