Motivated by the remarkable performance of nonlinear approximation schemes in the solution od complex problems (e.g. functional optimization problems from control theory, text pronounciation, etc.), we discuss some issues on the rates of approximation by linear and nonlinear approximators in certain spaces of functions. We demontrate some qualitative arguments which support better experimental performance of nonlinear approximators with respect to the linear ones. Finally, using the concept of variation of a function with respect to a set, we present bounds to the rate of the approximation error achievable by nonlinear approximators in certain spaces of functions and show that they are better than the rates achieved by linear approximation in the same spaces. This motivates using the nonlinear approximators for solving complex problems.

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