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Type‐2 computability on spaces of integrable functions
Author(s) -
Kunkle Daren
Publication year - 2004
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200310109
Subject(s) - mathematics , orthonormal basis , integrable system , computability , real line , pure mathematics , convolution (computer science) , locally integrable function , hermite polynomials , lp space , type (biology) , fourier transform , measure (data warehouse) , discrete mathematics , mathematical analysis , banach space , ecology , physics , quantum mechanics , database , machine learning , artificial neural network , computer science , biology
Using Type‐2 theory of effectivity, we define computability notions on the spaces of Lebesgue‐integrable functions on the real line that are based on two natural approaches to integrability from measure theory. We show that Fourier transform and convolution on these spaces are computable operators with respect to these representations. By means of the orthonormal basis of Hermite functions in L 2 , we show the existence of a linear complexity bound for the Fourier transform. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)