On Fourier coefficients

Let $\{\varphi_n\}$ be an orthonormal system of functions on the interval $[0,1]$, and let the function $f\in L^2(0, 1)$. We investigate the question of the convergence or divergence (depending on the smoothness of the function $f$) of series of the form
$$ \sum_{n = 1}^\infty|(f, \varphi_n)|^{\alpha_n}, $$
where $\alpha_n\uparrow2$ or $\alpha_n\to\alpha$ with $\alpha\in[0,2)$.
It is shown that in a certain sense, the assertions obtained are definitive for the Haar system.
Bibliography: 14 titles.