Geodesic
Parseval frames of serial shifts of a function in spaces of trigonometric polynomials
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2018), pp. 30-36

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The work establishes possible dimensions of Parseval's frame in the space of trigonometric polynomials of the form $T_Q(x)=\sum\limits_{k\in Q}c_k e^{ikx}$ consisting of serial translations of a polynomial ($c_k\in\mathbb C$, where the finite set $Q\subset\mathbb Z$). Sufficient and necessary conditions for a system of serial translations to be a Parseval's frame are also established there. The result is applied to some particular cases. 
@article{VMUMM_2018_6_a3,
     author = {A. V. Fadeeva},
     title = {Parseval frames of serial shifts of a function in spaces of trigonometric polynomials},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {30--36},
     publisher = {mathdoc},
     number = {6},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2018_6_a3/}
}
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A. V. Fadeeva. Parseval frames of serial shifts of a function in spaces of trigonometric polynomials. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2018), pp. 30-36. http://geodesic.mathdoc.fr/item/VMUMM_2018_6_a3/