# Transitivity and Unimprovability of Imbedding Theorems. Compactness

• S. M. Nikol’skii
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 205)

## Abstract

Suppose given a system of numbers (1) ${\bf\it r} = (r_1,...,r_n) \ > {\bf 0}, \ {\bf\it p} = (p_1,...,p_n) \ (1{\mathop<\limits_=} \ p_l {\mathop<\limits_=}\ \infty)$ and numbers p′, p″, satisfying the inequalities (2) $$p_l \ {\mathop<\limits_=} \ {p^\prime} \ < \ p^{\prime\prime} \ {\mathop<\limits_=}\ \infty.$$ If the following conditions are satisfied : (3) $$\varrho_i^\prime = {r_i\chi\over \chi_i},$$ (4) $$\chi=1-{\mathop\sum\limits_{l=1}^n}{{1\over p_l}-{1\over p^\prime}\over r_l}>0,$$ (5) $$\chi_i=1-{\mathop\sum\limits_{l=1}^n}{{1\over p_l}-{1\over p_i}\over r_l}>0\quad (i=1,...,n),$$ then the imbedding theorem (6.8) holds: $$B_{p\theta}^r({\rm R}_n)\rightarrow B_{p^\prime\theta}^{p^\prime}({\rm R}_n)$$ ccomplishing the transition from the system of numbers (1) to the system of numbers $$\varrho^\prime=(\varrho_1^\prime,...,\varrho_n^\prime), \ p^\prime.$$ 1 S. M. Nikol’skii [3, 10].

## Keywords

Banach Space Entire Function Boundary Function Exponential Type Mixed Derivative
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.