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Eszter Gselmann

Publications and source records attributed to Eszter Gselmann.

At least 19 recordsLinked to original sources
arXiv2026

Perturbations of Cauchy differences

This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and also more general cases where the inhomogeneity depends on unknown functions \begin{align*} f(x+y)-f(x)-f(y)&= αx y \\[2.5mm] f(x+y)-f(x)-f(y)&= α(x y)\\[2.5mm] f(x+y)-f(x)-f(y)&= α(x)α(y). \end{align*} Our results extend previous work on the bilinearity of the Cauchy exponential difference by Alzer and Matkowski. We characterize solutions under various structural and regularity assumptions, including additive and exponential Cauchy differences, and show that solutions often reduce to additive functions, exponential polynomials, or combinations thereof. For Levi-Civita type equations, we provide explicit representations of solutions in terms of additive and exponential components. Furthermore, we determine conditions under which real-valued solutions exist and describe their forms. The paper concludes with open problems concerning generalized equations that cannot be solved by the methods presented here, suggesting directions for future research.

math.CA↗
arXiv2026

Second-order derivations of function spaces -- a characterization of second-order differential operators

Let $Ω\subset \mathbb{R}$ be a nonempty and open set, then for all $f, g, h\in \mathscr{C}^{2}(Ω)$ we have \begin{multline*} \diff{2}{x}(f\cdot g\cdot h) -f\diff{2}{x}(g\cdot h)-g\diff{2}{x}(f\cdot h)-h\diff{2}{x}(f\cdot g) + f\cdot g\diff{2}{x}h+f\cdot h\diff{2}{x}g+g\cdot h\diff{2}{x}f=0 \end{multline*} The aim of this paper is to consider the corresponding operator equation \[ D(f\cdot g \cdot h) - fD(g\cdot h) - gD(f\cdot h) - hD(f \cdot g) + f\cdot g D(h) + f\cdot h D(g) +g\cdot h D(f) =0 \] for operators $D\colon \mathscr{C}^{k}(Ω)\to \mathscr{C}(Ω)$, where $k$ is a given nonnegative integer and the above identity is supposed to hold for all $f, g, h \in \mathscr{C}^{k}(Ω)$. We show that besides the operators of first and second derivative, there are more solutions to this equation, and we characterize all solutions. Some special cases characterizing differential operators are also studied.

math.CA↗
arXiv2025

A direct and algebraic characterization of higher-order differential operators

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating multiple distinct operators. In contrast, we introduce a novel operator equation involving only a single $n$\textsuperscript{th}-order differential operator. We demonstrate that, under certain mild conditions, this equation serves to characterize such operators. Specifically, our results show that these higher-order differential operators can be identified as particular solutions to this single-operator identity. This approach provides a framework for understanding the algebraic structure of higher-order differential operators acting on function spaces.

math.CA↗
arXiv2025

On Iverson's law of similarity

Iverson (2006) proposed the law of similarity \[ ξ_{s}(λx)= γ(λ, s)ξ_{η(λ, s)}(x) \] for the sensitivity functions $ξ_{s}\, (s\in S)$. Compared to the former models, the generality of this one lies in that here $γ$ and $η$ can also depend on the variables $λ$ and $s$. In the literature, this model (or its special cases) is usually considered together with a given psychophysical representation (e.g. Fechnerian, subtractive, or affine). Our goal, however, is to study at first Iverson's law of similarity on its own. At first we show that if certain mild assumptions are fulfilled, then $ξ$ can be written in a rather simple form containing only one-variable functions. The obtained form proves to be very useful when we assume some kind of representation. Motivated by Hsu and Iverson (2016), in the second part of the third section we study the above model assuming that the mapping $η$ is multiplicatively translational. First, we show how these mappings can be characterized. Later on we turn to the examination of the so-called power law. According to our results, the corresponding function $ξ$ then does not have a Fechnerian representation, but it do have a subtractive representation. As an application of the results of the subsection, we close the paper with the study of the shift invariance property.

math.CA↗
arXiv2024

Characterizations of second-order differential operators

{Let $N, k$ be positive integers with $k\geq 2$, and $Ω\subset \mathbb{R}^{N}$ be a domain.} By the well-known properties of the Laplacian and the gradient, we have \[ Δ(f\cdot g)(x)=g(x) Δf(x)+f(x) Δg(x)+2\langle \nabla f(x), \nabla g(x)\rangle \] for all $f, g\in \mathscr{C}^{k}(Ω, \mathbb{R})$. {Due to the results of H.~König and V.~Milman, Operator relations characterizing derivatives. Birkhäuser / Springer, Cham, 2018.,} the converse is also true, i.e. this operator equation characterizes the Laplacian and the gradient under some assumptions. Thus the main aim of this paper is to provide an extension of this result and to study the corresponding equation \[ T(f\cdot g)= fT(g)+T(f)g+2B(A(f), A(g)) \qquad \left(f, g\in P\right), \] where $Q$ and $R$ are commutative rings, $P$ is a subring of $Q$ and $T\colon P\to Q$ and $A\colon P\to R$ are additive, while $B\colon R\times R\to Q$ is a symmetric and bi-additive. Related identities with one function will also be considered.

math.CA↗
arXiv2024

A functional equation for monomial functions

Let $\mathbb{F}\subset \mathbb{K}$ be fields with characteristic zero, $n$ be a positive integer and $κ\in \mathbb{K}$. In this paper, we determine those monomials $f\colon \mathbb{F}\to \mathbb{K}$ of degree $n$ for which \[ f(x^{2})= κ\cdot x^{n}f(x) \] holds for all $x\in \mathbb{F}$. We show that similar to the classical results, where additive functions were considered, the monomial functions in the equation can be represented with the aid of homomorphisms and higher-order derivations.

