When do Sobolev spaces form a Hilbert scale?
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- by Andreas Neubauer PDF
- Proc. Amer. Math. Soc. 103 (1988), 557-562 Request permission
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 ${{\mathbf {R}}^n}$.References
- Heinz W. Engl and Andreas Neubauer, Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind, Constructive methods for the practical treatment of integral equations (Oberwolfach, 1984) Internat. Schriftenreihe Numer. Math., vol. 73, Birkhäuser, Basel, 1985, pp. 120–141. MR 882562
- Heinz W. Engl and Andreas Neubauer, Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales, Proc. Amer. Math. Soc. 102 (1988), no. 3, 587–592. MR 928985, DOI 10.1090/S0002-9939-1988-0928985-X
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- S. G. Kreĭn and Ju. I. Petunin, Scales of Banach spaces, Uspehi Mat. Nauk 21 (1966), no. 2 (128), 89–168 (Russian). MR 0193499 J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and application. I, Springer-Verlag, Berlin and New York, 1972.
- John Locker and P. M. Prenter, Regularization with differential operators. I. General theory, J. Math. Anal. Appl. 74 (1980), no. 2, 504–529. MR 572669, DOI 10.1016/0022-247X(80)90145-6
- M. A. Naĭmark, Lineĭ nye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
- Frank Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Applicable Anal. 18 (1984), no. 1-2, 29–37. MR 762862, DOI 10.1080/00036818408839508
- Andreas Neubauer, An a posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates, SIAM J. Numer. Anal. 25 (1988), no. 6, 1313–1326. MR 972456, DOI 10.1137/0725074
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 557-562
- MSC: Primary 46E35; Secondary 46M35
- DOI: https://doi.org/10.1090/S0002-9939-1988-0943084-9
- MathSciNet review: 943084