Geodesic
Nonharmonic Fourier series without the Riemann--Lebesgue property
Izvestiya. Mathematics , Tome 45 (1995) no. 3, pp. 545-557

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We prove that in the class of separated sequences $\lambda_n$ there exists a sequence whose real parts decrease arbitrarily slowly to $-\infty$, so that for some continuous function $f$ on $[0,1]$ the general term of the nonharmonic Fourier series $f(t)\sim\sum c_ne^{\lambda_nt}$ diverges to infinity as $n=n_k\to\infty$ for all $t\in(0,1)$. 
@article{IM2_1995_45_3_a5,
     author = {A. M. Sedletskii},
     title = {Nonharmonic {Fourier} series without the {Riemann--Lebesgue} property},
     journal = {Izvestiya. Mathematics },
     pages = {545--557},
     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_45_3_a5/}
}
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A. M. Sedletskii. Nonharmonic Fourier series without the Riemann--Lebesgue property. Izvestiya. Mathematics , Tome 45 (1995) no. 3, pp. 545-557. http://geodesic.mathdoc.fr/item/IM2_1995_45_3_a5/