Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension

This report studies noise-shaped one-bit coefficients in normalized discrete polynomial Fourier extension. For first-order Sigma-Delta quantization, the error is written as e_k=u_k-q_k=Δv_k with a uniformly bounded state. Discrete summation by parts then yields variation estimates for complex weights and an O(N^{-1}) approximation rate on compact parameter sets. For the parabolic phase ϕ_{x,t}(ξ)=xξ+tξ^2, the bound is expressed through J(x,t)=\int_0^1 |x+2tξ|dξ, and the uniform N^{-1} rate is shown to be sharp over the admissible input class. Higher-order finite-record identities are derived with all endpoint traces retained. Under endpoint compatibility, or after explicit boundary correction, an rth-order noise-shaped error e=Δ^r v gives O(N^{-r}) decay for sufficiently smooth weights and O(N^{-(r-1+α)}) decay for C^{r-1,α} weights. Exact L^2 orthogonality identities, fourth-moment formulas, local kernel estimates, and oscillatory transfer bounds are also established. Extensions to polynomial phases, multidimensional parameter families, growing observation regions, and correlated state models are included.