Pith. Sign in

RECORD

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Paul Catala, Mathias Hockmann, Stefan Kunis, Markus Wageringel

Numerical Analysis · Submitted 20 Mar 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2203.10531 (version 2)
Pith IDOG6DKPMT, Pith's permanent record of this paper and its review
CitationMarkus Wageringel, Mathias Hockmann, Paul Catala, Stefan Kunis. "Approximation and Interpolation of Singular Measures by Trigonometric Polynomials." arXiv:2203.10531 (2022). https://pith.science/paper/OG6DKPMT
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,298 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.29 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.56 · compares with prior results 0.17 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Complex signed measures of finite total variation are a powerful signal model in many applications. background 1.00 0.04 0.02 asserted 1.00
s1 Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. background 1.00 0.08 0.04 asserted 0.90
s2 Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. contribution 0.92 0.64 0.07 asserted 0.90
s3 We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. result 0.90 0.95 0.70 asserted 1.00
s4 A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside. result 0.63 0.91 0.15 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.