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Computability and continuity in metric partial algebras equipped with computability structures

✍ Scribed by Fredrik Dahlgren

Publisher
John Wiley and Sons
Year
2004
Tongue
English
Weight
251 KB
Volume
50
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

In this paper we give an axiomatisation of the concept of a computability structure with partial sequences on a many‐sorted metric partial algebra, thus extending the axiomatisation given by Pour‐El and Richards in [9] for Banach spaces. We show that every Banach‐Mazur computable partial function from an effectively separable computable metric partial Σ‐algebra A to a computable metric partial Σ‐algebra B must be continuous, and conversely, that every effectively continuous partial function with semidecidable domain and which preserves the computability of a computably enumerable dense set must be computable. Finally, as an application of these results we give an alternative proof of the first main theorem for Banach spaces first proved by Pour‐El and Richards. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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We present a new variant of the simultaneous Stone-Weierstrass approximation of a function and its partial derivatives, when the function takes its values in a Banach space, and provide an explicit and direct computation of this approximation. In the particular case of approximation by means of poly