Search

Let $G$ be a group, let $A$ be an Abelian group, and let $n$ be an integer such that $n\backslash ge\; -1$. In the paper, the sets $\backslash Phi\_n(G,A)$ of functions from $G$ into $A$ of degree not greater than $n$ are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from $G$ into $A$ is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if $G$ is finite, then the study of the set $\backslash Phi\_n(G,A)$ is reduced to that of the set $\backslash Phi\_n(G/O^p(G),A\_p)$ for primes $p$ dividing $|G/G\text{'}|$. Here $O^p(G)$ stands for the $p$-coradical of the group $G$, $A\_p$ for the $p$-component of $A$, and $G\text{'}$ for the commutator subgroup of $G$. [ABSTRACT FROM AUTHOR]

The following problem is considered: obtain the asymptotic properties of a function $u$ from the asymptotic properties of the integral $\backslash int\_0^r\{u(t)\}\backslash ,dt$. As is well known, this can be done under additional constraints on the function $u(t)$. In this paper, we obtain a theorem in which these constraints are weaker than in other well-known versions of such theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study the spaces $B\_\{pq\}^s(G)$ and $L\_\{pq\}^s(G)$ of functions with positive exponent of smoothness $s>\; \backslash nomathbreak\; 0$, defined on a domain $G\backslash subset\; \backslash nomathbreak\; \backslash Bbb\; R^n$. For a domain $G$ with specific geometric properties, we establish the embedding $B\_\{pp\}^s(G)=L\_\{pp\}^s(G)\backslash subset\; L\_q(G)$, 1

New dimension functions for topological spaces are introduced in the spirit of Nagata’s approach. Expressions for the new functions in terms of covering dimension include the Bruijning—Nagata and Hashimoto—Hattori formulas. [ABSTRACT FROM AUTHOR]

Suppose that Δ is the set of functions x: I → ℝ on a finite interval I such that the divided differences [ x; t0,..., t s] of order s ∈ ℕ of these functions are nonnegative for all collections from ( s +1) different points t0,..., t s ∈ I. For all s ∈ ℕ and 1 ≤ p ≤ ∞, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of L p for classes $$\Delta _ + ^s B_p : = \Delta _ + ^s \cap B_p$$ , where B p is the unit ball in L p. We also establish the asymptotics of pseudodimensional widths in L p of these classes of functions. [ABSTRACT FROM AUTHOR]