Geodesic
Function systems obtained using translates and dilates of a~single function in the paces~$E_\varphi$ with $\lim_{t\to\infty}\frac{\varphi(t)}t=0$
Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 389-402

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We solve Ul'yanov's problem [1] of representing elements of class $\varphi(L)$ as series with respect to arbitrary function systems. 
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     author = {V. I. Filippov},
     title = {Function systems obtained using translates and dilates of a~single function in the paces~$E_\varphi$ with $\lim_{t\to\infty}\frac{\varphi(t)}t=0$},
     journal = {Izvestiya. Mathematics },
     pages = {389--402},
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     volume = {65},
     number = {2},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a5/}
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V. I. Filippov. Function systems obtained using translates and dilates of a~single function in the paces~$E_\varphi$ with $\lim_{t\to\infty}\frac{\varphi(t)}t=0$. Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 389-402. http://geodesic.mathdoc.fr/item/IM2_2001_65_2_a5/