On the piecewise-concave approximations of functions

TL;DR

This paper proves that piecewise-concave functions can approximate various classes of functions, including multivariate twice differentiable and Lipschitz-continuous functions, with arbitrary precision over bounded sets.

Contribution

It establishes the theoretical property that piecewise-concave functions can approximate specific classes of functions to any desired accuracy.

Findings

Proves approximation capability for multivariate twice differentiable functions.

Shows approximation for univariate Lipschitz-continuous functions.

Extends results to multivariate separable Lipschitz-continuous functions.

Abstract

The piecewise-concave function may be used to approximate a wide range of other functions to arbitrary precision over a bounded set. In this short paper, this property is proven for three function classes: (a) the multivariate twice continuously differentiable function, (b) the univariate Lipschitz-continuous function, and (c) the multivariate separable Lipschitz-continuous function.