AI 中文总结
研究离散多项式傅里叶扩展中噪声整形一位系数，利用一阶Sigma-Delta量化及分部离散求和得出近似率等，推导高阶恒等式，建立多种公式和估计，还包括对多种情况的扩展，给出不同权重下误差衰减率。
AI 中文摘要
本报告研究归一化离散多项式傅里叶扩展中的噪声整形一位系数。对于一阶Sigma-Delta量化，误差可写为$e_k = u_k - q_k = \Delta v_k$，且状态有界。通过分部离散求和得出复权重的变化估计以及在紧致参数集上的$O(N^{-1})$近似率。对于抛物线相位$\phi_{x,t}(\xi)=x\xi + t\xi^2$，其界通过$J(x,t)=\int_0^1 |x + 2t\xi|d\xi$表示，且在允许输入类上均匀的$N^{-1}$率是精确的。推导了高阶有限记录恒等式并保留所有端点迹。在端点兼容性下或经过显式边界校正后，$r$阶噪声整形误差$e = \Delta^r v$对于足够光滑的权重给出$O(N^{-r})$衰减，对于$C^{r - 1,\alpha}$权重给出$O(N^{-(r - 1 + \alpha)})$衰减。还建立了精确的$L^2$正交恒等式、四阶矩公式、局部核估计和振荡转移界。包括了对多项式相位、多维参数族、增长观测区域和相关状态模型的扩展。
英文摘要
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.
