Analytic Regularity and Approximation Limits of Coefficient-Constrained Shallow Networks

Battle Count: 79

This is a purely theoretical mathematics paper in approximation theory with no direct connection to quantitative trading, financial modeling, or market analysis. While neural networks are used in quantitative finance, this paper addresses fundamental mathematical limitations of analytic activation networks and does not involve any financial applications, time series, or market data.

This is a purely theoretical paper with no implementation component. There are no algorithms to implement, no code to write, and no computational procedures. The contribution is entirely in the form of mathematical theorems, propositions, lemmas, and proofs. The 'complexity' is in understanding the mathematical arguments, not in any software implementation.

The paper is entirely theoretical with complete mathematical proofs. All results are deterministic and self-contained, relying on classical approximation theory (Bernstein-type inequalities, polynomial approximation bounds). No computational experiments or data are required. The proofs can be verified by any reader with appropriate mathematical background.