Bases Consisting of Rational Functions of Uniformly Bounded Degrees or More General Functions

We prove in this paper the existence of a Schauder basis for C[0, 1] consisting of rational functions of uniformly bounded degrees. This solves an open question of some years standing concerning the possible existence of such bases. This result follows from a more general construction of bases on R and [0, 1]. We prove that the new bases are unconditional bases for Lp, 1<p<∞, and Besov spaces. On [0, 1], they are Schauder bases for C[0, 1] as well. The new bases are utilized for nonlinear approximation.