11.2 Nonlinear approximation and sparse wavelet expansions

Historically, nonlinear approximation developed from the work of several mathematicians in Central and Eastern Europe on rational approximation. Let f be a function defined on a closed and bounded (compact) interval I. To fix our ideas, assume that f belongs to L ²( I). For each positive integer N, one looks for a rational fraction g _N ( x) = P ( x)/ Q( x) with degree ? N (defined as the maximum of the degrees of the polynomials P and Q) that gives the best approximation to fin the L ²( I) norm. Thus one seeks, for each value of N, to minimize ? f ? g _N ? ₂ with the constraints g _N = P/Q, deg P ? N, and deg Q ? N. No hypothesis is made about the position of the poles of g _N. Since the set R _N of rational fractions g _N = P/ Q that are examined in seeking the minimum is not a linear subspace of L ²( I), the algorithm defining the best approximation is not linear. Furthermore, the function g _N is not unique; it is, however, unique if the approximation is measured in the uniform ( L ^?) norm. (See [226] for a complete discussion of rational...