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On an ad hoc computability structure in a Hilbert space. (English) Zbl 1052.03039

The author re-examines the “ad-hoc” computability structure of M. B. Pour-El and J. I. Richards [Computability in analysis and physics. Springer-Verlag, Berlin (1989; Zbl 0678.03027)] which was to take an effective generating set as a slightly modified orthonormal basis. He shows that applying the Poincairé-Wigner orthogonalization process to the Pour-El-Richards (loc. cit.) generating set gives an orthonormal effective generating set which yields a third natural computability structure.

MSC:

03F60 Constructive and recursive analysis
46S30 Constructive functional analysis
03D80 Applications of computability and recursion theory
03D45 Theory of numerations, effectively presented structures
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Citations:

Zbl 0678.03027
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References:

[1] Daubechies, I.: Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics, no.,61). Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992).
[2] Komatsu, H.: Fractional powers of operators. Pacific J. Math., 19 , 285-346 (1966). · Zbl 0154.16104 · doi:10.2140/pjm.1966.19.285
[3] Pour-El, Marian B., and Richards, J. Ian: Computability in Analysis and Physics. Springer-Verlag, Berline-Heidelberg (1989). · Zbl 0678.03027
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