On certain generalizations of the Levi-Civita and Wilson functional equations

Almira, J. M.; Shulman, Ekaterina

doi:10.1007/s00010-017-0489-4

Cited by 4 publications

(6 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The case of Abelian topological groups was also discussed by Laird and by Székelyhidi, see [8], [15,Theorem 10.1], [16]. (As for the history of Theorem A, see also [2] and the references given there.) The result was generalized for measurable (instead of continuous) functions defined on locally compact Abelian groups, see [5,14] and [15,Theorem 10.2].…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

See 1 more Smart Citation

The Levi-Civita equation in function classes

Laczkovich

2019

Aequat. Math.

View full text Add to dashboard Cite

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form n i=1 p i • m i , where m 1 ,. .. , mn are exponentials belonging to V , and p 1 ,. .. , pn are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra A of complex valued functions such that whenever an exponential m belongs to A, then m −1 ∈ A. As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary σ-algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…, y n ] be the polynomial obtained from f by substitution (2). It is easy to check that substitution (2) induces a bijection from (2).…”

Section: Theorem 4 (I) Every Homogeneous Polynomialmentioning

confidence: 99%

The Levi-Civita equation in function classes

Laczkovich

2019

Aequat. Math.

View full text Add to dashboard Cite

show abstract

“…As Theorem 3 shows, there are some particular cases of equation (5) which characterize continuous polynomials in R d . In this section we present a new proof of Theorem 3 which, in contrast to the one included in [6], is not based on the very technical tools used in [11]. We also present a new proof of a classical result by Ghurye and Olkin about characterizations of Gaussian distributions (see Theorem 9 below), under certain additional restrictions.…”

Section: Polynomials As Solution Sets Of Certain Functional Equationsmentioning

confidence: 99%

“…

In [6] the functional equationwas studied, both in the classical context of continuous complex valued functions and in the framework of complex valued Schwartz distributions, where these equations were properly introduced in two different ways. The solution sets of these equations are, typically, exponential polynomials and, in some particular cases, they reduce to ordinary polynomials.

…”

mentioning

confidence: 99%

Characterization of polynomials as solutions of certain functional equations

Almira¹

2018

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

Abstract. In [6] the functional equationa i pyqv i pxq with x, y P R d and b i , c i P GL d pCq, was studied, both in the classical context of continuous complex valued functions and in the framework of complex valued Schwartz distributions, where these equations were properly introduced in two different ways. The solution sets of these equations are, typically, exponential polynomials and, in some particular cases, they reduce to ordinary polynomials. In this paper we present several characterizations of ordinary polynomials as the solution sets of certain related functional equations. Some of these equations are important because of their connection with the Characterization Problem of distributions in Probability Theory.

show abstract

“…Some recent results can be found in [1,2] and the references therein. In [3], the authors prove some new properties of exponential polynomials. In the present paper we join this research by proving further results concerning properties of generalized exponential polynomials on hypergroups.…”

Section: It Is Clear That a Linear Subspace X Of C(h) Is A Variety Ifmentioning

confidence: 99%