Geodesic
Recovery of integrable functions and trigonometric series
Sbornik. Mathematics, Tome 212 (2021) no. 6, pp. 843-858

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Classes $\Gamma$ of $L_1$-functions with fixed rate of decrease of their Fourier coefficients are considered. For each class $\Gamma$, it is shown that there exists a (recovery) set $G$ with arbitrarily small measure such that any function in $\Gamma$ can be recovered from its values on $G$. A formula for evaluation of the Fourier coefficients of such a function from its values on $G$ is given. In addition, it is shown that, for any $L_1$-function, a function-specific recovery set can be found. The problem of recovery of general trigonometric series from the Zygmund classes which converge to summable functions on such sets $G$ is also solved. Bibliography: 10 titles.
Keywords: trigonometric series, Fourier series, recovery problem, $V$-set. 
@article{SM_2021_212_6_a3,
     author = {M. G. Plotnikov},
     title = {Recovery of integrable functions and trigonometric series},
     journal = {Sbornik. Mathematics},
     pages = {843--858},
     publisher = {mathdoc},
     volume = {212},
     number = {6},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2021_212_6_a3/}
}
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M. G. Plotnikov. Recovery of integrable functions and trigonometric series. Sbornik. Mathematics, Tome 212 (2021) no. 6, pp. 843-858. http://geodesic.mathdoc.fr/item/SM_2021_212_6_a3/