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    188宝金博页面版: Generalized boundary value problems for ordinary differential operators and least squares solutions

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    内容提示： JOUKNAL Ok MATHEMATICAL ANALYSIS ASD APPLK‘A~TIONS 110, 23@-246 ( 1985) Generalized Boundary Value Problems for Ordinary Differential Operators and Least Squares Solutions ULRICH TIPPENHAUER Fuchbereich Marhematik der L’nicersiliit Kaiserslaurern. 6750 Kaiserslaurern, West German? Submitred bx C. I.. Dolph Let R = (a, b)c R. k E N and W:(Q) be the Sobolev space. For dilTerential operators L = Et_, u,D’ of order k with a,~ C”(0). a,(x)> 0 and for any n con- tinuous linear functionals B,, I < i $ n. on W...

    文档格式：PDF | 页数：17 | 浏览次数：25 | 上传日期：2015-12-08 10:46:39 | 文档星级：
    JOUKNAL Ok MATHEMATICAL ANALYSIS ASD APPLK‘A~TIONS 110, 23@-246 ( 1985) Generalized Boundary Value Problems for Ordinary Differential Operators and Least Squares Solutions ULRICH TIPPENHAUER Fuchbereich Marhematik der L’nicersiliit Kaiserslaurern. 6750 Kaiserslaurern, West German? Submitred bx C. I.. Dolph Let R = (a, b)c R. k E N and W:(Q) be the Sobolev space. For dilTerential operators L = Et_, u,D’ of order k with a,~ C”(0). a,(x)> 0 and for any n con- tinuous linear functionals B,, I < i $ n. on W:(Q). least square solutions no D(L) = (c E W<(Q) 1 (u, B,) = 0, 1 < i G n} are constructed, e.g., 11 Lu - gII&, = inf{IILv-RiL?,n,IGED(L)), where g E L’(Q). This construction is achieved Crst by constructing the representatives of the functionals B, by the reproducing kernel of the whole Sobolev space W;(Q) and then by determining the reproducing kernel of the subspace D(L). Moreover. it makes use of the Moore-Penrose inverse For matrices and yields the explicit determination of the generalized Green’s function relative to the extensive class of boundary conditions B,E W:(n)‘. I <i cn, above. 1 IV85 Acadcmlc Press Inc I. INTRODUCTION The purpose of this paper is to present a method to construct least squares solutions for linear differential operators on subspaces of Sobolev spaces, which are defined by continuous linear functionals. Let Wk,(Q) be the Sobolev space where Q= (a, h)c 68 and ke N. Furthermore let L = If= 0 a, D’ be a differential operator of order k with U,E C?(Q), a,(x)>0 and B,, 1 < id n, continuous linear functionals on W:(Q), e.g., B, E W:(Q)‘. For a given function g E L’(Q) there exists at least one function .feD(L)= {UE W:(Q) I (u, B,) =0, 1 <i<n) with the property lI~f-gllf,~,,,=inf(ilL~-g~11.2~n) I v~D(Lll where (., .) denotes the canonical duality (W:(Q), W:(Q)‘). The construc- tion of a particular least squares solution f for L under the “generalized 230 0022-247X:85 S3.00

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