Nonlinear L1 approximation of smooth functions

Abstract

In this paper we shall consider certain aspects of nonlinear approximation of smooth functions in the L, norm on an interval [a, b] of the real line. In particular it will be shown that under certain circumstances (which would include many practical situations) second derivative techniques are applicable. This leads to checkable sufficient conditions for a local best approximation and makes applicable certain numerical techniques such as Newton’s method. We also consider the unicity problem and extend certain results in [l, 21 given for L, spaces with p > 2 to the setting of this paper. The approximation problem we consider is as follows. An open set S of Euclidean N-space EN is given and a map A(*): S + C]a, b] such that the map x + A”(x, ., .) is continuous on S where A”(x, -, .) is the second FrCchet derivative of the map A at the point X. Moreover we will assume that