On Constructive Approximation of Nonlinear Operators

Suppose $K_{_Y}$ and $K_{_X}$ are the image and the preimage of a nonlinear operator $\f:K_{_Y}\rightarrow K_{_X}$. It is supposed that the cardinality of each $K_{_Y}$ and $K_{_X}$ is $N$ and $N$ is large. % large sets of observed and reference signals, respectively, each containing $N$ signals. We provide an approximation to the map $\f$ that requires a prior information only on { a few elements} $p$ from $K_{_Y}$, where $p\ll N$, but still effectively represents $\f(K_{_Y})$. It is achieved under quite non-restrictive assumptions. The device behind the proposed method is based on a special extension of the piecewise linear interpolation technique to the case of sets of stochastic elements. The proposed technique provides {a single} operator that transforms any element from the {arbitrarily large } set $K_Y$. The operator is determined in terms of pseudo-inverse matrices so that it always exists.