On sequences of Fourier coefficients of functions of Hölder classes

The following theorem is proved. Let $\{\psi_l(t)\}$ be an arbitrary complete orthonormal system on $[0,1]$ and let $1/2<\alpha<1$. Then an $f(t)\in C_\beta$ exists for all $\beta<\alpha$ such that $\sum_{k=1}^\infty|c_k(f)|^p=\infty$, $p=2/(1+2\alpha)$, where $c_k(f)=\int\limits_0^1f\psi_k\,dt$.