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

The article discusses the criterion for the basis property of perturbed exponential systems in Lebesgue function spaces with variable exponent. It obtains the criteria for the basis property on equiconvergence and compact sets with trigonometric series in exponential systems. It points out that the range of systems covered by these criteria is larger than the range of exponential systems. It discusses the hardy classes with variable exponent, its theorem, equations, and solutions. The Reimann problem is the classes is also discussed.