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

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 \(s\), \(0\leq s< 1\).


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


Zbl 0678.03027
Full Text: DOI


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