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Aleksandra Świątczak

Publications and source records attributed to Aleksandra Świątczak.

5 recordsLinked to original sources
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↗
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

Delta-Sincov mappings in Banach algebras

We study solutions and approximate solutions of the multiplicative Sincov equation $$T(f, h) = T(f, g)T(g, h)$$ for mapping $T$ taking values in a commutative Banach algebra.

math.FA↗
arXiv2020

On the $c_0$-equivalence and permutations of series

Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given arbitrary sequence $(b_n)$ of real numbers there exists a permutation $σ\colon \N \to \N$ such that $σ(n) = n$ for every $n \notin A$ and $(b_n)$ is $c_0$-equivalent to a subsequence of the sequence of partial sums of the series $\sum\limits_{n=1}^\infty a_{σ(n)}$. Moreover, we discuss a connection between our main result with the classical Riemann series theorem.

math.FA↗