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

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 $|x-a_j|^{α_j}$ with rational exponents $α_j\in(0,1)$ 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.

💡 Research Summary

The paper addresses the longstanding challenge of efficiently approximating continuous but non‑differentiable functions that possess algebraic cusp singularities of the form (|x-a_j|^{\alpha_j}) with rational exponents (\alpha_j\in(0,1)). Classical approximation theory guarantees density of algebraic polynomials (Weierstrass theorem) but offers only algebraic convergence rates for such irregular functions, because the lack of smoothness near the cusps limits the effectiveness of single‑layer polynomial approximations.

To overcome this limitation, the authors propose a deep composite polynomial architecture consisting of two distinct layers. The inner layer is a division‑free iterative scheme that computes fractional powers without using division or roots. For a single cusp with exponent (\alpha=r/s), the iteration
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