Jackson-type theorem in the weak $L_{1}$-space

Abstract

The weak $L_{1}$-space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak $L_{1}$-space. The difficulty of working with the weak $L_{1}$-space is that the weak $L_{1}$-space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak $L_{1}$-space, studied its properties, found a criterion for convergence to zero of the modulus of continuity of the function from the weak $L_{1}$-space, and proved in this space an analogue of the Jackson-type theorem.