Exponential Convergence of Deep Composite Polynomial Approximation for Cusp-Type Functions

TL;DR

This paper introduces a deep composite polynomial approximation method for cusp-type functions that achieves exponential convergence, significantly outperforming classical algebraic rates, with confirmed numerical efficiency.

Contribution

The paper presents a novel constructive scheme combining fractional power polynomial iteration with analytic polynomial fitting, achieving exponential convergence for non-differentiable cusp functions.

Findings

Exponential decay of approximation error with respect to parameter count.

Numerical experiments confirm theoretical exponential convergence.

Deep composite polynomials outperform classical methods for cusp functions.

Abstract

We investigate deep composite polynomial approximations of continuous but non-differentiable functions with algebraic cusp singularities. The functions in focus consist of finitely many cusp terms of the form $|x-a_j|^{\alpha_j}$ with rational exponents $\alpha_j\in(0,1)$ on a real-analytic background. We propose a constructive approximation scheme that combines a division-free polynomial iteration for fractional powers with an outer layer for the analytic polynomial fitting. Our main result shows that this composite structure achieves exponential convergence in the the number of scalar coefficients in the inner and outer polynomial layers. Specifically, the $L^p([-1,1])$ approximation error, decays exponentially with respect to the parameter budget, in contrast to the algebraic rates obtained by classical single-layer polynomial approximation for cusp-type functions. Numerical experiments for both single and multiple cusp configurations confirm the theoretical rates and demonstrate the parameter efficiency of deep composite polynomial constructions.