Moment functions and exponential monomials on commutative hypergroups

Fechner, Żywilla; Gselmann, Eszter; Székelyhidi, László

doi:10.1007/s00010-021-00827-5

Aequat. Math.

2021

DOI: 10.1007/s00010-021-00827-5

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Moment functions and exponential monomials on commutative hypergroups

Żywilla Fechner

Eszter Gselmann

László Székelyhidi

Abstract: The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.

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Cited by 2 publications

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Moment functions of higher rank on some types of hypergroups

Fechner,

Gselmann,

Székelyhidi

2023

Semigroup Forum

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We consider moment functions of higher order. In our earlier paper, we have already investigated the moment functions of higher order on groups. The main purpose of this work is to prove characterization theorems for moment functions on the multivariate polynomial hypergroups and on the Sturm–Liouville hypergroups. In the first case, the moment generating functions of higher rank are partial derivatives (taken at zero) of the composition of generating polynomials of the hypergroup and functions whose coordinates are given by the formal power series. On Sturm–Liouville hypergroups the moment functions of higher rank are restrictions of even smooth functions that also satisfy certain boundary value problems. The second characterization of moment functions of higher rank on Sturm–Liouville hypergroups is given by means of an exponential family. In this case, the moment functions of higher rank are partial derivatives of an appropriately modified exponential family again taken at zero.

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Moment functions of higher rank on some types of hypergroups

Fechner,

Gselmann,

Székelyhidi

2023

Semigroup Forum

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Spectral synthesis via moment functions on hypergroups

2022

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In this paper, we continue the discussion about relations between exponential polynomials and generalized moment functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite-dimensional variety is spanned by moment functions? Let 𝑚 be an exponential on 𝑋. In our former paper, we have proved that if the linear space of all 𝑚-sine functions in the variety of an 𝑚-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions generated by 𝑚. In this paper, we show that this may happen also in cases where the 𝑚-sine functions span a more than one-dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in 𝑑 variables, describe exponentials on it and give the characterization of the so-called 𝑚-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in 𝑑 variables is the polynomial ring in 𝑑 variables. Finally, using the Ehrenpreis–Palamodov Theorem, we show that every exponential polynomial on the polynomial hypergroup in 𝑑 variables is a linear combination of moment functions contained in its variety.

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Moment functions and exponential monomials on commutative hypergroups

Abstract: The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.

Cited by 2 publications

References 5 publications

Moment functions of higher rank on some types of hypergroups

Moment functions of higher rank on some types of hypergroups

Spectral synthesis via moment functions on hypergroups

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