arXiv subjects

E. V. Shulman

Publications and source records attributed to E. V. Shulman.

2 recordsLinked to original sources

arXiv2017

On polynomial functions on non-conmmutative groups

Let $G$ be a topological group. We investigate relations between two classes of "polynomial like" continuous functions on $G$ defined, respectively, by the conditions (1) $Δ_h^{n+1}f=0$ for every $h \in G$, and (2) $Δ_{h_{n+1}} Δ_{h_{n}}\cdots Δ_{h_{1}}f=0$, for every $h_1,\cdots, h_{n+1} \in G$. It is shown that for many (but not all) groups these classes coincide. We consider also Montel type versions of the above conditions - when (1) and (2) hold only for steps $h$ in a generating subset of $G$. Our approach is based on the study of the counterparts of the discussed classes for general representations of groups (instead of the regular representation).

math.CA↗

arXiv2017

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

We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)= \sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in {GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued functions and in the framework of complex-valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets are, typically, exponential polynomials and, in some particular cases, related to so called characterization problem of the normal distribution in Probability Theory, they reduce to ordinary polynomials.

math.CA↗