Unconditional Convergence of General Fourier Series

We consider problems of unconditional convergence of Fourier series of $\operatorname {Lip}1$ functions with respect to general orthonormal systems (ONSs). Sufficient conditions on the functions of an ONS are found under which the Fourier series of every $\operatorname {Lip}1$ function with respect to this system converges unconditionally. We show that some of the obtained results are sharp. We also prove that from any ONS $(\varphi _n)$ one can extract a subsequence $(\varphi _{n_k})$ with respect to which the Fourier series of every $\operatorname {Lip}1$ function converges unconditionally.