We revisit the classical problem of band-limited signal reconstruction -- a variant of the *Set Query* problem -- which asks to efficiently reconstruct (a subset of) a $d$-dimensional Fourier-sparse signal ($\|\hat{x}(t)\|_0 \leq k$), from minimum noisy samples of $x(t)$ in the time domain. We present a unified framework for this problem, by developing a theory of sparse Fourier transforms over *lattices*, which can be viewed as a "semi-continuous" version of SFT, in-between discrete and continuous domains. Using this framework, we obtain the following results: $\bullet$ *High-dimensional Fourier sparse recovery* We present a sample-optimal discrete Fourier Set-Query algorithm with $O(k^{\omega+1})$ reconstruction time in one dimension, independent of the signal's length ($n$) and $\ell_\infty$-norm ($R^* \approx \|\hat{x}\|_\infty$). This complements the state-of-art algorithm of [Kap17], whose reconstruction time is $\tilde{O}(k \log^2 n \log R^*)$, and is limited to low-dimensions. By contrast, our algorithm works for arbitrary $d$ dimensions, mitigating the $\exp(d)$ blowup in decoding time to merely linear in $d$. $\bullet$ *High-accuracy Fourier interpolation* We design a polynomial-time $(1+ \sqrt{2} +\epsilon)$-approximation algorithm for continuous Fourier interpolation. This bypasses a barrier of all previous algorithms [PS15, CKPS16] which only achieve $>100$ approximation for this problem. Our algorithm relies on several new ideas of independent interest in signal estimation, including high-sensitivity frequency estimation and new error analysis with sharper noise control. $\bullet$ *Fourier-sparse interpolation with optimal output sparsity* We give a $k$-Fourier-sparse interpolation algorithm with optimal output signal sparsity, improving on the approximation ratio, sample complexity and runtime of prior works [CKPS16, CP19].

翻译：我们重新审视了频带限制信号重建的经典问题 -- -- * Set Query* 问题的一个变体 -- -- 它要求高效重建( 一个子集) 美元维度 Fleier-spar smassy 信号 ( ⁇ hat{x} (t) ⁇ 0\leq k$), 由时间域中 $x(t) 最小的噪音样本 。 我们为这一问题提出了一个统一的框架, 通过开发一种在 * latiters * 上流度变异的理论, 它可以被视为SFT 的“ 半持续” 版本, 在离异和连续的域中 。 我们获得以下结果: $\ bull$ * 高度Fleier Set- Query 算法, $(komega+1) 在一个维度上提供重建时间, 与信号前程值(n) 美元 和 美元内部变异度变异度变异度变数 。