Geodesic
Luzin's Correction Theorem and the Coefficients of Fourier Expansions in the Faber--Schauder System
Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 172-178

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Suppose that $b_n\downarrow0$ and $\sum_{n=1}^{\infty}({b_n}/{n})=+\infty$. In this paper, it is proved that any measurable almost everywhere finite function on $[0,1]$ can be corrected on a set of arbitrarily small measure to a continuous function $\widetilde{f}$ so that the nonzero moduli $|A_n(\widetilde{f}\mspace{4mu})|$ of the Fourier–Faber–Schauder coefficients of the corrected function are $b_n$.
Keywords: Luzin's correction theorem, Faber–Schauder system, correcting function, Faber–Schauder spectrum. 
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     author = {M. G. Grigoryan and V. G. Krotov},
     title = {Luzin's {Correction} {Theorem} and the {Coefficients} of {Fourier} {Expansions} in the {Faber--Schauder} {System}},
     journal = {Matemati\v{c}eskie zametki},
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     volume = {93},
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     year = {2013},
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M. G. Grigoryan; V. G. Krotov. Luzin's Correction Theorem and the Coefficients of Fourier Expansions in the Faber--Schauder System. Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 172-178. http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a1/