Exponential Convergence of Deep Composite Polynomial Approximation for Cusp-Type 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 | αj with rational exponents α j ∈ (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 analytic polynomial fitting. Our main result shows that this composite structure achieves exponential convergence under a balanced allocation of inner depth and outer degree. More precisely, the L p ([−1, 1]) approximation error decays exponentially both in the number of fitted scalar coefficients and in the size of the layered arithmetic graph, with these two budgets being linearly equivalent for the constructed sequence. This contrasts with the algebraic degree convergence of 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.