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1
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Polynomial inequalities in Banach spaces

100%
Baran M.
Banach Center Publications
|
2015
|
tom 107
|
nr 1
23-42
EN
We point out relations between the injective complexification of a real Banach space and polynomial inequalities. In particular we prove a generalization of a classical Szegő inequality to the case of polynomial mappings between Banach spaces. As an application we observe a complex version of known Bernstein-Szegő type inequalities.
2
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Cauchy-Poisson transform and polynomial inequalities

100%
Baran M.
Annales Polonici Mathematici
|
2009
|
tom 95
|
nr 3
199-206
EN
We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in $ℝ^N$ is Hölder continuous then E admits a Szegö type inequality with weight function $dist(x,∂E)^{-(1-κ)}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.
3
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Markov inequality on sets with polynomial parametrization

100%
Baran M.
Annales Polonici Mathematici
|
1994-1995
|
tom 60
|
nr 1
69-79
EN
The main result of this paper is the following: if a compact subset E of $ℝ^n$ is UPC in the direction of a vector $v ∈ S^{n-1}$ then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve an earlier result of Pawłucki and Pleśniak [PP1].
4
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Homogeneous extremal function for a ball in ℝ²

100%
Baran M.
Annales Polonici Mathematici
|
1999
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tom 71
|
nr 2
141-150
EN
We point out relations between Siciak's homogeneous extremal function $Ψ_B$ and the Cauchy-Poisson transform in case $B$ is a ball in ℝ². In particular, we find effective formulas for $Ψ_B$ for an important class of balls. These formulas imply that, in general, $Ψ_B$ is not a norm in ℂ².
5
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Product property for capacities in $ℂ^{N}$

64%
Baran M., Bialas-Ciez L.
Annales Polonici Mathematici
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2012
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tom 106
|
nr 1
19-29
EN
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.
6
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Sets with the Bernstein and generalized Markov properties

64%
Baran M., Kowalska A.
Annales Polonici Mathematici
|
2014
|
tom 111
|
nr 3
259-270
EN
It is known that for $C^{∞}$ determining sets Markov's property is equivalent to Bernstein's property. We are interested in finding a generalization of this fact for sets which are not $C^{∞}$ determining. In this paper we give examples of sets which are not $C^{∞}$ determining, but have the Bernstein and generalized Markov properties.
7
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Markov's property for kth derivative

51%
Baran M., Milówka B., Ozorka P.
Annales Polonici Mathematici
|
2012
|
tom 106
|
nr 1
31-40
EN
Consider the normed space $(ℙ(ℂ^{N}),||·||)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_{g}: (ℙ(ℂ^{N}),||·||) ∋ f ↦ gf ∈ (ℙ(ℂ^{N}),||·||)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|∂/∂z_{j} P|| ≤ M(deg P)^{m} ||P||$, j = 1,...,N, $P ∈ ℙ(ℂ^{N})$, with positive constants M and m is equivalent to the inequality $||∂^{N}/∂z₁...∂z_{N} P|| ≤ M'(deg P)^{m'}||P||$, $P ∈ ℙ(ℂ^{N})$, with some positive constants M' and m'. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In addition, we give a nontrivial example of Markov's inequality in the Wiener algebra of absolutely convergent trigonometric series and show that the Banach algebra approach to Markov's property furnishes new tools in the study of polynomial inequalities.
8
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Siciak's extremal function of convex sets in $C^N$

44%
Baran M.
Annales Polonici Mathematici
|
1988
|
tom 48
|
nr 3
275-280
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