Linear and Nonlinear Approximation

Abstract

Let f be a continuous function on the interval [a, b] ⊂ ℝ |R which is to be approximated by an approximation function $Φ$ ∈ C[a, b]. $Φ$ shall be dependent on x ∈ [a, b] and on certain parameters $$ c_0 ,c_1 , \ldots ,c_n $$:$$ \Phi (x): = \Phi (x,c_0 ,c_1 , \ldots ,c_n ) = \Phi (x,c)\,for c = (c_0 ,c_1 , \ldots ,c_n )^T. $$The Parameters are to be determined in such a way that the distance between f and $Φ$ as functions on [a, b] is minimized in a manner still to be specified.