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### De Rham Cohomology of an Analytic Space.

Inventiones mathematicae

### Robin functions and extremal functions

Annales Polonici Mathematici

Given a compact set $K\subset {ℂ}^{N}$, for each positive integer n, let ${V}^{\left(n\right)}\left(z\right)$ = ${V}_{K}^{\left(n\right)}\left(z\right)$ := sup$1/\left(degp\right){V}_{p\left(K\right)}\left(p\left(z\right)\right)$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions ${V}_{K}^{\left(n\right)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, ${V}_{K}\left(z\right)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${||p||}_{K}\le 1$]. Our main result is that if K is regular, then all of the functions ${V}_{K}^{\left(n\right)}$ are continuous; and their associated Robin functions ${\varrho }_{{V}_{K}^{\left(n\right)}}\left(z\right):=limsu{p}_{|\lambda |\to \infty }\left[{V}_{K}^{\left(n\right)}\left(\lambda z\right)-log\left(|\lambda |\right)\right]$ increase to ${\varrho }_{K}:={\varrho }_{{V}_{K}}$ for all z outside a pluripolar set....

### The transfinite diameter of the real ball and simplex

Annales Polonici Mathematici

We calculate the transfinite diameter for the real unit ball ${B}_{d}:=x\in {ℝ}^{d}:|x|\le 1$ and the real unit simplex ${T}_{d}:=x\in {ℝ}_{+}^{d}:{\sum }_{j=1}^{d}{x}_{j}\le 1.$

### Polynomial interpolation and approximation in ${ℂ}^{d}$

Annales Polonici Mathematici

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.

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