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

• URL: http://arxiv.org/abs/2607.24868v1
• Date: Sun, 26 Jul 2026 21:51:41 GMT
• Title: Research Report on Noise-Shaped One-Bit Coefficients in Discrete Polynomial Fourier Extension
• Abstract summary: This report studies noise-shaped one-bit coefficients in normalized discrete Fourier extension.<n>For first-order Sigma- quantization, the error is written as $e_k=u_k-q_k=v_k$ with a uniformly bounded state.<n>For the parabolic $_x,t()=x+t2$, the bound is expressed through $J(x,t)=int_01 |x+2t|d$, and the uniform $N-1$ rate is shown to be sharp.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $r$th-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.

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