Constructive Approximation of Nonlinear Operators Based on Piecewise Interpolation Technique

Suppose KY and KX are the image and the preimage of a nonlinear operator KY → KX.It is supposed that the cardinality of each KY and KX is N and N is large. We provide anapproximation to the map F that requires prior information only on a few elements p fromKY, where p ≪ N, but still effectively represents F(KY). It is achieved under Lipschitzcontinuity assumptions. The device behind the proposed method is based on a specialextension of the piecewise linear interpolation technique to the case of sets of stochasticelements. The proposed technique provides a single operator that transforms any elementfrom the arbitrarily large set KY. The operator is determined in terms of pseudo-inversematrices so that it always exists.