RECORD
The hypergroup property and representation of Markov kernels
Probability · Submitted 25 Jan 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the case of Jacobi polynomials with non-symmetric parameters, the hypergroup property is nothing else than Gasper's theorem.
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
The abstract, sentence by sentence
For a given orthonormal basis $(f_n)$ on a probability measure space, we want to describe all Markov operators which have the $f_n$ as eigenvectors. We introduce for that what we call the hypergroup property. We study this property in three different cases. On finite sets, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. The representation theory of groups provides generic examples where these two properties are satisfied, although this group structure is not necessary in general. The hypergroup property also holds for Sturm--Liouville bases associated with log-concave symmetric measures on a compact interval, as stated in Achour--Trim\`eche's theorem. We give some criteria to relax this symmetry condition in view of extensions to a more general context. In the case of Jacobi polynomials with non-symmetric parameters, the hypergroup property is nothing else than Gasper's theorem. The proof we present is based on a natural interpretation of these polynomials as harmonic functions and is related to analysis on spheres. The proof relies on the representation of the polynomials as the moments of a complex variable.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For a given orthonormal basis $(f_n)$ on a probability measure space, we want to describe all Markov operators which have the $f_n$ as eigenvectors. | background 0.94 | 0.26 | 0.05 | asserted 0.52 |
| s1 | We introduce for that what we call the hypergroup property. | contribution 0.93 | 0.90 | 0.04 | asserted 0.90 |
| s2 | We study this property in three different cases. | contribution 0.90 | 0.81 | 0.03 | asserted 0.92 |
| s3 | On finite sets, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. | result 0.66 | 0.20 | 0.07 | asserted 1.00 |
| s4 | The representation theory of groups provides generic examples where these two properties are satisfied, although this group structure is not necessary in general. | background 0.87 | 0.11 | 0.07 | asserted 0.98 |
| s5 | The hypergroup property also holds for Sturm--Liouville bases associated with log-concave symmetric measures on a compact interval, as stated in Achour--Trim\`eche's theorem. | background 0.65 | 0.07 | 0.10 | asserted 1.00 |
| s6 | We give some criteria to relax this symmetry condition in view of extensions to a more general context. | contribution 0.81 | 0.80 | 0.05 | asserted 0.80 |
| s7 | In the case of Jacobi polynomials with non-symmetric parameters, the hypergroup property is nothing else than Gasper's theorem. | result 0.64 | 0.14 | 0.14 | asserted 1.00 |
| s8 | The proof we present is based on a natural interpretation of these polynomials as harmonic functions and is related to analysis on spheres. | method 0.91 | 0.78 | 0.06 | asserted 1.00 |
| s9 | The proof relies on the representation of the polynomials as the moments of a complex variable. | method 0.99 | 0.70 | 0.06 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
For a given orthonormal basis $(f_n)$ on a probability measure space, we want to describe all Markov operators which have the $f_n$ as eigenvectors. We introduce for that what we call the hypergroup property. We study this property in three different cases. On finite sets, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. The representation theory of groups provides generic examples where these two properties are satisfied, although this group structure is not necessary in general. The hypergroup property also holds for Sturm--Liouville bases associated with log-concave symmetric measures on a compact interval, as stated in Achour--Trim\`eche's theorem. We give some criteria to relax this symmetry condition in view of extensions to a more general context. In the case of Jacobi polynomials with non-symmetric parameters, the hypergroup property is nothing else than Gasper's theorem. The proof we present is based on a natural interpretation of these polynomials as harmonic functions and is related to analysis on spheres. The proof relies on the representation of the polynomials as the moments of a complex variable.
Human review
How the literature uses it
1 arXiv paper cite this in 7 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 1 paper
-
false For more details, the reader may consult [ref] , [ref] and [ref] , especially Section 2.4.
On the Markov sequence problem for Jacobi polynomials · Product formulas and the Markov sequence problem · 0.92
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.