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Effective Fine‐convergence of Walsh‐Fourier series
Author(s) -
Mori Takakazu,
Yasugi Mariko,
Tsujii Yoshiki
Publication year - 2008
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.200710063
Subject(s) - mathematics , lebesgue integration , dominated convergence theorem , fourier series , locally integrable function , function series , pure mathematics , mean value theorem (divided differences) , convergence (economics) , bounded function , sequence (biology) , dirichlet series , convergence tests , mathematical analysis , integrable system , dirichlet distribution , picard–lindelöf theorem , boundary value problem , rate of convergence , channel (broadcasting) , fixed point theorem , biology , economic growth , electrical engineering , economics , genetics , engineering
We define the effective integrability of Fine‐computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integrals such as the Bounded Convergence Theorem, the Dominated Convergence Theorem, and the Second Mean Value Theorem. It is also proved that the Walsh‐Fourier coefficients of an effectively integrable Fine‐computable function form a Euclidian computable sequence of reals which converges effectively to zero. This property of convergence is the effectivization of the Walsh‐Riemann‐Lebesgue Theorem. The article is closed with the effective version of Dirichlet's test. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)