On One Extremal Problem for a Seminorm on the Space l₁ with Weight

Abstract

Let $\alpha = \left\{ {\alpha _j } \right\}_{j \in \mathbb{N}}$ be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences $\xi = \left\{ {\xi _j } \right\}_{j \in \mathbb{N}}$ for which $\left\| \xi \right\|_{1,\alpha } :\; = \sum {_{j = 1}^\infty } \alpha _j \left| {\xi _j } \right| < + \infty$. We associate every sequence ξ from l _1,α with a sequence $\xi * = \left\{ {\left| {\xi _{\varphi (j)} } \right|} \right\}_{j \in \mathbb{N}}$, where ϕ(·) is a permutation of the natural series such that $\left| {\xi _{\varphi (j)} } \right| \geqslant \left| {\xi _{\varphi (j + 1)} } \right|$, j ∈ ℕ. If p is a bounded seminorm on l _1,α and $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, then

$$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$

Using this equality, we obtain several known statements.