math.NT↗
arXiv2024

Quadratic functions as solutions of polynomial equations

The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there are a lot of open questions. In some specific cases, according to classical results, the unknown additive functions are homomorphisms, derivations, or linear combinations of these. The question arises as to whether the solutions can be described even if the unknown functions are not assumed to be additive but to be generalized monomials. As a starting point, we will deal with quadratic functions in this paper. We aim to show that quadratic functions that are solutions to certain polynomial equations necessarily have a `special' form. Further, we also present a method to determine these special forms.

math.AC↗
arXiv2023

A characterization of differential operators in the ring of complex polynomials

The paper aims to provide a full characterization of all operators $T\colon \mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot g+f\cdot T(g) \] for all $f, g\in \mathscr{P}(\mathbb{C})$. We do not assume the linearity of $T$. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.

math.CA↗
arXiv2023

Operator relations characterizing higher-order differential operators

Let $r$ be a positive integer, $N$ be a nonnegative integer and $Ω\subset \mathbb{R}^{r}$ be a domain. Further, for all multi-indices $α\in \mathbb{N}^{r}$, $|α|\leq N$, let us consider the partial differential operator $D^α$ defined by \[ D^α= \frac{\partial^{|α|}}{\partial x_{1}^{α_{1}}\cdots \partial x_{r}^{α_{r}}}, \] where $α= (α_{1}, \ldots, α_{r})$. Here by definition we mean $D^{0}\equiv \mathrm{id}$. An easy computation shows that if $f, g\in \mathscr{C}^{N}(Ω)$ and $α\in \mathbb{N}^{r}, |α|\leq N$, then we have \[ \tag{$\ast$} D^α(f\cdot g) = \sum_{β\leq α}\binomαβD^β(f)\cdot D^{α- β}(g). \] This paper is devoted to the study of identity $(\ast)$ in the space $\mathscr{C}(Ω)$. More precisely, if $r$ is a positive integer, $N$ is a nonnegative integer and $Ω\subset \mathbb{R}^{r}$ is a domain, then we describe those mappings $T_α \colon \mathscr{C}(Ω)\to \mathscr{C}(Ω)$, $α\in \mathbb{N}^{r}, |α|\leq N$ that satisfy identity $(\ast)$ for all possible multi-indices $α\in \mathbb{N}^{r}$, $|α|\leq N$. Our main result says that if the domain is $\mathscr{C}(Ω)$, then the mappings $T_α$ are of a rather special form. Related results in the space $\mathscr{C}^{N}(Ω)$ are also presented.

math.CA↗
arXiv2023

Polynomial equations for additive functions I. The inner case

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered $$\sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left(x\in \mathbb{F}\right),$$ where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$.

math.CA↗
arXiv2023

Polynomial equations for additive functions II

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$. Using the theory of decomposable functions we describe the solutions as compositions of higher order derivations and field homomorphisms. In many cases we also give a tight upper bound for the order of the involved derivations. Moreover, we present the full description of the solutions in some important special cases, too.

math.CA↗
arXiv2022

Monomial functions, normal polynomials and polynomial equations

In this paper we consider generalized monomial functions $f, g\colon \mathbb{F}\to \mathbb{C}$ (of possibly different degree) that also fulfill \[ f(P(x))= Q(g(x)) \qquad \left(x\in \mathbb{F}\right), \] where $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ are given (classical) polynomials.

math.AC↗
arXiv2022

Generalized derivations and generalized exponential monomials on hypergroups

In one of our former papers {\it Endomorphisms of the measure algebra of commutative hypergroups arXiv:2204.07499 we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra.

math.RA↗
arXiv2022

Endomorphisms and derivations of the measure algebra of commutative hypergroups

Endomorphisms of the measure algebra of commutative hypergroups are investigated. We focus on derivations and higher order derivations which are closely related to moment function sequences of higher rank. We describe the exact connection between those higher order derivations which are endomorphisms of the measure algebra if it is considered as a module over the ring of continuous functions.

math.FA↗
arXiv2022

Moment functions of higher rank on polynomial hypergroups

In this paper we consider generalized moment functions of higher order. These functions are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. In our former paper we investigated the properties of generalized moment functions of higher order on commutative groups. In particular, we proved the characterization of generalized moment functions on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and a sequence additive functions. In the present paper we continue the study of generalized moment function sequences of higher order in the more abstract setting, namely we consider functions defined on a hypergroup. We characterize these functions on the polynomial hypergroup in one variable by means of partial derivatives of a composition of polynomials generating the polynomial hypergroup and an analytic function. As an example, we give an explicit formula for moment generating functions of rank at most two on the Tchebyshev hypergroup.

math.CO↗
arXiv2021

Polynomial identities satisfied by generalized polynomials

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ be polynomials. Our aim is to prove characterization theorems for generalized polynomials $f\colon \mathbb{F}\to \mathbb{C}$ of degree two that also fulfill equation \[ f(P(x))= Q(f(x)) \] for each $x\in \mathbb{F}$. As it turns out, the difficulty of such problems heavily depends on that we consider the above equation for generalized polynomials or for (normal) polynomials. Therefore, firstly we study the connection between these two notions.

math.AC↗
arXiv2021

Moment functions and exponential monomials on commutative hypergroups

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.

math.SP↗
arXiv2021

Spectral synthesis via moment functions on hypergroups

In this paper we continue the discussion about relations between exponential polynomials and generalized moment generating 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 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.

math.SP↗