Geodesic
Majorants and uniqueness of series in the Franklin system
Matematičeskie zametki, Tome 59 (1996) no. 4, pp. 521-545

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It is proved that if a series in the Franklin system converges almost everywhere to a function $f(t)$ and the distribution function of the majorant of partial sums satisfies the condition $$ \operatorname{mes}\bigl\{t\in[0,1]:s(t)>\lambda\bigr\} =o\biggl(\frac 1\lambda\biggr) $$ as $\lambda\to\infty$, then this series is a Fourier series for Lebesgue integrable functions $f(t)$. In the general case the coefficients of the series are reconstructed by means of an $A$-integral. 
@article{MZM_1996_59_4_a4,
     author = {G. G. Gevorkyan},
     title = {Majorants and uniqueness of series in the {Franklin} system},
     journal = {Matemati\v{c}eskie zametki},
     pages = {521--545},
     publisher = {mathdoc},
     volume = {59},
     number = {4},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_4_a4/}
}
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G. G. Gevorkyan. Majorants and uniqueness of series in the Franklin system. Matematičeskie zametki, Tome 59 (1996) no. 4, pp. 521-545. http://geodesic.mathdoc.fr/item/MZM_1996_59_4_a4/