Polynomials of least deviation from zero in Sobolev \(p\)-norm (Q2117580)

The authors study \textit{polynomials of least deviation from zero} (classical situation: determine existence, uniqueness and characterize the monic polynomial \(P_n\) of degree \(n\in\mathbb{Z}_{+}\) with \(\|P_n\|=\inf_{Q_n\in\mathbb{P}'_n}\,\|Q_n\|\), where \(\|\cdot\|\) is the norm in the linear space of polynomials \(\mathbb{P}\) and \(\mathbb{P}'_n\) the subset of polynomials of degree \(n\in\mathbb{Z}_{+}\)) with respect to the norm defined as follows: \begin{itemize} \item \(1\leq p<\infty\), \item \(\mu=(\mu_0,\mu_1,\ldots,\mu_m)\) with \(m\in\mathbb{Z}_{+}\) a vector of measures where \(\mu_k\) is a positive, finite Borel measure satisfying \(\operatorname{supp}\mu_k\subset\mathbb{R}\), \item \(\mathbb{P}\subset L^1 (\mu_k)\) for \(k=0,1,\ldots,m\), \item \(\Delta_k\) is the convex hull of \(\operatorname{supp}\mu_k\), \end{itemize} Then, for a function \(f\) (with \(f^{(k)}\) the \(k\)th derivative) \[\|f\|_{p,\mu}=\left(\sum_{k=0}^m\,\|f^{(k)}\|^{p}_{k,p}\right)^{1/p}=\left(\sum_{k=0}^m\,\int_{\Delta_k}\,|f{(k)}|^pd\mu_k\right)^{1/p}.\tag{1}\] In a previous paper [Bull. Math. Sci. 11, No. 1, Article ID 1950019, 18 p. (2021; Zbl 1465.42001)] the authors proved Theorem 1. Consider the following: Sobolev \(p\)-norm \((1)\) for \(1<p<\infty\). Then the monic polynomial \(P_n\) is the \(n\)th Sobolev minimal polynomial if and only if \[\langle P_n,q\rangle=\sum_{k=0}^m\,\int_{\Delta_k}\,q^{(k)}\operatorname{sgn}\left(P_n^{(k)}\right)\left|P_n^{(k)}\right|^{p-1}d\mu_k=0,\] for every polynomial \(q\in\mathbb{P}_{n-1}\). The layout of the paper is now as follows: \S1. Introduction (3 pages): Historical background and definitions of special norms (continuous Sobolev, discrete Sobolev) and the special norm that will be used in the sequel in the so-called \textit{discrete case}. Given \(N\in\mathbb{Z}_{+},\,\Omega=\{c_1,\ldots,c_N\}\subset\mathbb{C},\ \{m_0,\ldots,m_N\}\subset\mathbb{Z}_{+}\) and \(m=\max\{m_0,\ldots,m_N\}\). \begin{itemize} \item[1.] \(\mu_0=\mu+\sum_{j=1}^N\,A_{j,0}\delta_{c_j}\), where \(A_{j,0}\geq 0,\ \mu\) a finite positive Borel measure, \(\operatorname{supp}\mu\subset\mathbb{R}\) with infinitely mass points, \(\mathbb{P}\subset L^1(\mu)\) and \(\delta_x\) denotes the Dirac measure with mass point one at the point \(x\). \item[2.] For \(k=1,\ldots,m\) we have \(\mu_k=\sum_{j=1}^N\,A_{j,k}\delta_{c_j}\) where \(A_{j,k}\geq 0,\ A{j,m_j}>0\) and \(A_{j,k}=0\) if \(m_j<k\leq m\). \end{itemize} \S2. Polynomials of least deviation from zero when \(p=1\) (6 pages): Extension of Theorem 1 to the case \(p=1\). In Theorem 2 a general condition for sufficiency is given, that is shown not to be necessary (examples 2 and 3), nor does it guarantee uniqueness (example 1). Finally their Theorem 3 establishes a necessary and sufficient condition under which the formula in Theorem 1 characterizes minimality with respect to (1) when \(p=1\). Theorem 3. Let \(\mu=(\mu_0,\mu_1,\ldots,\mu_m)\) be a continuous standard vector measure. Then \(P_n\) is an \(n\)th Sobolev minimal polynomial with respect to \(\|\cdot\|_{1,\mu}\) if and only if \[\langle P_n,q\rangle_{1,\mu}=\sum_{k=0}^m\,\int_{\Delta_k}\,q^{(k)}\operatorname{sgn}\left(P_n^{(k)}\right)d\mu_k=0,\ \forall q\in\mathbb{P}_{n-1}.\] \S3. Lacunary and non-lacunary discrete Sobolev norms (\(5\frac{1}{2}\) pages): The focus is now discrete Sobolev norms (for every \(k=1,\ldots,m\) the measure \(\mu_k\) is supported on finitely many points) and the norm takes the form \[\|f\|_{p,\mu}=\left(\sum_{k=0}^m\,\int_{\Delta_k}\,\left|f^{(k)}\right|^p d\mu_k\right)^{1/p}= \left(\int_{\Delta}\,|f|^pd\mu + \sum_{j=1}^N \sum_{k=0}^{m_j}\,A_{j,k}|f^{(k)}(c_j)|^p\right)^{1/p}.\tag{2}\] The main results are as follows: Theorem 4. If (2) is essentially non-lacunary, then the set of zeros of a minimal polynomial sequence is uniformly bounded. Theorem 5. Consider a Sobolev \(p\)-norm (2), such that \(\mu\in\mathbf{Reg}\) and \(\Delta\) is abounded real interval. If \(\{P_n\}\) is the sequence of monic minimal polynomials with respect to (2), then for all \(j\geq 0\) \[\lim_{n\rightarrow\infty}\,\|P_n^{(j)}\|_{\Delta}^{1/n} =\operatorname{cap} (\Delta)\text{ and }w-\lim_{n\rightarrow\infty}\,\theta \left(P_n^{(j)}\right)= \omega_{\Delta}\text{ in the weak topology of measures}.\] \S4. Sequentially ordered discrete Sobolev norm (7 pages): The norm (2) is said to be \textit{sequentially ordered} if \[\Delta_k\cap\mathbf{Co}\left(\cup_{i=0}^{k-1}\Delta_i\right)^{\circ}=\emptyset,\ k=1,2,\ldots,m,\] and we have \[\Delta_0=\mathbf{Co}(\Delta\cup\{c_j\,:\,A_{j,0}>0\}).\] Furthermore \(d^{\ast}=|\{A_{j,k}>0\,:\, j=1,\ldots,N,\ k=0,1,\ldots,m_j\}|\) whre \(|U|\) is the cardinality of a set \(U\). Now the new result is the following: Theorem 7. Let \(\mu\) be a standard vector measure and \(1<p\leq \infty\). If \(\|\cdot\|_{p,\mu}\) ia a sequentially ordered Sobolev norm as in (2), where \(\mu\) is taken in such a way that \(c_j\not\in\Delta^{\circ}\), then \(P_n\) has at least \(n-d^{\ast}\) changes of sign on \(\Delta^{\circ}\)