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Reproducing kernel Hilbert spaces generated by the binomial coefficients

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Daniel Alpay, Palle Jorgensen

Complex Variables · Submitted 7 Dec 2012

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

The authors' main claim We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space.

how it is stated · asserted 1.00 read the record →

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arXiv ID1212.1564 (version 3)
Pith IDBGYNP7MI, Pith's permanent record of this paper and its review
CitationDaniel Alpay, Palle Jorgensen. "Reproducing kernel Hilbert spaces generated by the binomial coefficients." arXiv:1212.1564 (2012). https://pith.science/paper/BGYNP7MI
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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.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,845 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions and with the quantum case.

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 We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. contribution 0.92 0.80 0.08 asserted 0.94
s1 We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. contribution 0.63 0.94 0.12 asserted 1.00
s2 Finally, we mention connections with the theory of discrete analytic functions and with the quantum case. interpretation 0.70 0.49 0.10 asserted 0.72

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from proof or derivation 1.00.

What was examined, and what was not

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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • 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

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions and with the quantum case.

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