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

Yoshikawa, Atsushi

Proc. Japan Acad., Ser. A 79, No. 3, 65-70 (2003).

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.

Reviewer: R. Downey (Wellington)

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)

Keywords:

effective generating set; orthonormal basis; Poincairé-Wigner orthogonalization; computability structure

Citations:

Zbl 0678.03027

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
[3]	Pour-El, Marian B., and Richards, J. Ian: Computability in Analysis and Physics. Springer-Verlag, Berline-Heidelberg (1989). · Zbl 0678.03027