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arXiv 2007.04061math.CA
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多项式和数列的离散加权Markov-Bernstein不等式

A discrete weighted Markov--Bernstein inequality for polynomials and sequences

Dimitar K. Dimitrov, Geno P. Nikolov

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15 pages
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摘要：

对于参数$\,c\in(0,1)\,$和$\,β>0$，令$\,\ell_{2}(c,β)\,$是在$\,\mathbb{N}\,$上定义的实函数的希尔伯特空间（即，实数序列），对于该序列，我们研究了最好的$$ \| f \|_{c,β}^2 := \sum_{k=0}^{\infty}\frac{(β)_k}{k!}\,c^k\,[f(k)]^2<\infty\,. $$（即，最小可能的）离散Markov-Bernstein不等式$$ \|ΔP\|_{c,β}\leq γ_n(c,β)\,\|P\|_{c,β}\,,\quad P\in\mathcal{P}_n\,, $$中的常数$\,γ_n(c,β)\,$，其中$\,\mathcal{P}_n\,$是次数至多为$\,n\,$和$\,Δf(x):=f(x+1)-f(x)\,$的实代数多项式的集合。我们证明了：（i）对于每个$\,n\in \mathbb{N}\,$和$\displaystyle \lim_{n\to\infty}γ_n(c,1)= 1+\frac{1}{\sqrt{c}}\,$，都有$\displaystyle γ_n(c,1)\leq 1+\frac{1}{\sqrt{c}}\,$. (ii)对于每个固定的$\,c\in (0,1)\,$，$\,γ_n(c,β)\,$是$\,(0,\infty)\,$中$\,β\,$的单调递减函数。(iii)对于每个固定的$\,c\in (0,1)\,$和$\,β>0\,$，最佳Markov-Bernstein常数$\,γ_n(c,β)\,$关于$\,n$一致有界。证明了$\,\ell_{2}(c,β)\,$中序列的一个类似的Markov-Bernstein不等式。我们还建立了最好的Markov-Bernstein常数$\,γ_n(c,β)\,$和某些显式给定的Jacobi矩阵的最小特征值之间的关系。

英文摘要：

For parameters $\,c\in(0,1)\,$ and $\,β>0$, let $\,\ell_{2}(c,β)\,$ be the Hilbert space of real functions defined on $\,\mathbb{N}\,$ (i.e., real sequences), for which $$ \| f \|_{c,β}^2 := \sum_{k=0}^{\infty}\frac{(β)_k}{k!}\,c^k\,[f(k)]^2<\infty\,. $$ We study the best (i.e., the smallest possible) constant $\,γ_n(c,β)\,$ in the discrete Markov-Bernstein inequality $$ \|ΔP\|_{c,β}\leq γ_n(c,β)\,\|P\|_{c,β}\,,\quad P\in\mathcal{P}_n\,, $$ where $\,\mathcal{P}_n\,$ is the set of real algebraic polynomials of degree at most $\,n\,$ and $\,Δf(x):=f(x+1)-f(x)\,$. We prove that: (i) $\displaystyle γ_n(c,1)\leq 1+\frac{1}{\sqrt{c}}\,$ for every $\,n\in \mathbb{N}\,$ and $\displaystyle \lim_{n\to\infty}γ_n(c,1)= 1+\frac{1}{\sqrt{c}}\,$. (ii) For every fixed $\,c\in (0,1)\,$, $\,γ_n(c,β)\,$ is a monotonically decreasing function of $\,β\,$ in $\,(0,\infty)\,$. (iii) For every fixed $\,c\in (0,1)\,$ and $\,β>0\,$, the best Markov-Bernstein constants $\,γ_n(c,β)\,$ are bounded uniformly with respect to $\,n$. A similar Markov-Bernstein unequality is proved for sequences in $\,\ell_{2}(c,β)\,$. We also establish a relation between the best Markov-Bernstein constants $\,γ_n(c,β)\,$ and the smallest eigenvalues of certain explicitly given Jacobi matrices.

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