Wyniki wyszukiwania

1

Polynomial interpolation and approximation in $ℂ^d$

100%

Bloom T., Bos L., Calvi J.-P., Levenberg N.

Annales Polonici Mathematici

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2012

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tom 106

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nr 1

53-81

EN

We update the state of the subject approximately 20 years after the publication of T. Bloom, L. Bos, C. Christensen, and N. Levenberg, Polynomial interpolation of holomorphic functions in ℂ and ℂⁿ, Rocky Mountain J. Math. 22 (1992), 441-470. This report is mostly a survey, with a sprinkling of assorted new results throughout.

2

Robin functions and extremal functions

100%

Bloom T., Levenberg N., Ma'u S.

Annales Polonici Mathematici

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2003

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tom 80

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nr 1

55-84

EN

Given a compact set $K ⊂ ℂ^{N}$, for each positive integer n, let $V^{(n)}(z)$ = $V^{(n)}_{K}(z)$ := sup{$1/(deg p) V_{p(K)}(p(z))$: p holomorphic polynomial, 1 ≤ deg p ≤ n}. These "extremal-like" functions $V^{(n)}_{K}$ are essentially one-variable in nature and always increase to the "true" several-variable (Siciak) extremal function, $V_{K}(z)$:= max[0, sup{1/(deg p) log|p(z)|: p holomorphic polynomial, $||p||_{K} ≤ 1$}]. Our main result is that if K is regular, then all of the functions $V^{(n)}_{K}$ are continuous; and their associated Robin functions $ϱ_{V^{(n)}_{K}}(z) := limsup_{|λ|→∞} [V^{(n)}_{K}(λz) - log(|λ|)]$ increase to $ϱ_{K} := ϱ_{V_{K}}$ for all z outside a pluripolar set.

3

The transfinite diameter of the real ball and simplex

100%

Bloom T., Bos L., Levenberg N.

Annales Polonici Mathematici

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2012

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tom 106

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nr 1

83-96

EN

We calculate the transfinite diameter for the real unit ball $B_d:={x∈ ℝ^d : |x|≤1}$ and the real unit simplex $T_d:={x∈ℝ^d_+ : ∑_{j=1}^dx_j≤1}.$

Ograniczanie wyników

3 Annales Polonici Mathematici

3 Bloom T.

3 Levenberg N.

2 Bos L.

1 Calvi J.-P.

1 Ma'u S.

2 2012

1 2003

Polynomial interpolation and approximation in $ℂ^d$

Robin functions and extremal functions

The transfinite diameter of the real ball and simplex