A criterion for the basis property of perturbed exponential systems in Lebesgue spaces with variable exponent

Exponential systems in Lebesgue function spaces with variable exponent are considered. The basis properties of these exponential systems are studied in the spaces under certain conditions on the function. The space endowed with the norm is Banach and the set of compactly supported infinitely differentiable functions is found to be everywhere dense. The singular integral is found to be piecewise Hölder curve in the complex plane and acts boundedly for certain conditions. A system is proved to be complete and minimal but not necessarily forms a basis in this space. It is also proved that the systems are biorthonormal and that the norms of projections are uniformly bounded. It is shown that the exponential system is complete and minimal but does not form a basis in this space.