We show that, given a set E ⊂ 𝕋 of measure zero, the set of continuous functions whose Fourier series expansion is divergent at any point t ∈ E is dense-algebrable, i.e. there exists an infinite-dimensional, infinitely generated dense subalgebra of 𝓒(𝕋) every non-zero element of which has a Fourier series expansion divergent in E.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-1-5,
author = {Richard M. Aron and David P\'erez-Garc\'\i a and Juan B. Seoane-Sep\'ulveda},
title = {Algebrability of the set of non-convergent Fourier series},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {83-90},
zbl = {1102.42001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-1-5}
}
Richard M. Aron; David Pérez-García; Juan B. Seoane-Sepúlveda. Algebrability of the set of non-convergent Fourier series. Studia Mathematica, Tome 173 (2006) pp. 83-90. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-1-5/