On the coefficients of everywhere convergent series in some rearranged orthonormal systems

In this paper the author establishes the existence of a series $\sum a_{\nu_k}\cos\nu_kx+b_{\nu_k}\sin\nu_kx$ in some rearranged trigonometric system, which converges everywhere to a Lebesgue integrable function and whose coefficients $a_{\nu_k}$ and $b_{\nu_k}$, $k=1,2,\dots$, are not the Fourier–Lebesgue coefficients of this function. Similar results are established for the Haar system and some other systems.
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