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When do Sobolev spaces form a Hilbert scale?

Author: Andreas Neubauer
Journal: Proc. Amer. Math. Soc. 103 (1988), 557-562
MSC: Primary 46E35; Secondary 46M35
MathSciNet review: 943084
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Abstract: In this paper we show that the usual Sobolev spaces $ {\left( {{H^s}\left( \Omega \right)} \right)_{s \in {\mathbf{R}}}}$ are no Hilbert scale in the sense of Krein-Petunin, if $ \Omega $ is an open bounded subset of $ {{\mathbf{R}}^n}$.

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  • [1] H. W. Engl and A. Neubauer, Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind, Constructive Methods for the Practical Treatment of Integral Equations, G. Hämmerlin and K. H. Hoffman (eds.), Birkhäuser, Basel, 1985, pp. 120-141. MR 882562
  • [2] -, Convergence rates for Tikhonov-regularization in finite-dimensional subspaces of Hilbert scales, Proc. Amer. Math. Soc. 102 (1988), 587-592. MR 928985 (89c:65067)
  • [3] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Math., no. 24, Pitman, Boston, Mass., 1985. MR 775683 (86m:35044)
  • [4] S. G. Krein and J. I. Petunin, Scales of Banach spaces, Russian Math. Surveys 21 (1966), 85-160. MR 0193499 (33:1719)
  • [5] J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and application. I, Springer-Verlag, Berlin and New York, 1972.
  • [6] J. Locker and P. M. Prenter, Regularization with differential operators. I, J. Math. Anal. Appl. 74 (1980), 504-529; II: SIAM J. Numer. Anal. 17 (1980), 247-267. MR 567272 (83j:65062b)
  • [7] M. A. Naimark, Linear differential operators, Part II, Harrap, London, 1968. MR 0353061 (50:5547)
  • [8] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal. 18 (1984), 29-37. MR 762862 (86e:65081)
  • [9] A. Neubauer, An $ a$-posteriori parameter choice for Tikhonov-regularization in Hilbert scales leading to optimal convergence rates, SIAM J. Numer. Anal. (to appear). MR 972456 (90b:65108)

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Article copyright: © Copyright 1988 American Mathematical Society

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