CRITERION FOR BASICITY PROPERTIES OF THE WEIGHTED EXPONENTIAL SYSTEM WITH EXCESS IN THE GRAND LEBESGUE SPACES

Abstract

In this work, we consider the exponential system {rho(t)e(int)} n is an element of Z with degenerate coefficient rho(.) in the grand Lebesgue space L-p)(-pi, pi), 1 < p < +infinity. This space is non-separable and therefore, we define the subspace Gp)(-pi, pi) subset of L-p)(-pi, pi) in which the infinitely differentiable functions are densely located. The basicity of such systems in classical Lebesgue spaces has been studied quite well in the works on various mathematics. We establish criteria for the approximative properties (completeness, minimality, basicity) of this system in G(p))(-pi, pi), 1 < p < +infinity. Moreover, we consider the defective system {rho(t)e(int)}(n not equal n0) and obtain similar results regarding this system in the same space.