Nonlinear approximation theory

The initial investigations into nonlinear approximation problems were conducted by P. L. Chebyshev, whose contributions are foundational to the theory of uniform approximation. His ideas paved the way for the development of best uniform approximation theories for rational functions and polynomials within a cohesive framework. The distinction between linear and rational approximation emerged in the 1960s, alongside the exploration of alternative approaches to nonlinear approximation. The introduction of tools such as nonlinear functional analysis and topological methods revealed that linearization alone is inadequate for fully addressing nonlinear families. Notably, the use of global analysis and the examination of flows within the family of approximating functions brought forth innovative concepts previously absent in approximation theory, which have since proved significant across various analytical fields. Additionally, techniques developed for nonlinear approximation are frequently applicable to linear approximation problems. A key example is the resolution of moment problems through rational approximation. The pursuit of optimal quadrature formulae or the quest for the best linear spaces often necessitates the consideration of spline functions with free nodes. The appendix of this work addresses the renowned challenge of best interpolation by polynomials.