On converse Marcinkiewicz-Zygmund inequalities in \(L_p,p>1\)
In this paper some converse Marcinkiewicz-Zygmund inequalities are investigated. In particular, the following result is proved: Theorem: Let \(n\geq 1\) and let \(\{t_j\}^n_{j= 1}\) satisfy \(-1\leq t_1< t_2<\cdots< t_n\leq 1\) and let \(t_j:= -1\) for \(j\leq 0\), \(t_j:= 1\) for \(j>n\). Let \(\nu:[-1, 1]\to [0,\infty)\) be measurable and let \(\mu(t):= 0\) for \(t\not\in [-1,1]\). Let \(\Pi_n(t)\) be a polynomial of degree \(n\) whose zeros are \(\{t_j\}^n_{j= 1}\), normalized by the condition \(|\Pi_n(t)\nu(t)|\leq 1\) for \(t\in [-1,1]\). Let \(L\geq 1\) and \(\delta_j:= t_{j+ L}- t_{j-L}\), \(1\leq j\leq n\). Assume there exists \(\alpha> 0\) such that for \(1\leq j, k\leq n\) with \(|j-k|\geq L\), \[ |t_j- t_k|\geq \alpha|j-k|^{1/3} [1+\log|j-k|]^{2/3} \delta_j. \] Let \(\varepsilon> 0\), \(1< p<\infty\). Thus for \(P\in{\mathcal P}_{n-1}=\{\)the family of polynomials of degree less or equal \(n-1\}\), \[ \int^1_{-1}|P(t) \nu(t)|^p dt\leq C \sum^n_{j=1}|P(t_j)|^p \Biggl\{\int^{t_j+ \varepsilon\delta_j}_{t_j- \varepsilon\delta_j}|l_j(t) \nu(t)|^p dt+{\delta_j\over [\delta_j|\Pi_n(t_j)|]^p} \Biggr\}, \] where \(\{l_j(t)\}\) are the fundamental polynomials of Lagrange interpolation at the \(\{t_j\}\) that is \(L_n(P)(t)= \sum^n_{j=1} P(t_j)l_j(t)\). The constant \(C\) depends on \(L\), \(\alpha\), \(\varepsilon\), but is independent of \(\nu\), \(\{t_j\}^n_{j= 1}\), \(n\), \(P\).
scientific article; zbMATH DE number 1398363