Two intimately related topics are studied in this dissertation, namely, n-term rational approximation in Lp( R2 ) (0 p infinity) and anisotropic Franklin bases on compact polygonal domains in R2 .; One of the main results relates n-term rational approximation in Lp to nonlinear approximation from a broad class of piecewise polynomials over multilevel triangulations allowing a lot of flexibility and, in particular, arbitrarily sharp angles. This relationship and the existing estimates for spline approximation give a Jackson estimate for n-term rational approximation in terms of a minimal smoothness norm over a large collection of anisotropic Besov type spaces (B-spaces).; Franklin systems generated by Courant elements over multilevel nested triangulations on compact polygonal domains E ⊂ R2 are introduced and explored. Anisotropic triangulations allowing arbitrarily sharp angles are included. It is shown that these Franklin systems are unconditional bases for Lp(E), 1 p infinity, and Schauder bases for C( E).
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