Given a modulus of continuity $\omega$ and $q\in [1,\infty [$ then $H_q^{\omega }$ denotes the space of all functions $f$ with the period $1$ on $ℝ$ that are locally integrable in power $q$ and whose integral modulus of continuity of power $q$ (see(1)) is majorized by a multiple of $\omega$. The moduli of continuity $\omega$ are characterized for which $H_q^{\omega }$ contains “many” functions with infinite “essential” variation on an interval of length $1$.

[For the entire collection see Zbl 0742.00067.]Let ${𝔤}_k$ be the Lie algebra $𝔤l(k,𝒞)$, and let $U_k$ be the universal enveloping algebra for ${𝔤}_k$. Let $Z_k$ be the center of $U_k$. The authors consider the chain of Lie algebras ${𝔤}_n\supset {𝔤}_{n-1}\supset \cdots \supset {𝔤}_1$. Then $Z=\langle Z_k\mid k=1,2,\cdots n\rangle$ is an associative algebra which is called the Gel’fand-Zetlin subalgebra of $U_n$. A ${𝔤}_n$ module $V$ is called a $GZ$-module if $V={\sum }_x\oplus V\left(x\right)$, where the summation is over the space of characters of $Z$ and $V\left(x\right)=\{v\in V\mid {(a-x\left(a\right))}^{m}v=0$, $m\in {𝒵}_{+}$, $a\in 𝒵\}$. The authors describe several properties of $GZ$- modules. For example, they prove that if $V\left(x\right)=0$...

n the present paper the authors study some families of functions from a complex linear space $X$ into a complex linear space $Y$. They introduce the notion of $(j,k)$-symmetrical function ($k=2,3,\cdots$; $j=0,1,\cdots ,k-1$) which is a generalization of the notions of even, odd and $k$-symmetrical functions. They generalize the well know result that each function defined on a symmetrical subset $U$ of $X$ can be uniquely represented as the sum of an even function and an odd function.