Fourier coefficients of functions with respect to general othonormal systems from classes \( Lip (\alpha)\)

Let \(\{\varphi_n\}^\infty_{n=1}\) be a real-valued orthonormal system in \(L^2[0,1]\), \(0<\alpha\leq 1\), \(0<\varepsilon<\alpha\), and let \(\{c_n(f)\}^\infty_{n=1}\) be a sequence of Fourier coefficients of \(f\in L^2[0,1]\) with respect to \(\{\varphi_n\}^\infty_{n=1}\). We prove a sufficient condition on \(\{\varphi_n\}^\infty_{n=1}\) such that the series \(\sum^\infty_{k=1}k^{2(\alpha-\varepsilon)}c^2_k(f)\) converges for any \(f\in Lip (\alpha)\). We check that the trigonometric system and the Haar system satisfy this condition. On the other hand, the condition is not fulfilled in general.