On a method of constructing conditional bases of exponentials in some Banach spaces of analytic functions

In this note a method of constructing conditional exponential bases in some Banach spaces of analytic functions is presented. The method of constructing basic families due to A. Figà-Talamanca is generalized. Concrete applications and examples of conditional basic families in the space of multipliers of power series with the sequence of Taylor coefficients from $l^p$, in the space of multipliers of Cauchy type integrals, and in other spaces are given.