Several functional equations defined on groups arising from stochastic distance measures.

Several functional equations related to stochastic distance measures have been widely studied when defined on the real line. This dissertation generalizes several of those results to functions defined on groups and fields. Specifically, we consider when the domain is an arbitrary group, G, and the range is the field of complex numbers, C. We begin by looking at the linear functional equation f(pr, qs)+f(ps, qr) = 2f(p, q)+2f(r, s) for all p, q, r, s, € G. The general solution f : G x G → C is given along with a few specific examples. Several generalizations of this equation are also considered and used to determine the general solution f, g, h, k : G x G → C of the functional equation f(pr, qs) + g(ps, qr) = h(p, q) + k(r, s) for all p, q, r, s € G. We then consider the non-linear functional equation f(pr, qs) + f(ps, qr) = f(p, q) f(r, s). The solution f : G x G → C is given for all p, q, r, s € G when f is an abelian function. It is followed by the structure of the general solution, f, dependent upon how the function acts on the center of the group. Several generalizations of the equation are also considered. The general structure of the solution f, g, h : G x G → C of the functional equation f(pr, qs) + f(ps, qr) = g(p, q) h(r, s) is given for all p, q, r, s € G, dependent upon how the function h acts on the center of the group. Future plans related to these equations will be given

Characterizations of derivations

The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations. In Chapter 2 we collect all the definitions and results regarding derivations that are essential while studying this area. In Chapter 3 we intend to show that derivations can be characterized by one single functional equation. More exactly, we study here the following problem. Let $Q$ be a commutative ring and let $P$ be a subring of $Q$. Let $\lambda, \mu\in Q\setminus\left\{0\right\}$ be arbitrary, $f\colon P\rightarrow Q$ be a function and consider the equation $$\lambda\left[f(x+y)-f(x)-f(y)\right]+ \mu\left[f(xy)-xf(y)-yf(x)\right]=0 \quad \left(x, y\in P\right).$$ In this chapter it will be proved that under some assumptions on the rings $P$ and $Q$, derivations can be characterized via the above equation. Chapter 4 is devoted to the additive solvability of a system of functional equations. Moreover, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n} \colon\mathbb{R}\to\mathbb{R}$ of the above system of equations is characterized. Finally, the closing chapter deals with the following problem. Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and for the additive function $f\colon \mathbb{R}\to \mathbb{R}$, the mapping $$\varphi(x)=f\left(\xi(x)\right)-\xi'(x)f(x)$$ fulfills some regularity condition on its domain. Is it true that in such a case $f$ is a sum of a derivation and a linear function

On the function spaces of general weights

The aim of this paper is twofold. Firstly, we chatacterize the Besov spaces $\dot{B}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and the Triebel-Lizorkin spaces $\dot{F}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ for $q=\infty$. Secondly, under some suitable assumptions on the $p$-admissible weight sequence $\{t_{k}\}$, we prove that \begin{equation*} \dot{A}_{p,q}(\mathbb{R}^{n},\{t_{k}\})=\dot{A}_{p,q}(\mathbb{R} ^{n},t_{j}),\quad j\in \mathbb{Z}, \end{equation*} in the sense of equivalent quasi-norms, with $\dot{A}$ $\in \{\dot{B},\dot{F}\}$. Moreover, we find a necessary and sufficient conditions for the coincidence of the spaces $\dot{A}_{p,q}(\mathbb{R}^{n},t_{i}),i\in \{1,2\}$.Comment: We add Theorem 3.34 and corollaries 3.37 and 3.42. arXiv admin note: substantial text overlap with arXiv:2009.12223, arXiv:2106.00621, arXiv:2009.0363

Characterizations of second-order differential operators

If $k\geq 2$ is a positive integer, $\Omega \subset \mathbb{R}^{N}$ is a domain then by the well-known properties of the Laplacian and the gradient, we have $$\Delta(f\cdot g)=g \Delta f+f \Delta g+2\langle \nabla f, \nabla g\rangle$$ for all $f, g\in \mathscr{C}^{k}(\Omega, \mathbb{R})$. Due to the results of K\"onig--Milman \cite{KonMil18}, the converse is also true under some assumptions. Thus the main aim is 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 and $P$ is a subring of $Q$, further $T\colon P\to Q$ and $A\colon P\to R$ are additive mappings, while $B\colon R\times R\to Q$ is a symmetric and bi-additive mapping. Related identities with one function will also be considered which turn also suitable for characterizing second-order differential operators

A p-adic Montel theorem and locally polynomial functions

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if $$\Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all} x\in\mathbb{Q}_p,$$ and $|h_0|_p=p^{-N_0}$ then, for all $x_0\in \mathbb{Q}_p$, the restriction of $f$ over the set $x_0+p^{N_0}\mathbb{Z}_p$ coincides with a polynomial $p_{x_0}(x)=a_0(x_0)+a_1(x_0)x+...+a_m(x_0)x^m$. Motivated by this result, we compute the general solution of the functional equation with restrictions given by {equation} \Delta_h^{m+1}f(x)=0 \ \ (x\in X \text{and} h\in B_X(r)=\{x\in X:\|x\|\leq r\}), {equation} whenever $f:X\to Y$, $X$ is an ultrametric normed space over a non-Archimedean valued field $(\mathbb{K},|...|)$ of characteristic zero, and $Y$ is a $\mathbb{Q}$-vector space. By obvious reasons, we call these functions uniformly locally polynomial.Comment: 12 pages, submitted to a journa