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Miyamoto, Shoki; Yoshikawa, Atsushi

Computable sequences in the Sobolev spaces. (English) Zbl 1060.46058

Proc. Japan Acad., Ser. A 80, No. 3, 15-17 (2004).
Summary: M. B. Pour-El and J. I. Richards [“Computability in analysis and physics”, (Perspectives in Mathematical Logic, Springer Berlin) (1989; Zbl 0678.03027)] discussed computable smooth functions with noncomputable first derivatives. We show that a similar result holds in the case of Sobolev spaces by giving a non-computable \({\mathcal H}^1(0,1)\)-element which, however, is computable in any of larger Sobolev spaces \({\mathcal H}^s(0,1)\) for any computable \(s\), \(0\leq s< 1\).

Cited in 1 Document

MSC:

46S30 Constructive functional analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
03F60 Constructive and recursive analysis

Keywords:

effective and non-effective convergence

Citations:

Zbl 0678.03027
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\textit{S. Miyamoto} and \textit{A. Yoshikawa}, Proc. Japan Acad., Ser. A 80, No. 3, 15--17 (2004; Zbl 1060.46058)
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References:

[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65, Academic Press, New York-London (1975). · Zbl 0314.46030
[2] Bergh, J., and Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin-New York (1976). · Zbl 0344.46071
[3] Calderón, A. P.: Intermediate spaces and interpolation, the complex method. Studia Math., 24 , 113 -190 (1964). · Zbl 0204.13703
[4] Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften, no. 256, Springer-Verlag, Berlin-Heidelberg-New York (1983). · Zbl 0521.35001
[5] Pour-El, Marian B., and Richards, J. Ian: Computability in Analysis and Physics. Perspective in Mathematical Logic. Springer-Verlag, Berlin (1989). · Zbl 0678.03027
[6] Yoshikawa, A.: Interpolation functor and computability. Theoret. Comput. Sci., 284 , 487-498 (2002). · Zbl 1051.03051
[7] Zhong, N.: Computability structure of the Sobolev spaces and its applications. Theoret. Comput. Sci., 219 , 487-510 (1999). · Zbl 0916.68044
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