Geodesic
Singularities of carleman type for subsystems of a~trigonometric system
Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 515-520.

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We prove that for arbitrary $\varepsilon>0$ there exists a sequence of positive integers $\{n_k\}$ such that a) the system $\{\cos n_kX,\sin n_kX\}$ is a basis with respect to the $C[-\pi,\pi]$ norm in the closure of its linear hull, and b) a continuous function $f(x)$ belonging to the closure of the linear hull of the system can be found such that its Fourier coefficients $a_n$ and $b_n$ satisfy the relation $$ \sum{n=1}^\infty|a_n|^{2-\varepsilon}+|b_n|^{2-\varepsilon}=\infty. $$ 
@article{MZM_1974_15_4_a0,
     author = {S. F. Lukomskii},
     title = {Singularities of carleman type for subsystems of a~trigonometric system},
     journal = {Matemati\v{c}eskie zametki},
     pages = {515--520},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a0/}
}
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S. F. Lukomskii. Singularities of carleman type for subsystems of a~trigonometric system. Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 515-520. http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a0/