[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85942-en":3,"doc-seo-85942-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85942,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Optimal Extrapolation Bounds for Sparse Fourier Sums","Optimal extrapolation inequality for k-sparse Fourier sums with arbitrary real frequencies is proved without any frequency separation assumption. For g(t)=∑_{j=1}^k v_j e^{iλ_j t}, the result bounds |g(x)| for x≥1 purely from the observed energy ∥g∥_{L^2[-1,1]}. In the edge regime x=1+δ (0≤δ≤1), the bound becomes an explicit Chebyshev-scale estimate refined over prior outside-window growth results. The exponential dependence is shown optimal up to constants and polynomial factors in k, with algorithmic gains for clustered-frequency recovery and sharp exterior prediction guarantees in sparse Fourier feature spaces.","arXiv :2607 . 1050 1v 1 [ cs .DS] 11 Jul 2026  \nOptimal Extrapolation Bounds for Sparse Fourier Sums  \nRuizhe Zhang  \n[rzzhang@purdue. edu](rzzhang@purdue. edu)  \nPurdue University  \nAbstract  \nWe prove an optimal extrapolation theorem for k-sparse Fourier sums over arbitrary real frequencies, without any separation assumption, bounding how large such a sum can be just outside an interval on which its energy is observed. For every g(t) = Pkj=1 vj eiλjt with λj ∈ Rand every x ≥ 1,  \n|g(x)| ≤ kO(1) exp(O(k arcoshx))∥g∥L2 [−1 , 1] .  \nIn the endpoint regime, this refines to the explicit bound  \n|g(1 + δ)| ≤ O(k)exp(O(k √δ  ))∥g∥L2 [−1 , 1], 0 ≤ δ ≤ 1.  \nThis improves on the exp(O(k2 log k·δ)) growth estimate of Chen and Price (ICALP 2019), and the exponential scaling is optimal up to constants and polynomial factors in k.  \nAs an algorithmic consequence, we improve the cluster-center resolution of Chen–Price’s clustered-frequency recovery algorithm by a factor of k, while preserving its sample complexity up to logarithmic factors. We also obtain exterior leverage-score and transfer bounds for sparse Fourier feature spaces, converting in-domain active-regression guarantees into essentially sharp prediction guarantees just outside the sampling interval.  \n1 Introduction  \nA signal with only k significant spectral components admits a concise representation as a superposition of k complex exponentials, and exploiting this structure is a central theme in sparse Fourier algorithms. In the discrete setting, Fourier sparsity enables sublinear-time algorithms for identifying and estimating the dominant Fourier coefficients [GGI+02 , Iwe10 , HIKP12b, HIKP12a, IK14 , Kap16 , NSW19] . In the continuous, off-grid setting, where the frequencies may be arbitrary real numbers, it underlies sparse Fourier recovery and interpolation algorithms [PS15, CKPS16 , CP19b, SSWZ23]; closely related super-resolution problems ask to recover sparse point sources from bandlimited measurements [Don92, CFG14 , DN15 , Moi15 , LLF20 , LZGL22 , DELZ24] .  \nA central difficulty in the continuous setting is the absence of a frequency gap. When several frequencies are closer than the inverse length of the observation window, the corresponding exponential atoms become nearly linearly dependent: the associated Fourier–Vandermonde systems become ill-conditioned, and the individual frequencies may no longer be stably identifiable. In the limiting case of a k-fold frequency collision, suitably rescaled finite differences of the exponential atoms converge to {1, t, ... , tk−1}, so the clustered Fourier model approaches the space of algebraic polynomials of degree at most k − 1. In this regime, the natural algorithmic goal is not to recover the individual frequencies but to reconstruct the entire signal with L 2-error guarantees [CKPS16, CP19a, SSWZ23], or to estimate the location of the frequency cluster [CP19b] .  \nIn this paper, we study a deterministic extrapolation problem that underlies this clustered regime. Let  \nk  \ng (t) =Xvjeiλjt, λj ∈ R,  \nj=1  \nbe a k-Fourier-sparse signal whose L2-energy is known on [−1, 1] . Consider the following natural question:  \nHow large can g become at a point x > 1 outside the observation interval?  \nChen and Price [CP19b] introduced an outside-window growth estimate as a key ingredient in their algorithm for locating the center of a clustered Fourier signal. In the edge regime x = 1 + δ, their Lemma 8.5 gives, in our notation,  \n|g(1 + δ)| ≤ poly(k)exp 􀀀O (k2 log k · δ)􀀁 ∥g∥L2[−1 , 1] . (1)  \nTheir proof first derives a linear recurrence involving d = O (k2 log k) shifted values of the signal. It then propagates the estimate across steps of length Θ(1/d), paying a constant multiplicative loss at each step. Reaching distance δ therefore requires O (dδ) propagation steps, which produces the exponent in (1) .  \nThe same frequency-collision picture, however, suggests that this exponent is not tight. As the frequencies coalesce, k-term ex","cbCaipa3I4pmIyLC","https://ap.wps.com/l/cbCaipa3I4pmIyLC","pdf",501495,5,1,34,"English","en",105,"# Introduction\n## Chebyshev-scale extrapolation inequality\n## Applications of the extrapolation inequality","[{\"question\":\"What problem does the document address?\",\"answer\":\"It studies how large a k-sparse Fourier sum can become at points x\\u003e1 outside an observation interval where the L2 energy on [-1,1] is known.\"},{\"question\":\"What is the main theorem’s extrapolation bound?\",\"answer\":\"For g(t)=∑_{j=1}^k v_j e^{iλ_j t} with real frequencies and any x≥1, the document gives a Chebyshev-scale inequality bounding |g(x)| in terms of k and ∥g∥_{L^2[-1,1]}.\"},{\"question\":\"How does the result improve previous work and lead to algorithmic consequences?\",\"answer\":\"It refines earlier outside-window growth estimates by achieving the Chebyshev-scale exponential behavior rather than the prior larger growth rate, improving cluster-center resolution in the clustered-frequency recovery algorithm while preserving sample complexity up to logarithmic 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problem does the document address?","Question",{"text":76,"@type":77},"It studies how large a k-sparse Fourier sum can become at points x>1 outside an observation interval where the L2 energy on [-1,1] is known.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main theorem’s extrapolation bound?",{"text":81,"@type":77},"For g(t)=∑_{j=1}^k v_j e^{iλ_j t} with real frequencies and any x≥1, the document gives a Chebyshev-scale inequality bounding |g(x)| in terms of k and ∥g∥_{L^2[-1,1]}.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the result improve previous work and lead to algorithmic consequences?",{"text":85,"@type":77},"It refines earlier outside-window growth estimates by achieving the Chebyshev-scale exponential behavior rather than the prior larger growth rate, improving cluster-center resolution in the clustered-frequency recovery algorithm while preserving sample complexity up to logarithmic 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