The Complex Equilibrium Measure of a Symmetric Convex Set in R n

Bedford, Eric; Taylor, B. A.

doi:10.2307/2000210

Cited by 21 publications

(49 citation statements)

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“…Here, for a symmetric convex body E in R n , E * := {y ∈ R n : x · y ≤ 1, for all x ∈ E} is also a symmetric convex body in R n , called the polar of E. The quantity δ B (x, y) is continuous on K o × R n and for each fixed x ∈ K o , y → δ B (x, y) is a norm on R n ; i.e., δ B (x, y) ≥ 0, δ B (x, λy) = λδ B (x, y) for λ ≥ 0, and δ B (x, y 1 + y 2 ) ≤ δ B (x, y 1 ) + δ B (x, y 2 ) (see [3]). …”

Section: Introductionmentioning

confidence: 99%

“…For a symmetric convex body, i.e., K = −K, Bedford and Taylor [3] showed the existence of the limit (1.2) and proved the formula (1.3) using the description of the Monge-Ampère solution given by Lundin [10]. The present paper relies on the description of V K given in [6], [7] for general convex bodies K. [6] showed the existence, through each point z ∈ C n \ K, of a holomorphic curve on which V K is harmonic, while [7] showed that for many K (all K in R 2 ) these curves give a continuous foliation of C n \ K by holomorphic curves.…”

Section: Introductionmentioning

confidence: 99%

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Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities

Burns

Levenberg

Ma`u

et al. 2010

Trans. Amer. Math. Soc.

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Abstract. We use geometric methods to calculate a formula for the complex Monge-Ampère measure (dd c V K ) n , for K R n ⊂ C n a convex body and V K its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area ellipses determining the complex Monge-Ampère solution V K .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities

Burns

Levenberg

Ma`u

et al. 2010

Trans. Amer. Math. Soc.

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“…An essential role is played by the global extremal function. Bedford and Taylor (1986) gave precise estimates for the measure (dd c V K ) n when K is compact and contained in R n and gave an exact expression for it when K is convex and symmetric. Zeriahi (1996) investigated the global extremal function on nonsingular algebraic varieties and extended results in C n to that case.…”

Section: The Global Extremal Functionmentioning

confidence: 99%

“…This capacity was first introduced by Ko lodziej (2003) and corresponds to the relative capacity of Bedford and Taylor (1982). The global extremal function is also defined:…”

Section: Capacities Defined By the Relative Extremal Functionmentioning

confidence: 99%

Vyacheslav Zakharyuta’s complex analysis

Kiselman¹

2009

Functional Analysis and Complex Analysis

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“…There is a well-developed theory of capacities in C n [3], [6], [17], [18], [20], but for our purposes these are difficult to estimate, especially as there is no longer such a simple relationship between potentials and logs of polynomials. We prefer to use product capacity and Favarov's capacity (a close cousin of Ronkin's γ-capacity), as discussed by Cegrell [6, p.86, p.81].…”

Section: §1 Introductionmentioning

confidence: 99%

On the size of lemniscates of polynomials in one and several variables

Cuyt

Driver

Lubinsky

1996

Proc. Amer. Math. Soc.

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Abstract. In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set E := z : |z| ≤ r and |P (z)| ≤ n , for a polynomial P of degree ≤ n. Usually, P is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of E, in terms of Hausdorff contents, planar Lebesgue measure m 2 , or logarithmic capacity cap. Here we normalize P L∞ |z|≤r = 1 and show that cap(E) ≤ 2r and m 2 (E) ≤ π(2r ) 2 are the sharp estimates for the size of E. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on C n or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in r and . §1. IntroductionIn the convergence theory of Padé approximation, and more generally rational interpolation, an essential ingredient is an estimate on the size of the lemniscatewhere P is a polynomial of degree ≤ n. There are several ways to provide this estimate. Cartan's Lemma shows that if P is normalized to be monic of degree n, then we can cover this set by a union of ≤ n balls B j , 1 ≤ j ≤ , whose diameters d(B j ) satisfy, for a given α > 0,The remarkable thing about the estimate is its independence of the degree of P . See [1, p. 194

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The Complex Equilibrium Measure of a Symmetric Convex Set in R n

Cited by 21 publications

References 0 publications

Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities

Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities

Vyacheslav Zakharyuta’s complex analysis

On the size of lemniscates of polynomials in one and several variables

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