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Effective Fine-convergence of Walsh-Fourier series

✍ Scribed by Takakazu Mori; Mariko Yasugi; Yoshiki Tsujii

Publisher
John Wiley and Sons
Year
2008
Tongue
English
Weight
182 KB
Volume
54
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

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)

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