Direct and Inverse Approximation Theorems for the p-Version of the Finite Element Method in the Framework of Weighted Besov Spaces. Part I: Approximability of Functions in the Weighted Besov Spaces

This is the first of a series devoted to the approximation theory of the p-version of the finite element method in two dimensions in the framework of the Jacobi-weighted Besov spaces, which provides the p-version with a solid mathematical foundation. In this paper, we establish a mathematical framework of the Jacobi-weighted Besov and Sobolev spaces and analyze the approximability of the functions in the framework of these spaces, particularly, singular functions of $r^\gamma$-type and $r^\gamma \log ^\nu r$-type. These spaces and the corresponding approximation properties are of fundamental importance to the proof of the optimal convergence for the p-version in two dimensions in part II and to various sharp inverse approximation theorems in part III.