Basis Properties of Trigonometric Systems in Weighted Morrey Spaces

In this paper, the basis properties (completeness, minimality and basicity) of the system of exponents are investigated in weighted Morrey spaces, where the weight function is defined as a product of power functions. Although the same properties of the system of exponents, as well as their perturbations, are well studied in weighted Lebesgue spaces, the situation changes cardinally in Morrey spaces. For instance, since Morrey spaces are not separable, the first difficulty arises concerning the formulation of the problem: to find the “suitable” subspace, in which the above mentioned properties have a “chance” to be true. Another difficulty, that frustrates the “usual” attempts is that, the infinite differentiable functions (even continuous functions) are not dense in Morrey spaces. Nevertheless, there are works that study these problems. For example, in [8], the basis properties of the system of exponents in Morrey space have been studied. Also, in [9, 7] the basis properties of the perturbed systems of exponents in Morrey space have been investigated. On the other hand, some approximation problems have been investigated in Morrey-Smirnov classes in [22].