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Polynomials of least deviation from zero in Sobolev \(p\)-norm. (English) Zbl 1489.30003

The authors study 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:
\(1\leq p<\infty\),
\(\mu=(\mu_0,\mu_1,\ldots,\mu_m)\) with \(m\in\mathbb{Z}_{+}\) a vector of measures where \(\mu_k\) is a positive, finite Borel measure satisfying \(\operatorname{supp}\mu_k\subset\mathbb{R}\),
\(\mathbb{P}\subset L^1 (\mu_k)\) for \(k=0,1,\ldots,m\),
\(\Delta_k\) is the convex hull of \(\operatorname{supp}\mu_k\),

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:
§1. 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 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 \(m=\max\{m_0,\ldots,m_N\}\).
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 \(x\).
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 \(m_j<k\leq m\).

§2. 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 \[\langle P_n,q\rangle_{1,\mu}=\sum_{k=0}^m\,\int_{\Delta_k}\,q^{(k)}\operatorname{sgn}\left(P_n^{(k)}\right)d\mu_k=0,\ \forall q\in\mathbb{P}_{n-1}.\]
§3. 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 \(j\geq 0\) \[\lim_{n\rightarrow\infty}\,\|P_n^{(j)}\|_{\Delta}^{1/n} =\operatorname{cap} (\Delta)\text{ and }w-\lim_{n\rightarrow\infty}\,\theta \left(P_n^{(j)}\right)= \omega_{\Delta}\text{ in the weak topology of measures}.\]
§4. Sequentially ordered discrete Sobolev norm (7 pages):
The norm (2) is said to be 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 \(\Delta^{\circ}\)

MSC:

30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Citations:

Zbl 1465.42001
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References:

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