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 ek=uk-qk=Delta vk 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 φx,t(xi)=xxi+txi2, the bound is expressed through J(x,t)=int01 |x+2txi|dxi, 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=Deltar v gives O(N-r) decay for sufficiently smooth weights and O(N-(r-1+α)) decay for Cr-1,α weights. Exact L2 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.