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Schoenberg's Problem on Positive Definite Functions

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Alexander Koldobsky

Functional Analysis · Submitted 19 Oct 1992

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

The authors' main claim If $n \ge 3$, $q>2$ and $\beta > 0$ then the function $\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{\beta/q})$\ is not positive definite.

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arXiv IDmath/9210207 (version 1)
Pith ID37PTH45E, Pith's permanent record of this paper and its review
CitationAlexander Koldobsky. "Schoenberg's Problem on Positive Definite Functions." arXiv:math/9210207 (1992). https://pith.science/paper/37PTH45E
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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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,023 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.19 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.58 · compares with prior results 0.18 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

If $n \ge 3$, $q>2$ and $\beta > 0$ then the function $\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{\beta/q})$\ is not positive definite. This result gives an answer to a question posed by I.J.~Schoenberg in 1938. This text is an authorized English translation of the paper published in Russian in Algebra and Analysis 3(1991), \#3, p.78--85.

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 If $n \ge 3$, $q>2$ and $\beta > 0$ then the function $\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{\beta/q})$\ is not positive definite. result 1.00 0.84 0.95 asserted 1.00
s1 This result gives an answer to a question posed by I.J.~Schoenberg in 1938. interpretation 0.58 0.49 0.05 asserted 1.00
s2 This text is an authorized English translation of the paper published in Russian in Algebra and Analysis 3(1991), \#3, p.78--85. background 0.98 0.12 0.02 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

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  • full text
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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • 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

If $n \ge 3$, $q>2$ and $\beta > 0$ then the function $\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{\beta/q})$\ is not positive definite. This result gives an answer to a question posed by I.J.~Schoenberg in 1938. This text is an authorized English translation of the paper published in Russian in Algebra and Analysis 3(1991), \#3, p.78--85.

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