Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Siciak's extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

100%

Bialas-Ciez L.

Annales Polonici Mathematici

2012

tom 106

nr 1

41-51

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E ⊂ ℂ^{N}$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$. Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $lim_{n→∞} Mₙ(E)^{1/n} = 1$ where $Mₙ(E) := sup{|| |grad P| ||_{E}/||P||_{E}}$, the supremum is taken over all polynomials P of N variables of total degree at most n and $||·||_{E}$ is the uniform norm on E. The above condition is fulfilled e.g. for all regular (in the sense of the continuity of the pluricomplex Green function) compact sets in $ℂ^{N}$.

Product property for capacities in $ℂ^{N}$

63%

Baran M., Bialas-Ciez L.

Annales Polonici Mathematici

2012

tom 106

nr 1

19-29

The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν}(E₁ × E₂) = min(C_{ν₁}(E₁),C_{ν₂}(E₂))$, where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N₁+N₂}$, ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$, denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes in $ℝ^{2N}$. We prove that $C(E) = ω_{E}/2$ for a ball E in $ℂ^{N}$, while $C(E) = ω_{E}/4$ if E is a convex symmetric body in $ℝ^{N}$. This gives a generalization of known formulas in ℂ. Moreover, we show by an example that the last equality is not true for an arbitrary convex body.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Bialas-Ciez L.

1 Baran M.

2 2012

Wyniki wyszukiwania

Siciak's extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

Product property for capacities in $ℂ^{N}$