The basis property of a perturbed trigonometric system (Q5935966)

Consider the perturbed trigonometric system \[ \chi_n(\theta):= \sin[(n+ \beta/2)\theta+ \gamma/2]- (\alpha/n)\cos[(n+ \beta/2)\theta+ \gamma/2], \] where \(n= 1,2,\dots\)\ . The following theorem is proved: If \(p> 1\), \[ -1/p< \gamma/\pi< 2-(1/p)\quad\text{and}\quad (1/p)- 2< (\gamma/\pi)+ \beta< 1/p, \] then the system \(\{\chi_n(\theta)\}\) is a basis in the space \(L_p(0,\pi)\) for all \(\alpha\), except possibly for a countable set without limit points. An example is presented in which the set of exceptional \(\alpha\) is infinite.