Sequences of rational deviations

A.A. Pekarskiĭ [1] proved that any strictly monotone sequence realizes as the sequence of the least rational deviations in the space $C^{\mathbb C}[0,1]$ of complex continuous functions with the uniform norm. It is not known whether a result of this sort is true for the space $L_2^{\mathbb C}[0,1]$. However, it turned out that Euclidian norm in general does not allow rational deviations to be arbitrary. In [2], we have shown that monotone sequences with large jumps at the beginning cannot be realized as sequences of rational deviations in the Hardy space $H^2(\Im z>0)$ in the upper half-plane. We consider this problem as a particular case of seeking an element of a Hilbert space having prescribed $m$-term deviations with respect to a given dictionary, which in turn is a variation of the well-known Bernstein lethargy problem.
References
[1] Pekarskiĭ A. A., “Existence of a function with given best uniform rational approximations.” Vestsī Akad. Navuk Belarusī Ser. Fīz. Mat. Navuk, 1999, (1): 23–26.
[2] Borodin P., Kopecká E., “Sequences of m-term deviations in Hilbert space.” J. Approx. Theory, 2022, 105821.