On Extremal Functions and V. Markov Type Polynomial Inequality for Certain Subsets of $${\mathbb {R}}^N

• Published in: Constructive approximation 2024-08, Vol.60 (1), p.135-163
• Main Authors: Baran, Mirosław, Sroka, Grzegorz
• Format: Article
• Language: English
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-023-09653-1

We introduce a polynomial extremal function $$\Phi (E,{\mathbb {F}},z)$$ Φ ( E , F , z ) which is one of possible generalizations of the classical Siciak extremal function, restricted to subspaces $${\mathbb {F}}$$ F of the linear space of all polynomials of N variables that are invariant under differentiation. We show that the so-called HCP condition in this situation: $$\log \Phi (E,{\mathbb {F}},z)\le A\text { dist}(z,E)^s,\ z\in {\mathbb {C}}^N$$ log Φ ( E , F , z ) ≤ A dist ( z , E ) s , z ∈ C N is equivalent to a generalization of the classical V. Markov’s inequality: $$||D^\alpha P||_E\le A_1^{|\alpha |}\frac{(\deg P)^{m|\alpha |}}{(|\alpha |!)^{m-1}}||P||_E,\ P\in {\mathbb {F}}$$ | | D α P | | E ≤ A 1 | α | ( deg P ) m | α | ( | α | ! ) m - 1 | | P | | E , P ∈ F with dependence $$m=1/s$$ m = 1 / s . The situation is similar to the basic case (cf. [8]) $${\mathbb {F}}={\mathbb {K}}[z_1,\dots ,z_N],$$ F = K [ z 1 , ⋯ , z N ] , where a V. Markov’s type inequality was introduced and the above-mentioned equivalence was proved. As a byproduct, we prove new results related to V. Markov’s inequality for an important class of subsets of $${\mathbb {R}}^N$$ R N , which are then applied to obtain the first versions of this inequality for some thin sets, such as spheres in $${\mathbb {R}}^{N+1}$$ R N + 1 and Euclidean spheres in particular. Furthermore, we prove an interesting fact on the polynomial convex hull of the circle $$S^1$$ S 1 , as a subset of $${\mathbb {R}}^2$$ R 2 .