On approximations for functions in the space of uniformly convergent Fourier series

JOURNAL OF APPROXIMATION THEORY

卷:249

Article

Boche, Holger^1,2  Pohl, Volker¹ 

[1] Tech Univ Munich, Inst Theoret Informat Technol, Arcisstr 21, D-80333 Munich, Germany

[2] MCQST, D-80799 Munich, Germany

关键词: Approximation;    Fourier series;    Sampling;    Turing computable;    Uniform recovery;   

DOI  :  10.1016/j.jat.2019.105307

来源: Elsevier

【 摘 要 】

This paper studies the possibility of approximating functions in the space of all uniformly convergent symmetric and non-symmetric Fourier series from finitely many samples of the given function. It is shown that no matter what approximation method is chosen, there always exists a residual subset such that the approximation method diverges for all functions from this subset. This general result implies that there exists no method to effectively calculate the Fourier series expansion on a digital computer for all functions from the space of uniformly convergent Fourier series. In particular, there exists no Turing computable approximation method in these spaces. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free